CINXE.COM

<!DOCTYPE html><html lang="en-US" class="html_stretched responsive av-preloader-disabled html_header_top html_logo_left html_main_nav_header html_menu_right html_custom html_header_sticky html_header_shrinking_disabled html_mobile_menu_phone html_header_searchicon html_content_align_center html_header_unstick_top_disabled html_header_stretch_disabled html_av-submenu-hidden html_av-submenu-display-click html_av-overlay-side html_av-overlay-side-classic html_av-submenu-noclone html_entry_id_23309 av-cookies-no-cookie-consent av-no-preview av-default-lightbox html_text_menu_active av-mobile-menu-switch-default"><head><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><meta charset="UTF-8" /> <script async src="https://www.googletagmanager.com/gtag/js?id=UA-2209856-4"></script> <script>window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-2209856-4');</script> <script data-optimized="1" src="https://www.physicsforums.com/insights/wp-content/litespeed/js/2592eef92248026c6738059400d0d827.js?ver=0d827" defer type="text/javascript"></script> <meta name="viewport" content="width=device-width, initial-scale=1"><meta name='robots' content='index, follow, max-image-preview:large, max-snippet:-1, max-video-preview:-1' /><style>img:is([sizes="auto" i], [sizes^="auto," i]) { contain-intrinsic-size: 3000px 1500px }</style><title>An Introduction to the Traditional Abacus of China and Japan</title><meta name="description" content="In this article, we will focus on the traditional abacuses in China and Japan, also known as suanpans (China) and sorobans (Japan)." /><link rel="canonical" href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/" /><meta property="og:locale" content="en_US" /><meta property="og:type" content="article" /><meta property="og:title" content="An Introduction to the Traditional Abacus of China and Japan" /><meta property="og:description" content="In this article, we will focus on the traditional abacuses in China and Japan, also known as suanpans (China) and sorobans (Japan)." /><meta property="og:url" content="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/" /><meta property="og:site_name" content="Physics Forums Insights" /><meta property="article:publisher" content="https://www.facebook.com/physicsforums" /><meta property="article:published_time" content="2019-01-19T22:15:54+00:00" /><meta property="article:modified_time" content="2024-08-17T21:56:42+00:00" /><meta property="og:image" content="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png" /><meta property="og:image:width" content="240" /><meta property="og:image:height" content="135" /><meta property="og:image:type" content="image/png" /><meta name="author" content="YoungPhysicist" /><meta name="twitter:card" content="summary_large_image" /><meta name="twitter:creator" content="@physicsforums" /><meta name="twitter:site" content="@physicsforums" /><meta name="twitter:label1" content="Written by" /><meta name="twitter:data1" content="YoungPhysicist" /><meta name="twitter:label2" content="Est. reading time" /><meta name="twitter:data2" content="6 minutes" /> <script type="application/ld+json" class="yoast-schema-graph">{"@context":"https://schema.org","@graph":[{"@type":"Article","@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#article","isPartOf":{"@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/"},"author":{"name":"YoungPhysicist","@id":"https://www.physicsforums.com/insights/#/schema/person/b4abb5c26f30ae2f130244b4ec153d58"},"headline":"An Introduction to the Traditional Abacus of China and Japan","datePublished":"2019-01-19T22:15:54+00:00","dateModified":"2024-08-17T21:56:42+00:00","mainEntityOfPage":{"@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/"},"wordCount":907,"commentCount":7,"publisher":{"@id":"https://www.physicsforums.com/insights/#organization"},"image":{"@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#primaryimage"},"thumbnailUrl":"https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png","keywords":["science history"],"articleSection":["Education Articles"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#respond"]}]},{"@type":"WebPage","@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/","url":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/","name":"An Introduction to the Traditional Abacus of China and Japan","isPartOf":{"@id":"https://www.physicsforums.com/insights/#website"},"primaryImageOfPage":{"@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#primaryimage"},"image":{"@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#primaryimage"},"thumbnailUrl":"https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png","datePublished":"2019-01-19T22:15:54+00:00","dateModified":"2024-08-17T21:56:42+00:00","description":"In this article, we will focus on the traditional abacuses in China and Japan, also known as suanpans (China) and sorobans (Japan).","breadcrumb":{"@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#primaryimage","url":"https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png","contentUrl":"https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png","width":240,"height":135,"caption":"abacus introduction"},{"@type":"BreadcrumbList","@id":"https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://www.physicsforums.com/insights/"},{"@type":"ListItem","position":2,"name":"An Introduction to the Traditional Abacus of China and Japan"}]},{"@type":"WebSite","@id":"https://www.physicsforums.com/insights/#website","url":"https://www.physicsforums.com/insights/","name":"Physics Forums Insights","description":"Science and Math Articles, Tutorials and Guides","publisher":{"@id":"https://www.physicsforums.com/insights/#organization"},"alternateName":"PF Insights","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://www.physicsforums.com/insights/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https://www.physicsforums.com/insights/#organization","name":"Physics Forums","url":"https://www.physicsforums.com/insights/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https://www.physicsforums.com/insights/#/schema/logo/image/","url":"https://www.physicsforums.com/insights/wp-content/uploads/2018/05/facebook_image-1.png","contentUrl":"https://www.physicsforums.com/insights/wp-content/uploads/2018/05/facebook_image-1.png","width":205,"height":200,"caption":"Physics Forums"},"image":{"@id":"https://www.physicsforums.com/insights/#/schema/logo/image/"},"sameAs":["https://www.facebook.com/physicsforums","https://x.com/physicsforums","https://www.linkedin.com/company/physics-forums","https://www.pinterest.com/physicsforums"]},{"@type":"Person","@id":"https://www.physicsforums.com/insights/#/schema/person/b4abb5c26f30ae2f130244b4ec153d58","name":"YoungPhysicist","description":"From Taipei Taiwan interested in Quantum Mechanics, String theory, Chemistry, Biology, Astro physics, Calculus","url":"https://www.physicsforums.com/insights/author/youngphysicist/"}]}</script> <link rel="alternate" type="application/rss+xml" title="Physics Forums Insights » Feed" href="https://www.physicsforums.com/insights/feed/" /> <script type="text/javascript" id="wpp-js" src="https://www.physicsforums.com/insights/wp-content/plugins/wordpress-popular-posts/assets/js/wpp.js?ver=7.2.0" data-sampling="0" data-sampling-rate="100" data-api-url="https://www.physicsforums.com/insights/wp-json/wordpress-popular-posts" data-post-id="23309" data-token="5534ff0b56" data-lang="0" data-debug="1"></script> <link data-optimized="1" rel='stylesheet' id='xpress_wp_admin_style-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/59568f3369115d72ade0de8186a4674e.css?ver=4674e' type='text/css' media='all' /><link data-optimized="1" rel='stylesheet' id='wp-block-library-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/ad10b27e06583ae315af8368433fa1f7.css?ver=b9180' type='text/css' media='all' /><style id='global-styles-inline-css' type='text/css'>:root{--wp--preset--aspect-ratio--square: 1;--wp--preset--aspect-ratio--4-3: 4/3;--wp--preset--aspect-ratio--3-4: 3/4;--wp--preset--aspect-ratio--3-2: 3/2;--wp--preset--aspect-ratio--2-3: 2/3;--wp--preset--aspect-ratio--16-9: 16/9;--wp--preset--aspect-ratio--9-16: 9/16;--wp--preset--color--black: #000000;--wp--preset--color--cyan-bluish-gray: #abb8c3;--wp--preset--color--white: #ffffff;--wp--preset--color--pale-pink: #f78da7;--wp--preset--color--vivid-red: #cf2e2e;--wp--preset--color--luminous-vivid-orange: #ff6900;--wp--preset--color--luminous-vivid-amber: #fcb900;--wp--preset--color--light-green-cyan: #7bdcb5;--wp--preset--color--vivid-green-cyan: #00d084;--wp--preset--color--pale-cyan-blue: #8ed1fc;--wp--preset--color--vivid-cyan-blue: #0693e3;--wp--preset--color--vivid-purple: #9b51e0;--wp--preset--color--metallic-red: #b02b2c;--wp--preset--color--maximum-yellow-red: #edae44;--wp--preset--color--yellow-sun: #eeee22;--wp--preset--color--palm-leaf: #83a846;--wp--preset--color--aero: #7bb0e7;--wp--preset--color--old-lavender: #745f7e;--wp--preset--color--steel-teal: #5f8789;--wp--preset--color--raspberry-pink: #d65799;--wp--preset--color--medium-turquoise: #4ecac2;--wp--preset--gradient--vivid-cyan-blue-to-vivid-purple: linear-gradient(135deg,rgba(6,147,227,1) 0%,rgb(155,81,224) 100%);--wp--preset--gradient--light-green-cyan-to-vivid-green-cyan: linear-gradient(135deg,rgb(122,220,180) 0%,rgb(0,208,130) 100%);--wp--preset--gradient--luminous-vivid-amber-to-luminous-vivid-orange: linear-gradient(135deg,rgba(252,185,0,1) 0%,rgba(255,105,0,1) 100%);--wp--preset--gradient--luminous-vivid-orange-to-vivid-red: linear-gradient(135deg,rgba(255,105,0,1) 0%,rgb(207,46,46) 100%);--wp--preset--gradient--very-light-gray-to-cyan-bluish-gray: linear-gradient(135deg,rgb(238,238,238) 0%,rgb(169,184,195) 100%);--wp--preset--gradient--cool-to-warm-spectrum: linear-gradient(135deg,rgb(74,234,220) 0%,rgb(151,120,209) 20%,rgb(207,42,186) 40%,rgb(238,44,130) 60%,rgb(251,105,98) 80%,rgb(254,248,76) 100%);--wp--preset--gradient--blush-light-purple: linear-gradient(135deg,rgb(255,206,236) 0%,rgb(152,150,240) 100%);--wp--preset--gradient--blush-bordeaux: linear-gradient(135deg,rgb(254,205,165) 0%,rgb(254,45,45) 50%,rgb(107,0,62) 100%);--wp--preset--gradient--luminous-dusk: linear-gradient(135deg,rgb(255,203,112) 0%,rgb(199,81,192) 50%,rgb(65,88,208) 100%);--wp--preset--gradient--pale-ocean: linear-gradient(135deg,rgb(255,245,203) 0%,rgb(182,227,212) 50%,rgb(51,167,181) 100%);--wp--preset--gradient--electric-grass: linear-gradient(135deg,rgb(202,248,128) 0%,rgb(113,206,126) 100%);--wp--preset--gradient--midnight: linear-gradient(135deg,rgb(2,3,129) 0%,rgb(40,116,252) 100%);--wp--preset--font-size--small: 1rem;--wp--preset--font-size--medium: 1.125rem;--wp--preset--font-size--large: 1.75rem;--wp--preset--font-size--x-large: clamp(1.75rem, 3vw, 2.25rem);--wp--preset--spacing--20: 0.44rem;--wp--preset--spacing--30: 0.67rem;--wp--preset--spacing--40: 1rem;--wp--preset--spacing--50: 1.5rem;--wp--preset--spacing--60: 2.25rem;--wp--preset--spacing--70: 3.38rem;--wp--preset--spacing--80: 5.06rem;--wp--preset--shadow--natural: 6px 6px 9px rgba(0, 0, 0, 0.2);--wp--preset--shadow--deep: 12px 12px 50px rgba(0, 0, 0, 0.4);--wp--preset--shadow--sharp: 6px 6px 0px rgba(0, 0, 0, 0.2);--wp--preset--shadow--outlined: 6px 6px 0px -3px rgba(255, 255, 255, 1), 6px 6px rgba(0, 0, 0, 1);--wp--preset--shadow--crisp: 6px 6px 0px rgba(0, 0, 0, 1);}:root { --wp--style--global--content-size: 800px;--wp--style--global--wide-size: 1130px; }:where(body) { margin: 0; }.wp-site-blocks > .alignleft { float: left; margin-right: 2em; }.wp-site-blocks > .alignright { float: right; margin-left: 2em; }.wp-site-blocks > .aligncenter { justify-content: center; margin-left: auto; margin-right: auto; }:where(.is-layout-flex){gap: 0.5em;}:where(.is-layout-grid){gap: 0.5em;}.is-layout-flow > .alignleft{float: left;margin-inline-start: 0;margin-inline-end: 2em;}.is-layout-flow > .alignright{float: right;margin-inline-start: 2em;margin-inline-end: 0;}.is-layout-flow > .aligncenter{margin-left: auto !important;margin-right: auto !important;}.is-layout-constrained > .alignleft{float: left;margin-inline-start: 0;margin-inline-end: 2em;}.is-layout-constrained > .alignright{float: right;margin-inline-start: 2em;margin-inline-end: 0;}.is-layout-constrained > .aligncenter{margin-left: auto !important;margin-right: auto !important;}.is-layout-constrained > :where(:not(.alignleft):not(.alignright):not(.alignfull)){max-width: var(--wp--style--global--content-size);margin-left: auto !important;margin-right: auto !important;}.is-layout-constrained > .alignwide{max-width: var(--wp--style--global--wide-size);}body .is-layout-flex{display: flex;}.is-layout-flex{flex-wrap: wrap;align-items: center;}.is-layout-flex > :is(*, div){margin: 0;}body .is-layout-grid{display: grid;}.is-layout-grid > :is(*, div){margin: 0;}body{padding-top: 0px;padding-right: 0px;padding-bottom: 0px;padding-left: 0px;}a:where(:not(.wp-element-button)){text-decoration: underline;}:root :where(.wp-element-button, .wp-block-button__link){background-color: #32373c;border-width: 0;color: #fff;font-family: inherit;font-size: inherit;line-height: inherit;padding: calc(0.667em + 2px) calc(1.333em + 2px);text-decoration: none;}.has-black-color{color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-color{color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-color{color: var(--wp--preset--color--white) !important;}.has-pale-pink-color{color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-color{color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-color{color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-color{color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-color{color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-color{color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-color{color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-color{color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-color{color: var(--wp--preset--color--vivid-purple) !important;}.has-metallic-red-color{color: var(--wp--preset--color--metallic-red) !important;}.has-maximum-yellow-red-color{color: var(--wp--preset--color--maximum-yellow-red) !important;}.has-yellow-sun-color{color: var(--wp--preset--color--yellow-sun) !important;}.has-palm-leaf-color{color: var(--wp--preset--color--palm-leaf) !important;}.has-aero-color{color: var(--wp--preset--color--aero) !important;}.has-old-lavender-color{color: var(--wp--preset--color--old-lavender) !important;}.has-steel-teal-color{color: var(--wp--preset--color--steel-teal) !important;}.has-raspberry-pink-color{color: var(--wp--preset--color--raspberry-pink) !important;}.has-medium-turquoise-color{color: var(--wp--preset--color--medium-turquoise) !important;}.has-black-background-color{background-color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-background-color{background-color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-background-color{background-color: var(--wp--preset--color--white) !important;}.has-pale-pink-background-color{background-color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-background-color{background-color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-background-color{background-color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-background-color{background-color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-background-color{background-color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-background-color{background-color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-background-color{background-color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-background-color{background-color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-background-color{background-color: var(--wp--preset--color--vivid-purple) !important;}.has-metallic-red-background-color{background-color: var(--wp--preset--color--metallic-red) !important;}.has-maximum-yellow-red-background-color{background-color: var(--wp--preset--color--maximum-yellow-red) !important;}.has-yellow-sun-background-color{background-color: var(--wp--preset--color--yellow-sun) !important;}.has-palm-leaf-background-color{background-color: var(--wp--preset--color--palm-leaf) !important;}.has-aero-background-color{background-color: var(--wp--preset--color--aero) !important;}.has-old-lavender-background-color{background-color: var(--wp--preset--color--old-lavender) !important;}.has-steel-teal-background-color{background-color: var(--wp--preset--color--steel-teal) !important;}.has-raspberry-pink-background-color{background-color: var(--wp--preset--color--raspberry-pink) !important;}.has-medium-turquoise-background-color{background-color: var(--wp--preset--color--medium-turquoise) !important;}.has-black-border-color{border-color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-border-color{border-color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-border-color{border-color: var(--wp--preset--color--white) !important;}.has-pale-pink-border-color{border-color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-border-color{border-color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-border-color{border-color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-border-color{border-color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-border-color{border-color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-border-color{border-color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-border-color{border-color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-border-color{border-color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-border-color{border-color: var(--wp--preset--color--vivid-purple) !important;}.has-metallic-red-border-color{border-color: var(--wp--preset--color--metallic-red) !important;}.has-maximum-yellow-red-border-color{border-color: var(--wp--preset--color--maximum-yellow-red) !important;}.has-yellow-sun-border-color{border-color: var(--wp--preset--color--yellow-sun) !important;}.has-palm-leaf-border-color{border-color: var(--wp--preset--color--palm-leaf) !important;}.has-aero-border-color{border-color: var(--wp--preset--color--aero) !important;}.has-old-lavender-border-color{border-color: var(--wp--preset--color--old-lavender) !important;}.has-steel-teal-border-color{border-color: var(--wp--preset--color--steel-teal) !important;}.has-raspberry-pink-border-color{border-color: var(--wp--preset--color--raspberry-pink) !important;}.has-medium-turquoise-border-color{border-color: var(--wp--preset--color--medium-turquoise) !important;}.has-vivid-cyan-blue-to-vivid-purple-gradient-background{background: var(--wp--preset--gradient--vivid-cyan-blue-to-vivid-purple) !important;}.has-light-green-cyan-to-vivid-green-cyan-gradient-background{background: var(--wp--preset--gradient--light-green-cyan-to-vivid-green-cyan) !important;}.has-luminous-vivid-amber-to-luminous-vivid-orange-gradient-background{background: var(--wp--preset--gradient--luminous-vivid-amber-to-luminous-vivid-orange) !important;}.has-luminous-vivid-orange-to-vivid-red-gradient-background{background: var(--wp--preset--gradient--luminous-vivid-orange-to-vivid-red) !important;}.has-very-light-gray-to-cyan-bluish-gray-gradient-background{background: var(--wp--preset--gradient--very-light-gray-to-cyan-bluish-gray) !important;}.has-cool-to-warm-spectrum-gradient-background{background: var(--wp--preset--gradient--cool-to-warm-spectrum) !important;}.has-blush-light-purple-gradient-background{background: var(--wp--preset--gradient--blush-light-purple) !important;}.has-blush-bordeaux-gradient-background{background: var(--wp--preset--gradient--blush-bordeaux) !important;}.has-luminous-dusk-gradient-background{background: var(--wp--preset--gradient--luminous-dusk) !important;}.has-pale-ocean-gradient-background{background: var(--wp--preset--gradient--pale-ocean) !important;}.has-electric-grass-gradient-background{background: var(--wp--preset--gradient--electric-grass) !important;}.has-midnight-gradient-background{background: var(--wp--preset--gradient--midnight) !important;}.has-small-font-size{font-size: var(--wp--preset--font-size--small) !important;}.has-medium-font-size{font-size: var(--wp--preset--font-size--medium) !important;}.has-large-font-size{font-size: var(--wp--preset--font-size--large) !important;}.has-x-large-font-size{font-size: var(--wp--preset--font-size--x-large) !important;} :where(.wp-block-post-template.is-layout-flex){gap: 1.25em;}:where(.wp-block-post-template.is-layout-grid){gap: 1.25em;} :where(.wp-block-columns.is-layout-flex){gap: 2em;}:where(.wp-block-columns.is-layout-grid){gap: 2em;} :root :where(.wp-block-pullquote){font-size: 1.5em;line-height: 1.6;}</style><link data-optimized="1" rel='stylesheet' id='ez-toc-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/008d509d4fec340652d0312244582360.css?ver=d7f24' type='text/css' media='all' /><style id='ez-toc-inline-css' type='text/css'>div#ez-toc-container .ez-toc-title {font-size: 120%;}div#ez-toc-container .ez-toc-title {font-weight: 500;}div#ez-toc-container ul li {font-size: 95%;}div#ez-toc-container ul li {font-weight: 500;}div#ez-toc-container nav ul ul li {font-size: 90%;}div#ez-toc-container {background: #fff;border: 1px solid #ddd;}div#ez-toc-container p.ez-toc-title , #ez-toc-container .ez_toc_custom_title_icon , #ez-toc-container .ez_toc_custom_toc_icon {color: #4c4c4c;}div#ez-toc-container ul.ez-toc-list a {color: #066fac;}div#ez-toc-container ul.ez-toc-list a:hover {color: #235887;}div#ez-toc-container ul.ez-toc-list a:visited {color: #4594d1;} .ez-toc-container-direction {direction: ltr;}.ez-toc-counter ul{counter-reset: item ;}.ez-toc-counter nav ul li a::before {content: counters(item, '.', decimal) '. ';display: inline-block;counter-increment: item;flex-grow: 0;flex-shrink: 0;margin-right: .2em; float: left; }.ez-toc-widget-direction {direction: ltr;}.ez-toc-widget-container ul{counter-reset: item ;}.ez-toc-widget-container nav ul li a::before {content: counters(item, '.', decimal) '. ';display: inline-block;counter-increment: item;flex-grow: 0;flex-shrink: 0;margin-right: .2em; float: left; }</style><link data-optimized="1" rel='stylesheet' id='wordpress-popular-posts-css-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/f37de7c74c7905b881abee3714c59450.css?ver=b2ec6' type='text/css' media='all' /><link data-optimized="1" rel='stylesheet' id='avia-merged-styles-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/26482a11dcb0aabea248feffff00cc05.css?ver=0cc05' type='text/css' media='all' /> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/dddbaf834171fed716d265a90aea73b5.js?ver=a73b5" id="avia-head-scripts-js"></script> <style id="wpp-loading-animation-styles">@-webkit-keyframes bgslide{from{background-position-x:0}to{background-position-x:-200%}}@keyframes bgslide{from{background-position-x:0}to{background-position-x:-200%}}.wpp-widget-block-placeholder,.wpp-shortcode-placeholder{margin:0 auto;width:60px;height:3px;background:#dd3737;background:linear-gradient(90deg,#dd3737 0%,#571313 10%,#dd3737 100%);background-size:200% auto;border-radius:3px;-webkit-animation:bgslide 1s infinite linear;animation:bgslide 1s infinite linear}</style><link rel="profile" href="https://gmpg.org/xfn/11" /><link rel="alternate" type="application/rss+xml" title="Physics Forums Insights RSS2 Feed" href="https://www.physicsforums.com/insights/feed/" /><link rel="pingback" href="https://www.physicsforums.com/insights/xmlrpc.php" /><style type='text/css' media='screen'>#top #header_main > .container, #top #header_main > .container .main_menu .av-main-nav > li > a, #top #header_main #menu-item-shop .cart_dropdown_link{ height:70px; line-height: 70px; } .html_top_nav_header .av-logo-container{ height:70px; } .html_header_top.html_header_sticky #top #wrap_all #main{ padding-top:70px; }</style><link rel="icon" href="https://www.physicsforums.com/insights/wp-content/uploads/2016/08/favicon.ico" type="image/x-icon"><style type="text/css">.saboxplugin-wrap{-webkit-box-sizing:border-box;-moz-box-sizing:border-box;-ms-box-sizing:border-box;box-sizing:border-box;border:1px solid #eee;width:100%;clear:both;display:block;overflow:hidden;word-wrap:break-word;position:relative}.saboxplugin-wrap .saboxplugin-gravatar{float:left;padding:0 20px 20px 20px}.saboxplugin-wrap .saboxplugin-gravatar img{max-width:100px;height:auto;border-radius:0;}.saboxplugin-wrap .saboxplugin-authorname{font-size:18px;line-height:1;margin:20px 0 0 20px;display:block}.saboxplugin-wrap .saboxplugin-authorname a{text-decoration:none}.saboxplugin-wrap .saboxplugin-authorname a:focus{outline:0}.saboxplugin-wrap .saboxplugin-desc{display:block;margin:5px 20px}.saboxplugin-wrap .saboxplugin-desc a{text-decoration:underline}.saboxplugin-wrap .saboxplugin-desc p{margin:5px 0 12px}.saboxplugin-wrap .saboxplugin-web{margin:0 20px 15px;text-align:left}.saboxplugin-wrap .sab-web-position{text-align:right}.saboxplugin-wrap .saboxplugin-web a{color:#ccc;text-decoration:none}.saboxplugin-wrap .saboxplugin-socials{position:relative;display:block;background:#fcfcfc;padding:5px;border-top:1px solid #eee}.saboxplugin-wrap .saboxplugin-socials a svg{width:20px;height:20px}.saboxplugin-wrap .saboxplugin-socials a svg .st2{fill:#fff; transform-origin:center center;}.saboxplugin-wrap .saboxplugin-socials a svg .st1{fill:rgba(0,0,0,.3)}.saboxplugin-wrap .saboxplugin-socials a:hover{opacity:.8;-webkit-transition:opacity .4s;-moz-transition:opacity .4s;-o-transition:opacity .4s;transition:opacity .4s;box-shadow:none!important;-webkit-box-shadow:none!important}.saboxplugin-wrap .saboxplugin-socials .saboxplugin-icon-color{box-shadow:none;padding:0;border:0;-webkit-transition:opacity .4s;-moz-transition:opacity .4s;-o-transition:opacity .4s;transition:opacity .4s;display:inline-block;color:#fff;font-size:0;text-decoration:inherit;margin:5px;-webkit-border-radius:0;-moz-border-radius:0;-ms-border-radius:0;-o-border-radius:0;border-radius:0;overflow:hidden}.saboxplugin-wrap .saboxplugin-socials .saboxplugin-icon-grey{text-decoration:inherit;box-shadow:none;position:relative;display:-moz-inline-stack;display:inline-block;vertical-align:middle;zoom:1;margin:10px 5px;color:#444;fill:#444}.clearfix:after,.clearfix:before{content:' ';display:table;line-height:0;clear:both}.ie7 .clearfix{zoom:1}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-twitch{border-color:#38245c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-addthis{border-color:#e91c00}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-behance{border-color:#003eb0}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-delicious{border-color:#06c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-deviantart{border-color:#036824}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-digg{border-color:#00327c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-dribbble{border-color:#ba1655}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-facebook{border-color:#1e2e4f}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-flickr{border-color:#003576}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-github{border-color:#264874}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-google{border-color:#0b51c5}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-googleplus{border-color:#96271a}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-html5{border-color:#902e13}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-instagram{border-color:#1630aa}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-linkedin{border-color:#00344f}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-pinterest{border-color:#5b040e}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-reddit{border-color:#992900}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-rss{border-color:#a43b0a}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-sharethis{border-color:#5d8420}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-skype{border-color:#00658a}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-soundcloud{border-color:#995200}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-spotify{border-color:#0f612c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-stackoverflow{border-color:#a95009}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-steam{border-color:#006388}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-user_email{border-color:#b84e05}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-stumbleUpon{border-color:#9b280e}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-tumblr{border-color:#10151b}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-twitter{border-color:#0967a0}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-vimeo{border-color:#0d7091}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-windows{border-color:#003f71}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-whatsapp{border-color:#003f71}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-wordpress{border-color:#0f3647}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-yahoo{border-color:#14002d}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-youtube{border-color:#900}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-xing{border-color:#000202}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-mixcloud{border-color:#2475a0}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-vk{border-color:#243549}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-medium{border-color:#00452c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-quora{border-color:#420e00}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-meetup{border-color:#9b181c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-goodreads{border-color:#000}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-snapchat{border-color:#999700}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-500px{border-color:#00557f}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-mastodont{border-color:#185886}.sabox-plus-item{margin-bottom:20px}@media screen and (max-width:480px){.saboxplugin-wrap{text-align:center}.saboxplugin-wrap .saboxplugin-gravatar{float:none;padding:20px 0;text-align:center;margin:0 auto;display:block}.saboxplugin-wrap .saboxplugin-gravatar img{float:none;display:inline-block;display:-moz-inline-stack;vertical-align:middle;zoom:1}.saboxplugin-wrap .saboxplugin-desc{margin:0 10px 20px;text-align:center}.saboxplugin-wrap .saboxplugin-authorname{text-align:center;margin:10px 0 20px}}body .saboxplugin-authorname a,body .saboxplugin-authorname a:hover{box-shadow:none;-webkit-box-shadow:none}a.sab-profile-edit{font-size:16px!important;line-height:1!important}.sab-edit-settings a,a.sab-profile-edit{color:#0073aa!important;box-shadow:none!important;-webkit-box-shadow:none!important}.sab-edit-settings{margin-right:15px;position:absolute;right:0;z-index:2;bottom:10px;line-height:20px}.sab-edit-settings i{margin-left:5px}.saboxplugin-socials{line-height:1!important}.rtl .saboxplugin-wrap .saboxplugin-gravatar{float:right}.rtl .saboxplugin-wrap .saboxplugin-authorname{display:flex;align-items:center}.rtl .saboxplugin-wrap .saboxplugin-authorname .sab-profile-edit{margin-right:10px}.rtl .sab-edit-settings{right:auto;left:0}img.sab-custom-avatar{max-width:75px;}.saboxplugin-wrap .saboxplugin-gravatar img {-webkit-border-radius:50%;-moz-border-radius:50%;-ms-border-radius:50%;-o-border-radius:50%;border-radius:50%;}.saboxplugin-wrap .saboxplugin-gravatar img {-webkit-border-radius:50%;-moz-border-radius:50%;-ms-border-radius:50%;-o-border-radius:50%;border-radius:50%;}.saboxplugin-wrap .saboxplugin-gravatar img {-webkit-transition:all .5s ease;-moz-transition:all .5s ease;-o-transition:all .5s ease;transition:all .5s ease;}.saboxplugin-wrap .saboxplugin-gravatar img:hover {-webkit-transform:rotate(45deg);-moz-transform:rotate(45deg);-o-transform:rotate(45deg);-ms-transform:rotate(45deg);transform:rotate(45deg);}.saboxplugin-wrap {margin-top:0px; margin-bottom:0px; padding: 0px 0px }.saboxplugin-wrap .saboxplugin-authorname {font-size:18px; line-height:25px;}.saboxplugin-wrap .saboxplugin-desc p, .saboxplugin-wrap .saboxplugin-desc {font-size:14px !important; line-height:21px !important;}.saboxplugin-wrap .saboxplugin-web {font-size:14px;}.saboxplugin-wrap .saboxplugin-socials a svg {width:18px;height:18px;}</style> <script type="text/javascript">'use strict';var avia_is_mobile=!1;if(/Android|webOS|iPhone|iPad|iPod|BlackBerry|IEMobile|Opera Mini/i.test(navigator.userAgent)&&'ontouchstart' in document.documentElement){avia_is_mobile=!0;document.documentElement.className+=' avia_mobile '} else{document.documentElement.className+=' avia_desktop '};document.documentElement.className+=' js_active ';(function(){var e=['-webkit-','-moz-','-ms-',''],n='',o=!1,a=!1;for(var t in e){if(e[t]+'transform' in document.documentElement.style){o=!0;n=e[t]+'transform'};if(e[t]+'perspective' in document.documentElement.style){a=!0}};if(o){document.documentElement.className+=' avia_transform '};if(a){document.documentElement.className+=' avia_transform3d '};if(typeof document.getElementsByClassName=='function'&&typeof document.documentElement.getBoundingClientRect=='function'&&avia_is_mobile==!1){if(n&&window.innerHeight>0){setTimeout(function(){var e=0,o={},a=0,t=document.getElementsByClassName('av-parallax'),i=window.pageYOffset||document.documentElement.scrollTop;for(e=0;e<t.length;e++){t[e].style.top='0px';o=t[e].getBoundingClientRect();a=Math.ceil((window.innerHeight+i-o.top)*0.3);t[e].style[n]='translate(0px, '+a+'px)';t[e].style.top='auto';t[e].className+=' enabled-parallax '}},50)}}})();</script><style type="text/css">@font-face {font-family: 'entypo-fontello'; font-weight: normal; font-style: normal; font-display: swap; src: url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.woff2') format('woff2'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.woff') format('woff'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.ttf') format('truetype'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.svg#entypo-fontello') format('svg'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.eot'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.eot?#iefix') format('embedded-opentype'); } #top .avia-font-entypo-fontello, body .avia-font-entypo-fontello, html body [data-av_iconfont='entypo-fontello']:before{ font-family: 'entypo-fontello'; }</style></head><body id="top" class="post-template-default single single-post postid-23309 single-format-standard stretched avia-mobile-no-animations rtl_columns av-curtain-numeric arial-websave arial avia-responsive-images-support" itemscope="itemscope" itemtype="https://schema.org/WebPage" ><div id='wrap_all'><header id='header' class='all_colors header_color light_bg_color av_header_top av_logo_left av_main_nav_header av_menu_right av_custom av_header_sticky av_header_shrinking_disabled av_header_stretch_disabled av_mobile_menu_phone av_header_searchicon av_header_unstick_top_disabled av_bottom_nav_disabled av_header_border_disabled' aria-label="Header" data-av_shrink_factor='50' role="banner" itemscope="itemscope" itemtype="https://schema.org/WPHeader" ><div id='header_main' class='container_wrap container_wrap_logo'><div class='container av-logo-container'><div class='inner-container'><span class='logo avia-standard-logo'><a href='https://www.physicsforums.com/insights/' class='' aria-label='Physics_Forums_Insights_logo' title='Physics_Forums_Insights_logo'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzMDAiIGhlaWdodD0iMTAwIiB2aWV3Qm94PSIwIDAgMzAwIDEwMCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png" height="100" width="300" alt='Physics Forums Insights' title='Physics_Forums_Insights_logo' /><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png" height="100" width="300" alt='Physics Forums Insights' title='Physics_Forums_Insights_logo' /></noscript></a></span><nav class='main_menu' data-selectname='Select a page' role="navigation" itemscope="itemscope" itemtype="https://schema.org/SiteNavigationElement" ><div class="avia-menu av-main-nav-wrap"><ul role="menu" class="menu av-main-nav" id="avia-menu"><li role="menuitem" id="menu-item-27276" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-1"><a href="https://www.physicsforums.com/insights/trending-physics-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26559" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/physics-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics Articles</span></a></li><li role="menuitem" id="menu-item-26568" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-tutorials/physics-tutorials/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics Tutorials</span></a></li><li role="menuitem" id="menu-item-26564" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/physics-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics Guides</span></a></li><li role="menuitem" id="menu-item-26561" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-faqs/physics-faqs/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics FAQs</span></a></li></ul></li><li role="menuitem" id="menu-item-27288" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-2"><a href="https://www.physicsforums.com/insights/trending-math-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26558" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/math-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math Articles</span></a></li><li role="menuitem" id="menu-item-26567" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-tutorials/math-tutorials/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math Tutorials</span></a></li><li role="menuitem" id="menu-item-26563" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/math-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math Guides</span></a></li><li role="menuitem" id="menu-item-26560" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-faqs/math-faqs/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math FAQs</span></a></li></ul></li><li role="menuitem" id="menu-item-27287" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-3"><a href="https://www.physicsforums.com/insights/trending-bio-chem-tech-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Bio/Chem/Tech</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26556" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/biology-chemistry-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Bio/Chem Articles</span></a></li><li role="menuitem" id="menu-item-26566" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-tutorials/computer-science-tutorials/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Computer Science Tutorials</span></a></li><li role="menuitem" id="menu-item-26565" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/technology-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Technology Guides</span></a></li></ul></li><li role="menuitem" id="menu-item-27286" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-4"><a href="https://www.physicsforums.com/insights/trending-education-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Education</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26557" class="menu-item menu-item-type-taxonomy menu-item-object-category current-post-ancestor current-menu-parent current-post-parent"><a href="https://www.physicsforums.com/insights/science-math-articles/education-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Education Articles</span></a></li><li role="menuitem" id="menu-item-26562" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/education-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Education Guides</span></a></li></ul></li><li role="menuitem" id="menu-item-26569" class="menu-item menu-item-type-taxonomy menu-item-object-category menu-item-top-level menu-item-top-level-5"><a href="https://www.physicsforums.com/insights/scientist-interviews/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Interviews</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a></li><li role="menuitem" id="menu-item-26570" class="menu-item menu-item-type-taxonomy menu-item-object-category menu-item-top-level menu-item-top-level-6"><a href="https://www.physicsforums.com/insights/science-math-quizzes/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Quizzes</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a></li><li role="menuitem" id="menu-item-26571" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-top-level menu-item-top-level-7"><a href="https://www.physicsforums.com/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Forums</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a></li><li id="menu-item-search" class="noMobile menu-item menu-item-search-dropdown menu-item-avia-special" role="menuitem"><a aria-label="Search" href="?s=" rel="nofollow" data-avia-search-tooltip=" <search> <form role="search" action="https://www.physicsforums.com/insights/" id="searchform" method="get" class=""> <div> <input type="submit" value="" id="searchsubmit" class="button avia-font-entypo-fontello" title="Enter at least 3 characters to show search results in a dropdown or click to route to search result page to show all results" /> <input type="search" id="s" name="s" value="" aria-label='Search' placeholder='Search' required /> </div> </form> </search> " aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello'><span class="avia_hidden_link_text">Search</span></a></li><li class="av-burger-menu-main menu-item-avia-special " role="menuitem"> <a href="#" aria-label="Menu" aria-hidden="false"> <span class="av-hamburger av-hamburger--spin av-js-hamburger"> <span class="av-hamburger-box"> <span class="av-hamburger-inner"></span> <strong>Menu</strong> </span> </span> <span class="avia_hidden_link_text">Menu</span> </a></li></ul></div></nav></div></div></div><div class="header_bg"></div></header><div id='main' class='all_colors' data-scroll-offset='70'><div class='container_wrap container_wrap_first main_color sidebar_right'><div class='container template-blog template-single-blog '><main class='content units av-content-small alpha av-blog-meta-html-info-disabled av-main-single' role="main" itemscope="itemscope" itemtype="https://schema.org/Blog" ><article class="post-entry post-entry-type-standard post-entry-23309 post-loop-1 post-parity-odd post-entry-last single-big with-slider post-23309 post type-post status-publish format-standard has-post-thumbnail hentry category-education-articles tag-science-history" itemscope="itemscope" itemtype="https://schema.org/BlogPosting" itemprop="blogPost" ><div class="big-preview single-big" itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" ><a href="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png" title="abacus_introduction" ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png" class="wp-image-23328 avia-img-lazy-loading-23328 attachment-entry_with_sidebar size-entry_with_sidebar wp-post-image" alt="abacus introduction" decoding="async" /><noscript><img loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png" class="wp-image-23328 avia-img-lazy-loading-23328 attachment-entry_with_sidebar size-entry_with_sidebar wp-post-image" alt="abacus introduction" decoding="async" /></noscript></a></div><div class="blog-meta"></div><div class='entry-content-wrapper clearfix standard-content'><header class="entry-content-header" aria-label="Post: An Introduction to the Traditional Abacus of China and Japan"><h1 class='post-title entry-title ' itemprop="headline" >An Introduction to the Traditional Abacus of China and Japan<span class="post-format-icon minor-meta"></span></h1><span class="post-meta-infos"><time class="date-container minor-meta updated" itemprop="datePublished" datetime="2019-01-19T16:15:54-06:00" >January 19, 2019</time><span class="text-sep">/</span><span class="comment-container minor-meta"><a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comments" class="comments-link" >7 Comments</a></span><span class="text-sep">/</span><span class="blog-categories minor-meta">in <a href="https://www.physicsforums.com/insights/science-math-articles/education-articles/" rel="tag">Education Articles</a></span><span class="text-sep">/</span><span class="blog-author minor-meta">by <span class="entry-author-link" itemprop="author" ><span class="author"><span class="fn"><a href="https://www.physicsforums.com/insights/author/youngphysicist/" title="Posts by YoungPhysicist" rel="author">YoungPhysicist</a></span></span></span></span></span></header><div class="entry-content" itemprop="text" ><div><strong>Estimated Read Time:</strong> 4 minute(s)</div><div><strong>Common Topics:</strong> weight, soroban, suanpans, chinese, design</div><div><div id="ez-toc-container" class="ez-toc-v2_0_72 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction"><p class="ez-toc-title" style="cursor:inherit">Table of Contents</p> <label for="ez-toc-cssicon-toggle-item-67b562a7068dd" class="ez-toc-cssicon-toggle-label"><span class="ez-toc-cssicon"><span class="eztoc-hide" style="display:none;">Toggle</span><span class="ez-toc-icon-toggle-span"><svg style="fill: #4c4c4c;color:#4c4c4c" xmlns="http://www.w3.org/2000/svg" class="list-377408" width="20px" height="20px" viewBox="0 0 24 24" fill="none"><path d="M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z" fill="currentColor"></path></svg><svg style="fill: #4c4c4c;color:#4c4c4c" class="arrow-unsorted-368013" xmlns="http://www.w3.org/2000/svg" width="10px" height="10px" viewBox="0 0 24 24" version="1.2" baseProfile="tiny"><path d="M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z"/></svg></span></span></label><input type="checkbox" id="ez-toc-cssicon-toggle-item-67b562a7068dd" aria-label="Toggle" /><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-1" href="#Introduction" title="Introduction">Introduction</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-2" href="#Abacus-Etymology" title="Abacus Etymology">Abacus Etymology</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-3" href="#Suanpans-in-China" title="Suanpans in China">Suanpans in China</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-4" href="#Sorobans-in-Japan" title="Sorobans in Japan">Sorobans in Japan</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-5" href="#Modern-Suanpans-and-Sorobans" title="Modern Suanpans and Sorobans">Modern Suanpans and Sorobans</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-6" href="#Appearance" title="Appearance">Appearance</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-7" href="#How-to-express-numbers-with-soroban" title="How to express numbers with soroban">How to express numbers with soroban</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-8" href="#Always-clear-it-before-using-it" title="Always clear it before using it">Always clear it before using it</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-9" href="#Other-Cool-Facts" title="Other Cool Facts">Other Cool Facts</a></li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class="ez-toc-link ez-toc-heading-10" href="#Final-Thoughts" title="Final Thoughts">Final Thoughts</a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class="ez-toc-link ez-toc-heading-11" href="#More-Related-Articles" title="More Related Articles">More Related Articles</a></li></ul></li></ul></nav></div><h2><span class="ez-toc-section" id="Introduction"></span>Introduction<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Abacuses are commonly seen in ancient civilizations around the world as a convenient tool to do simple calculations. In this article, we will focus on the traditional abacuses in China and Japan, also known as suanpans (China) and sorobans (Japan).</span></p><h2><span class="ez-toc-section" id="Abacus-Etymology"></span>Abacus Etymology<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">The Chinese word「算盤」, which is pronounced as “suanpan”, literally means “calculating tray” or “calculating disk”. This word is sometimes used to mean planning things in Chinese, too.</span></p><p><span style="font-weight: 400;">The Japanese word 「そろばん」, which is pronounced as “soroban”, is a translation from Chinese.</span></p><h2><span class="ez-toc-section" id="Suanpans-in-China"></span>Suanpans in China<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Prototypes of suanpans started to appear in the Han dynasty (206BC ~ 220AD) in China, with a design similar to modern-day suanpans: One bead on top, and four beads down at the bottom. It is generally believed that its design was influenced by the Roman Empire, with whom they had traded goods and knowledge at that time. </span></p><p><span style="font-weight: 400;">The Romans used a small calculating device called a </span><a href="https://en.wikipedia.org/wiki/Roman_abacus" target="_blank"><span style="font-weight: 400;">Roman hand abacus.</span></a><span style="font-weight: 400;"> The hand abacus was composed of a calculating tray with grooved slots and sliding metal ball bearings. The suanpan improvement was to use beads on rods which was cheaper and easier to manufacture in bamboo. </span></p><p><span style="font-weight: 400;">Visual evidence of suanpan use can be found in one of the best-known of ancient Chinese paintings: “</span><a href="https://en.wikipedia.org/wiki/Along_the_River_During_the_Qingming_Festival" target="_blank"><span style="font-weight: 400;">Along the River During the Qingming Festival</span></a><span style="font-weight: 400;">” of the Song Dynasty. We guess the suanpan in the painting is used to calculate the prices of Chinese medicine powders.</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MjciIGhlaWdodD0iMjc3IiB2aWV3Qm94PSIwIDAgNTI3IDI3NyI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23313" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_painting.png" alt="abacus painting" width="527" height="277" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_painting.png 527w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_painting-450x237.png 450w" data-sizes="(max-width: 527px) 100vw, 527px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23313" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_painting.png" alt="abacus painting" width="527" height="277" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_painting.png 527w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_painting-450x237.png 450w" sizes="(max-width: 527px) 100vw, 527px" /></noscript></p><p><span style="font-weight: 400;">The Suanpan design changed in the Ming and Qing Dynasty(1368~</span><span style="font-weight: 400;"> 1912) to 2 beads on the top, and 5 beads at the bottom due to the use of hexadecimal weight units during that period, which is what modern Westerners now refer to as “suanpan”</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MjAiIGhlaWdodD0iMzQwIiB2aWV3Qm94PSIwIDAgNTIwIDM0MCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23317" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/suanpan_abacus.png" alt="suanpan abacus" width="520" height="340" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/suanpan_abacus.png 520w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/suanpan_abacus-450x294.png 450w" data-sizes="(max-width: 520px) 100vw, 520px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23317" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/suanpan_abacus.png" alt="suanpan abacus" width="520" height="340" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/suanpan_abacus.png 520w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/suanpan_abacus-450x294.png 450w" sizes="(max-width: 520px) 100vw, 520px" /></noscript></p><h2><span class="ez-toc-section" id="Sorobans-in-Japan"></span>Sorobans in Japan<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">During the Muromachi period in Japan(around the 14th century), Suanpans with a 2:5 bead design started to appear in Japan because of trading with China(Ming Dynasty). </span></p><p><span style="font-weight: 400;">The Japanese thought the 2:5 suanpan was unnecessarily complex and reduced the bead count to a 1:4 design, which happens to be the same as the old Chinese ones. They call it “sorobans”, which is the translation of “suanpan” from Chinese to Japanese.</span></p><h2><span class="ez-toc-section" id="Modern-Suanpans-and-Sorobans"></span>Modern Suanpans and Sorobans<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Suanpans had almost faded away completely in Chinese everyday life due to the economic switch to metric units of measure and the invention of digital calculators. It can usually only be found in museums and antique shops nowadays.</span></p><p><span style="font-weight: 400;">Sorobans on the other hand are still hanging around in many Asian regions because of their decimal design. Most elementary schools in Taiwan, China, and Japan have soroban classes of some sort.</span></p><p><span style="font-weight: 400;">Modern-day sorobans that can still be seen in most stationery shops in China, Taiwan, and Japan are suanpans that have a 1:4 bead design for decimal calculations with 23 columns of wooden or plastic beads. Some of them may also have a convenient clearing mechanism depending on the manufacturer.</span></p><h2><span class="ez-toc-section" id="Appearance"></span>Appearance<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Modern soroban with a 23-column design can handle two large calculations simultaneously:</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1NTEiIGhlaWdodD0iMjQwIiB2aWV3Qm94PSIwIDAgNTUxIDI0MCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23315" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/modern_soroban_abacus.png" alt="modern soroban abacus" width="551" height="240" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/modern_soroban_abacus.png 551w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/modern_soroban_abacus-450x196.png 450w" data-sizes="(max-width: 551px) 100vw, 551px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23315" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/modern_soroban_abacus.png" alt="modern soroban abacus" width="551" height="240" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/modern_soroban_abacus.png 551w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/modern_soroban_abacus-450x196.png 450w" sizes="(max-width: 551px) 100vw, 551px" /></noscript></p><p><span style="font-weight: 400;">The dots shown below, work like the thousands and decimal separators as in the display of “1,000,000”.</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1NTIiIGhlaWdodD0iMzEzIiB2aWV3Qm94PSIwIDAgNTUyIDMxMyI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23311" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_dots.png" alt="abacus dots" width="552" height="313" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_dots.png 552w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_dots-450x255.png 450w" data-sizes="(max-width: 552px) 100vw, 552px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23311" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_dots.png" alt="abacus dots" width="552" height="313" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_dots.png 552w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_dots-450x255.png 450w" sizes="(max-width: 552px) 100vw, 552px" /></noscript></p><p><span style="font-weight: 400;">The clearing button at the top zeros out all columns quickly in one step:</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1NTQiIGhlaWdodD0iMzE0IiB2aWV3Qm94PSIwIDAgNTU0IDMxNCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23310" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_clear_button.png" alt="abacus clear button" width="554" height="314" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_clear_button.png 554w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_clear_button-450x255.png 450w" data-sizes="(max-width: 554px) 100vw, 554px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23310" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_clear_button.png" alt="abacus clear button" width="554" height="314" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_clear_button.png 554w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_clear_button-450x255.png 450w" sizes="(max-width: 554px) 100vw, 554px" /></noscript></p><h2><span class="ez-toc-section" id="How-to-express-numbers-with-soroban"></span>How to express numbers with soroban<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Sorobans express numbers with a </span><a href="https://en.wikipedia.org/wiki/Bi-quinary_coded_decimal" target="_blank"><span style="font-weight: 400;">bi-quinary</span></a><span style="font-weight: 400;"> (two base 5s) system to store each decimal (base 10) digit:</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1NTUiIGhlaWdodD0iMjc5IiB2aWV3Qm94PSIwIDAgNTU1IDI3OSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23312" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_numbers.png" alt="abacus numbers" width="555" height="279" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_numbers.png 555w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_numbers-450x226.png 450w" data-sizes="(max-width: 555px) 100vw, 555px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23312" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_numbers.png" alt="abacus numbers" width="555" height="279" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_numbers.png 555w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_numbers-450x226.png 450w" sizes="(max-width: 555px) 100vw, 555px" /></noscript></p><p><span style="font-weight: 400;">The numbers should be read left to right, For instance:</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1NTUiIGhlaWdodD0iMzI2IiB2aWV3Qm94PSIwIDAgNTU1IDMyNiI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23316" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/reading_abacus.png" alt="reading abacus" width="555" height="326" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/reading_abacus.png 555w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/reading_abacus-450x264.png 450w" data-sizes="(max-width: 555px) 100vw, 555px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23316" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/reading_abacus.png" alt="reading abacus" width="555" height="326" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/reading_abacus.png 555w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/reading_abacus-450x264.png 450w" sizes="(max-width: 555px) 100vw, 555px" /></noscript></p><h2><span class="ez-toc-section" id="Always-clear-it-before-using-it"></span>Always clear it before using it<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">If your soroban has a clearing button, just press it. It will do the work for you.</span></p><p><span style="font-weight: 400;">If your soroban doesn’t have this feature, you can do it by hand: Put your thumb and index finger on the soroban’s middle beam, and sweep it across the device. That will sweep all the beads to their cleared position.</span></p><p><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1NTMiIGhlaWdodD0iMzE3IiB2aWV3Qm94PSIwIDAgNTUzIDMxNyI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" class="alignnone size-full wp-image-23314" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/clearing-abacus.png" alt="clearing abacus" width="553" height="317" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/clearing-abacus.png 553w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/clearing-abacus-450x258.png 450w" data-sizes="(max-width: 553px) 100vw, 553px" /><noscript><img decoding="async" class="alignnone size-full wp-image-23314" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/clearing-abacus.png" alt="clearing abacus" width="553" height="317" srcset="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/clearing-abacus.png 553w, https://www.physicsforums.com/insights/wp-content/uploads/2019/01/clearing-abacus-450x258.png 450w" sizes="(max-width: 553px) 100vw, 553px" /></noscript></p><h2><span class="ez-toc-section" id="Other-Cool-Facts"></span>Other Cool Facts<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Suanpans ( The 2:5 bead one) can be used to play checkers:</span></p><p><a href="https://en.wikipedia.org/wiki/Abacus_checkers/" target="_blank"><span style="font-weight: 400;">https://en.wikipedia.org/wiki/Abacus_checkers/</span></a></p><p><span style="font-weight: 400;">There was once a soroban versus an electronic calculator competition that took place in Japan in 1947:</span></p><p><a href="https://www.ee.ryerson.ca/~elf/abacus/abacus-contest.html" target="_blank"><span style="font-weight: 400;">https://www.ee.ryerson.ca/~elf/abacus/abacus-contest.html</span></a></p><p><span style="font-weight: 400;">The soroban won four rounds out of five rounds. Losing once, when performing a set of multiplication problems.</span></p><p><span style="font-weight: 400;">Besides basic arithmetic, a soroban can also do: </span></p><ol><li style="font-weight: 400;"><span style="font-weight: 400;">Square roots: </span></li></ol><p><a href="https://scienceblogs.com/goodmath/2006/10/02/square-root-on-the-abacus" target="_blank"><span style="font-weight: 400;">https://scienceblogs.com/goodmath/2006/10/02/square-root-on-the-abacus</span></a><span style="font-weight: 400;"> </span></p><ol><li style="font-weight: 400;"><span style="font-weight: 400;">Cube roots:</span></li></ol><p><a href="http://totton.idirect.com/soroban/Print_Cube_Root.pdf" target="_blank"><span style="font-weight: 400;">http://totton.idirect.com/soroban/Print_Cube_Root.pdf</span></a><span style="font-weight: 400;"> </span></p><ol><li style="font-weight: 400;"><span style="font-weight: 400;">Higher degree roots using Newton’s method:</span><span style="font-weight: 400;"><br /> </span><a href="https://math.stackexchange.com/ques...-sqrt-x-without-using-logarithm/538055#538055/" target="_blank"><span style="font-weight: 400;">https://math.stackexchange.com/ques…-sqrt-x-without-using-logarithm/538055#538055</span></a><span style="font-weight: 400;"> </span></li></ol><p> </p><h2><span class="ez-toc-section" id="Final-Thoughts"></span>Final Thoughts<span class="ez-toc-section-end"></span></h2><p><span style="font-weight: 400;">Suanpans have been the primary accounting tool for centuries of businessmen and traders and are still in active use today in many Asian countries. They can be a fun way to improve your calculating ability or to teach your kids about numbers while at the same time providing an environmentally friendly way of reducing your reliance on battery technology and calculators.</span></p><p><span style="font-weight: 400;">For anyone interested in how to calculate with suanpans or just know more about their history, here are some additional resources:</span></p><p><span style="font-weight: 400;">An online version of sorobans and suanpans: </span><a href="http://www.mathematik.uni-marburg.de/~thormae/lectures/ti1/code/abacus/" target="_blank"><span style="font-weight: 400;">http://www.mathematik.uni-marburg.de/~thormae/lectures/ti1/code/abacus/</span></a></p><p><span style="font-weight: 400;">Addition on soroban:</span></p><p><a href="https://scienceblogs.com/goodmath/2006/09/19/arithmetic-on-the-abacus-part" target="_blank"><span style="font-weight: 400;">https://scienceblogs.com/goodmath/2006/09/19/arithmetic-on-the-abacus-part</span></a></p><p><span style="font-weight: 400;">Multiplication on soroban:</span></p><p><a href="https://scienceblogs.com/goodmath/2006/09/21/using-the-abacus-part-2-multip-1" target="_blank"><span style="font-weight: 400;">https://scienceblogs.com/goodmath/2006/09/21/using-the-abacus-part-2-multip-1</span></a></p><p><span style="font-weight: 400;">Division on soroban:</span></p><p><a href="https://scienceblogs.com/goodmath/2006/09/25/division-on-the-abacus" target="_blank"><span style="font-weight: 400;">https://scienceblogs.com/goodmath/2006/09/25/division-on-the-abacus</span></a></p><p><span style="font-weight: 400;">Division can also be done digit by digit, which is easier and faster for me.</span></p><p><span style="font-weight: 400;">The two methods both have pros and cons though.</span></p><p><span style="font-weight: 400;">News for the soroban competition with a calculator:</span><span style="font-weight: 400;"><br /> </span><a href="https://www.wired.com/2009/11/1112abacus-beats-calculator/" target="_blank"><span style="font-weight: 400;">https://www.wired.com/2009/11/1112abacus-beats-calculator/</span></a></p><p><span style="font-weight: 400;">Some references to abacuses more generally:</span></p><p><a href="https://www.ee.ryerson.ca/~elf/abacus/history.html" target="_blank"><span style="font-weight: 400;">https://www.ee.ryerson.ca/~elf/abacus/history.html</span></a></p><p><a href="https://www.ee.ryerson.ca/~elf/abacus/index.html" target="_blank"><span style="font-weight: 400;">https://www.ee.ryerson.ca/~elf/abacus/index.html</span></a></p></div><div class="saboxplugin-wrap" itemtype="http://schema.org/Person" itemscope itemprop="author"><div class="saboxplugin-tab"><div class="saboxplugin-gravatar"><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDAiIGhlaWdodD0iMTAwIiB2aWV3Qm94PSIwIDAgMTAwIDEwMCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/649826.jpg" width="100" height="100" alt="Young Physicist" itemprop="image"><noscript><img decoding="async" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/649826.jpg" width="100" height="100" alt="Young Physicist" itemprop="image"></noscript></div><div class="saboxplugin-authorname"><a href="https://www.physicsforums.com/insights/author/youngphysicist/" class="vcard author" rel="author"><span class="fn">YoungPhysicist</span></a></div><div class="saboxplugin-desc"><div itemprop="description"><p>From Taipei Taiwan interested in Quantum Mechanics, String theory, Chemistry, Biology, Astro physics, Calculus</p></div></div><div class="clearfix"></div></div></div><style>.lwrp.link-whisper-related-posts{ margin-top: 25px; margin-bottom: 25px; } .lwrp .lwrp-title{ }.lwrp .lwrp-description{ } .lwrp .lwrp-list-container{ } .lwrp .lwrp-list-multi-container{ display: flex; } .lwrp .lwrp-list-double{ width: 48%; } .lwrp .lwrp-list-triple{ width: 32%; } .lwrp .lwrp-list-row-container{ display: flex; justify-content: space-between; } .lwrp .lwrp-list-row-container .lwrp-list-item{ width: calc(33% - 20px); } .lwrp .lwrp-list-item:not(.lwrp-no-posts-message-item){ margin-bottom: 25px; } .lwrp .lwrp-list-item img{ max-width: 100%; height: auto; object-fit: cover; aspect-ratio: 1 / 1; } .lwrp .lwrp-list-item.lwrp-empty-list-item{ background: initial !important; } .lwrp .lwrp-list-item .lwrp-list-link .lwrp-list-link-title-text, .lwrp .lwrp-list-item .lwrp-list-no-posts-message{ }@media screen and (max-width: 480px) { .lwrp.link-whisper-related-posts{ } .lwrp .lwrp-title{ }.lwrp .lwrp-description{ } .lwrp .lwrp-list-multi-container{ flex-direction: column; } .lwrp .lwrp-list-multi-container ul.lwrp-list{ margin-top: 0px; margin-bottom: 0px; padding-top: 0px; padding-bottom: 0px; } .lwrp .lwrp-list-double, .lwrp .lwrp-list-triple{ width: 100%; } .lwrp .lwrp-list-row-container{ justify-content: initial; flex-direction: column; } .lwrp .lwrp-list-row-container .lwrp-list-item{ width: 100%; } .lwrp .lwrp-list-item:not(.lwrp-no-posts-message-item){ } .lwrp .lwrp-list-item .lwrp-list-link .lwrp-list-link-title-text, .lwrp .lwrp-list-item .lwrp-list-no-posts-message{ }; }</style><div id="link-whisper-related-posts-widget" class="link-whisper-related-posts lwrp"><h3 class="lwrp-title">More Related Articles</h3><div class="lwrp-list-container"><div class="lwrp-list-multi-container"><ul class="lwrp-list lwrp-list-double lwrp-list-left"><li class="lwrp-list-item"><a href="https://www.physicsforums.com/insights/all-about-the-einstein-field-equations/" class="lwrp-list-link"><span class="lwrp-list-link-title-text">All About the Einstein Field Equations</span></a></li></ul><ul class="lwrp-list lwrp-list-double lwrp-list-right"></ul></div></div></div></div><footer class="entry-footer"><span class="blog-tags minor-meta"><strong>Tags:</strong><span> <a href="https://www.physicsforums.com/insights/tag/science-history/" rel="tag">science history</a></span></span><div class='av-social-sharing-box av-social-sharing-box-default av-social-sharing-box-fullwidth'><div class="av-share-box"><h5 class='av-share-link-description av-no-toc '>Share this entry</h5><ul class="av-share-box-list noLightbox"><li class='av-share-link av-social-link-facebook' ><a target="_blank" aria-label="Share on Facebook" href='https://www.facebook.com/sharer.php?u=https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/&t=An%20Introduction%20to%20the%20Traditional%20Abacus%20of%20China%20and%20Japan' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='' data-avia-related-tooltip='Share on Facebook' rel="noopener"><span class='avia_hidden_link_text'>Share on Facebook</span></a></li><li class='av-share-link av-social-link-twitter' ><a target="_blank" aria-label="Share on X" href='https://twitter.com/share?text=An%20Introduction%20to%20the%20Traditional%20Abacus%20of%20China%20and%20Japan&url=https://www.physicsforums.com/insights/?p=23309' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='' data-avia-related-tooltip='Share on X' rel="noopener"><span class='avia_hidden_link_text'>Share on X</span></a></li><li class='av-share-link av-social-link-whatsapp' ><a target="_blank" aria-label="Share on WhatsApp" href='https://api.whatsapp.com/send?text=https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='' data-avia-related-tooltip='Share on WhatsApp' rel="noopener"><span class='avia_hidden_link_text'>Share on WhatsApp</span></a></li><li class='av-share-link av-social-link-linkedin' ><a target="_blank" aria-label="Share on LinkedIn" href='https://linkedin.com/shareArticle?mini=true&title=An%20Introduction%20to%20the%20Traditional%20Abacus%20of%20China%20and%20Japan&url=https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='' data-avia-related-tooltip='Share on LinkedIn' rel="noopener"><span class='avia_hidden_link_text'>Share on LinkedIn</span></a></li><li class='av-share-link av-social-link-reddit' ><a target="_blank" aria-label="Share on Reddit" href='https://reddit.com/submit?url=https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/&title=An%20Introduction%20to%20the%20Traditional%20Abacus%20of%20China%20and%20Japan' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='' data-avia-related-tooltip='Share on Reddit' rel="noopener"><span class='avia_hidden_link_text'>Share on Reddit</span></a></li><li class='av-share-link av-social-link-mail' ><a aria-label="Share by Mail" href='mailto:?subject=An%20Introduction%20to%20the%20Traditional%20Abacus%20of%20China%20and%20Japan&body=https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='' data-avia-related-tooltip='Share by Mail'><span class='avia_hidden_link_text'>Share by Mail</span></a></li></ul></div></div></footer><div class='post_delimiter'></div></div><div class="post_author_timeline"></div><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/01/abacus_introduction.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>YoungPhysicist</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>YoungPhysicist</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-01-19T16:15:54-06:00" >2019-01-19 16:15:54</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-17 16:56:42</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>An Introduction to the Traditional Abacus of China and Japan</span></span></span></article><div class='single-big'></div><div class='related_posts clearfix av-related-style-full'><h5 class="related_title">You might also like</h5><div class="related_entries_container"><div class='av_one_half no_margin alpha relThumb relThumb1 relThumbOdd post-format-standard related_column'><a href='https://www.physicsforums.com/insights/lessons-from-the-bizzaro-universe/' class='relThumWrap noLightbox' title='Lessons From the Bizzaro Universe'><span class='related_image_wrap' ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMTgwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="180" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2018/04/bizarro_univerise-180x135.png" class="wp-image-19003 avia-img-lazy-loading-19003 attachment-square size-square wp-post-image" alt="bizarro univerise" title="bizarro_univerise" decoding="async" /><noscript><img loading="lazy" width="180" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2018/04/bizarro_univerise-180x135.png" class="wp-image-19003 avia-img-lazy-loading-19003 attachment-square size-square wp-post-image" alt="bizarro univerise" title="bizarro_univerise" decoding="async" /></noscript><span class='related-format-icon '><span class='related-format-icon-inner' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span></span></span><strong class="av-related-title">Lessons From the Bizzaro Universe</strong></a></div><div class='av_one_half no_margin relThumb relThumb2 relThumbEven post-format-standard related_column'><a href='https://www.physicsforums.com/insights/roger-babsons-anti-gravity-contest/' class='relThumWrap noLightbox' title='Roger Babson’s Anti-Gravity Contest and Modern Physics'><span class='related_image_wrap' ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMTgwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="180" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2016/04/antigravity-180x135.png" class="wp-image-9520 avia-img-lazy-loading-9520 attachment-square size-square wp-post-image" alt="antigravity" title="antigravity" decoding="async" /><noscript><img loading="lazy" width="180" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2016/04/antigravity-180x135.png" class="wp-image-9520 avia-img-lazy-loading-9520 attachment-square size-square wp-post-image" alt="antigravity" title="antigravity" decoding="async" /></noscript><span class='related-format-icon '><span class='related-format-icon-inner' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span></span></span><strong class="av-related-title">Roger Babson’s Anti-Gravity Contest and Modern Physics</strong></a></div><div class='av_one_half no_margin relThumb relThumb3 relThumbOdd post-format-standard related_column'><a href='https://www.physicsforums.com/insights/opinion-when-pro-scientists-explain-using-pop-science/' class='relThumWrap noLightbox' title='When Pro Scientists Explain Using Pop Science'><span class='related_image_wrap' ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMTgwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="180" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/04/pro-scientist-pop-science-180x135.png" class="wp-image-43125 avia-img-lazy-loading-43125 attachment-square size-square wp-post-image" alt="Carl Sagan" title="pro-scientist-pop-science" decoding="async" /><noscript><img loading="lazy" width="180" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/04/pro-scientist-pop-science-180x135.png" class="wp-image-43125 avia-img-lazy-loading-43125 attachment-square size-square wp-post-image" alt="Carl Sagan" title="pro-scientist-pop-science" decoding="async" /></noscript><span class='related-format-icon '><span class='related-format-icon-inner' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span></span></span><strong class="av-related-title">When Pro Scientists Explain Using Pop Science</strong></a></div><div class='av_one_half no_margin relThumb relThumb4 relThumbEven post-format-standard related_column'><a href='https://www.physicsforums.com/insights/historys-first-science-experiments/' class='relThumWrap noLightbox' title='Learn About History’s First Science Experiments'><span class='related_image_wrap' ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMTgwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="180" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2017/02/experimenthistory-180x135.png" class="wp-image-12395 avia-img-lazy-loading-12395 attachment-square size-square wp-post-image" alt="experimenthistory" title="experimenthistory" decoding="async" /><noscript><img loading="lazy" width="180" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2017/02/experimenthistory-180x135.png" class="wp-image-12395 avia-img-lazy-loading-12395 attachment-square size-square wp-post-image" alt="experimenthistory" title="experimenthistory" decoding="async" /></noscript><span class='related-format-icon '><span class='related-format-icon-inner' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span></span></span><strong class="av-related-title">Learn About History’s First Science Experiments</strong></a></div><div class='av_one_half no_margin omega relThumb relThumb5 relThumbOdd post-format-standard related_column'><a href='https://www.physicsforums.com/insights/assessing-physics-edmond-hamiltons-second-satellite/' class='relThumWrap noLightbox' title='Assessing the Physics in Edmond Hamilton’s “The Second Satellite”'><span class='related_image_wrap' ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMTgwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" loading="lazy" width="180" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2017/07/EdmondHamilton-180x135.png" class="wp-image-14732 avia-img-lazy-loading-14732 attachment-square size-square wp-post-image" alt="EdmondHamilton" title="EdmondHamilton" decoding="async" /><noscript><img loading="lazy" width="180" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2017/07/EdmondHamilton-180x135.png" class="wp-image-14732 avia-img-lazy-loading-14732 attachment-square size-square wp-post-image" alt="EdmondHamilton" title="EdmondHamilton" decoding="async" /></noscript><span class='related-format-icon '><span class='related-format-icon-inner' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span></span></span><strong class="av-related-title">Assessing the Physics in Edmond Hamilton’s “The Second Satellite”</strong></a></div></div></div><div class='comment-entry post-entry'><div class='comment_meta_container'><div class='side-container-comment'><div class='side-container-comment-inner'> <span class='comment-count'>7</span> <span class='comment-text'>replies</span> <span class='center-border center-border-left'></span> <span class='center-border center-border-right'></span></div></div></div><div class='comment_container'><ol class="commentlist" id="comments"><li class="comment even thread-even depth-1" id="li-comment-11248"><div id="comment-11248"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">Wrichik Basu</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11248"> <time > January 20, 2019 at 2:47 am </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><div>YoungPhysicist</div><p>Well, we usually do that like the… ah, what is this method called in English?</p><pre style="margin: 1em auto" title="Code"> 2 3 x 4 6 ---------- 1 3 8 9 2 --------------- 1 0 5 8 </pre><p>(That looks a bit odd cause the spacing can't display properly after I submitted this post, I don't know why)<br /> where we process it digit by digit, but from the<b> left to the right</b>, unlike the method I shown above, which is<b> right to the left</b>.<br /> thus your peoblem will be done in 12 steps:</p><ol><li>9×6</li><li>9×5</li><li>9×8</li><li>9×9</li><li>8×6</li><li>8×5</li><li>8×8</li><li>8×9</li><li>7×6</li><li>7×5</li><li>7×8</li><li>7×9</li></ol><p>Just make sure the calculations are placed on the right column.Yes, it was done like that; I can recall some of it. But visualising so many columns was (and still is) a difficult job for me. I can correctly visualise upto three columns on the abacus board currently; and anything more than that generally leads to an error. Maybe you can blame the calculator to some extent :wink:</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li><li class="comment byuser comment-author-jedishrfu odd alt thread-odd thread-alt depth-1" id="li-comment-11247"><div id="comment-11247"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">jedishrfu</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11247"> <time > January 20, 2019 at 2:08 am </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><p>I fixed the spacing using code tags which preserve the leading spaces and uses a mono spaced font.</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li><li class="comment even thread-even depth-1" id="li-comment-11246"><div id="comment-11246"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">YoungPhysicist</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11246"> <time > January 20, 2019 at 1:47 am </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><div>Wrichik Basu</div><p>(like 987×6598),Well, we usually do that like the… ah, what is this method called in English?</p><pre style="margin: 1em auto" title="Code"> 2 3 x 4 6 ---------- 1 3 8 9 2 --------------- 1 0 5 8 </pre><p>(That looks a bit odd cause the spacing can't display properly after I submitted this post, I don't know why)<br /> where we process it digit by digit, but from the<b> left to the right</b>, unlike the method I shown above, which is<b> right to the left</b>.<br /> thus your peoblem will be done in 12 steps:</p><ol><li>9×6</li><li>9×5</li><li>9×8</li><li>9×9</li><li>8×6</li><li>8×5</li><li>8×8</li><li>8×9</li><li>7×6</li><li>7×5</li><li>7×8</li><li>7×9</li></ol><p>Just make sure the calculations are placed on the right column.</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li><li class="comment odd alt thread-odd thread-alt depth-1" id="li-comment-11245"><div id="comment-11245"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">YoungPhysicist</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11245"> <time > January 20, 2019 at 1:30 am </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><div>Wrichik Basu</div><p>Nice first insight. Congratulations!</p><p>I learnt abacus when I was of <a href="https://www.physicsforums.com/members/649826/" class="username" data-user="649826, @YoungPhysicist">@YoungPhysicist</a>'s age. Though I have forgotten how to do the larger calculations (like 987×6598), while doing simple arithmetic, the technique of abacus now comes naturally. It was of great help in increasing concentration when I was learning.Yeah, It really does, and also a great advantge on tests and calculations to calculate quickly(though almost everyone classmate of mine also take abacus courses with me, so the edge is not that obvious.:rolleyes:</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li><li class="comment even thread-even depth-1" id="li-comment-11244"><div id="comment-11244"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">Wrichik Basu</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11244"> <time > January 20, 2019 at 1:26 am </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><p>Nice first insight. Congratulations!</p><p>I learnt abacus when I was of <a href="https://www.physicsforums.com/members/649826/" class="username" data-user="649826, @YoungPhysicist">@YoungPhysicist</a>'s age. Though I have forgotten how to do the larger calculations (like 987×6598), while doing simple arithmetic, the technique of abacus now comes naturally. It was of great help in increasing concentration when I was learning.</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li><li class="comment byuser comment-author-jedishrfu odd alt thread-odd thread-alt depth-1" id="li-comment-11243"><div id="comment-11243"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">jedishrfu</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11243"> <time > January 19, 2019 at 5:56 pm </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><p>Great job! Congratulations!</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li><li class="comment even thread-even depth-1" id="li-comment-11242"><div id="comment-11242"><article><div class="gravatar"></div><div class='comment_content'><header class="comment-header" aria-label="Comment Info" > <cite class="author_name heading"><cite class="comment_author_name">YoungPhysicist</cite></cite> <span class="says">says:</span><div class="comment-meta commentmetadata"> <a href="https://www.physicsforums.com/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#comment-11242"> <time > January 19, 2019 at 4:42 pm </time> </a></div></header><div class='comment_text entry-content-wrapper clearfix' ><p>Ya! My first insight! Can’t believe it!</p> <a rel="nofollow" class="comment-reply-login" href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">Log in to Reply</a></div></div></article></div></li></ol></div><div class="comment_container"><h3 class='miniheading '>Leave a Reply</h3><span class="minitext">Want to join the discussion?<br />Feel free to contribute!</span><div id="respond" class="comment-respond"><h3 id="reply-title" class="comment-reply-title">Leave a Reply <small><a rel="nofollow" id="cancel-comment-reply-link" href="/insights/an-introduction-to-the-traditional-abacus-of-china-and-japan/#respond" style="display:none;">Cancel reply</a></small></h3><p class="must-log-in">You must be <a href="https://www.physicsforums.com/insights/wp-login.php?redirect_to=https%3A%2F%2Fwww.physicsforums.com%2Finsights%2Fan-introduction-to-the-traditional-abacus-of-china-and-japan%2F">logged in</a> to post a comment.</p></div></div></div></main><aside class='sidebar sidebar_right smartphones_sidebar_active alpha units' aria-label="Sidebar" role="complementary" itemscope="itemscope" itemtype="https://schema.org/WPSideBar" ><div class="inner_sidebar extralight-border"><section id="custom_html-2" class="widget_text widget clearfix widget_custom_html"><div class="textwidget custom-html-widget"><h2 class="widgettitle">Trending Articles</h2><ul class="wpp-list wpp-sunset"><li class="" style="--item-position: 1;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/valentines-reflections-mathematical-matters-of-the-heart/" class="wpp-post-title" target="_self">Valentine’s Reflections: Mathematical Matters of the Heart</a></div></li><li class="" style="--item-position: 2;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/an-accurate-hookes-law-laboratory/" class="wpp-post-title" target="_self">An Example of An Accurate Hooke’s Law Laboratory</a></div></li><li class="" style="--item-position: 3;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/can-see-atom/" class="wpp-post-title" target="_self">Can We See an Atom?</a></div></li><li class="" style="--item-position: 4;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/what-is-a-tensor/" class="wpp-post-title" target="_self">What Is a Tensor? The mathematical point of view.</a></div></li><li class="" style="--item-position: 5;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/demystifying-chain-rule-calculus/" class="wpp-post-title" target="_self">Demystifying the Chain Rule in Calculus</a></div></li><li class="" style="--item-position: 6;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe-2/" class="wpp-post-title" target="_self">How to Determine the Change in Entropy</a></div></li><li class="" style="--item-position: 7;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/ac-power-analysis-part-1-basics/" class="wpp-post-title" target="_self">Intro to AC Power Analysis: Learn System Basics</a></div></li><li class="" style="--item-position: 8;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/" class="wpp-post-title" target="_self">What Planck Length Is and It’s Common Misconceptions</a></div></li><li class="" style="--item-position: 9;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/quantum-mechanics-and-the-famous-double-slit-experiment/" class="wpp-post-title" target="_self">Quantum Mechanics and the Famous Double-slit Experiment</a></div></li><li class="" style="--item-position: 10;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/misconceptions-virtual-particles/" class="wpp-post-title" target="_self">Top Misconceptions about Virtual Particles</a></div></li></ul></div><span class="seperator extralight-border"></span></section><section id="nav_menu-3" class="widget clearfix widget_nav_menu"><h3 class="widgettitle">Physics Forums</h3><div class="menu-sidebar-forums-container"><ul id="menu-sidebar-forums" class="menu"><li id="menu-item-26724" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26724"><a href="https://www.physicsforums.com/forums/classical-physics.61/">Classical Physics</a></li><li id="menu-item-26725" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26725"><a href="https://www.physicsforums.com/forums/atomic-and-condensed-matter.64/">Atomic and Condensed Matter</a></li><li id="menu-item-26726" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26726"><a href="https://www.physicsforums.com/forums/quantum-physics.62/">Quantum Physics</a></li><li id="menu-item-26727" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26727"><a href="https://www.physicsforums.com/forums/special-and-general-relativity.70/">Special and General Relativity</a></li><li id="menu-item-26728" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26728"><a href="https://www.physicsforums.com/forums/beyond-the-standard-model.66/">Beyond the Standard Model</a></li><li id="menu-item-26729" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26729"><a href="https://www.physicsforums.com/forums/high-energy-nuclear-particle-physics.65/">High Energy, Nuclear, Particle Physics</a></li><li id="menu-item-26731" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26731"><a href="https://www.physicsforums.com/forums/astronomy-and-astrophysics.71/">Astronomy and Astrophysics</a></li><li id="menu-item-26732" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26732"><a href="https://www.physicsforums.com/forums/cosmology.69/">Cosmology</a></li><li id="menu-item-26730" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26730"><a href="https://www.physicsforums.com/forums/other-physics-topics.111/">Other Physics Topics</a></li></ul></div><span class="seperator extralight-border"></span></section><section id="text-4" class="widget clearfix widget_text"><h3 class="widgettitle">Receive Insights Articles to Your Inbox</h3><div class="textwidget"><form style="border:0px solid #ccc;text-align:left;" action="https://feedburner.google.com/fb/a/mailverify" method="post" target="popupwindow" onsubmit="window.open('https://feedburner.google.com/fb/a/mailverify?uri=PhysicsForumsInsights', 'popupwindow', 'scrollbars=yes,width=550,height=520');return true"><p>Enter your email address:</p><p><input type="text" style="width:200px;border:2px solid #ccc;" name="email"/></p><input type="hidden" value="PhysicsForumsInsights" name="uri"/><input type="hidden" name="loc" value="en_US"/><input type="submit" value="Subscribe" /></p></form></div> <span class="seperator extralight-border"></span></section><section id="nav_menu-2" class="widget clearfix widget_nav_menu"><h3 class="widgettitle">Blog Information</h3><div class="menu-sidebar-container"><ul id="menu-sidebar" class="menu"><li id="menu-item-14015" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-14015"><a href="https://www.physicsforums.com/register">Become a Member!</a></li><li id="menu-item-14012" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-14012"><a href="https://www.physicsforums.com/insights/application/">Write for Us!</a></li><li id="menu-item-18348" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-18348"><a href="https://www.physicsforums.com/insights/postindex/">Table of Contents</a></li><li id="menu-item-18353" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-18353"><a href="https://www.physicsforums.com/insights/authors/">Blog Author List</a></li></ul></div><span class="seperator extralight-border"></span></section><section id="tag_cloud-3" class="widget clearfix widget_tag_cloud"><h3 class="widgettitle">Popular Topics</h3><div class="tagcloud"><a href="https://www.physicsforums.com/insights/tag/astronomy/" class="tag-cloud-link tag-link-307 tag-link-position-1" style="font-size: 9.1666666666667pt;" aria-label="astronomy (17 items)">astronomy<span class="tag-link-count"> (17)</span></a> <a href="https://www.physicsforums.com/insights/tag/black-holes/" class="tag-cloud-link tag-link-28 tag-link-position-2" style="font-size: 9.1666666666667pt;" aria-label="black holes (17 items)">black holes<span class="tag-link-count"> (17)</span></a> <a href="https://www.physicsforums.com/insights/tag/classical-physics/" class="tag-cloud-link tag-link-852 tag-link-position-3" style="font-size: 20.833333333333pt;" aria-label="classical physics (35 items)">classical physics<span class="tag-link-count"> (35)</span></a> <a href="https://www.physicsforums.com/insights/tag/cosmology/" class="tag-cloud-link tag-link-224 tag-link-position-4" style="font-size: 8pt;" aria-label="cosmology (16 items)">cosmology<span class="tag-link-count"> (16)</span></a> <a href="https://www.physicsforums.com/insights/tag/education/" class="tag-cloud-link tag-link-104 tag-link-position-5" style="font-size: 13.833333333333pt;" aria-label="education (23 items)">education<span class="tag-link-count"> (23)</span></a> <a href="https://www.physicsforums.com/insights/tag/electromagnetism/" class="tag-cloud-link tag-link-21 tag-link-position-6" style="font-size: 10.722222222222pt;" aria-label="electromagnetism (19 items)">electromagnetism<span class="tag-link-count"> (19)</span></a> <a href="https://www.physicsforums.com/insights/tag/general-relativity/" class="tag-cloud-link tag-link-333 tag-link-position-7" style="font-size: 10.722222222222pt;" aria-label="general relativity (19 items)">general relativity<span class="tag-link-count"> (19)</span></a> <a href="https://www.physicsforums.com/insights/tag/gravity/" class="tag-cloud-link tag-link-40 tag-link-position-8" style="font-size: 14.611111111111pt;" aria-label="gravity (24 items)">gravity<span class="tag-link-count"> (24)</span></a> <a href="https://www.physicsforums.com/insights/tag/interview/" class="tag-cloud-link tag-link-858 tag-link-position-9" style="font-size: 13.055555555556pt;" aria-label="interview (22 items)">interview<span class="tag-link-count"> (22)</span></a> <a href="https://www.physicsforums.com/insights/tag/mathematics/" class="tag-cloud-link tag-link-69 tag-link-position-10" style="font-size: 22pt;" aria-label="mathematics (38 items)">mathematics<span class="tag-link-count"> (38)</span></a> <a href="https://www.physicsforums.com/insights/tag/mathematics-self-study/" class="tag-cloud-link tag-link-484 tag-link-position-11" style="font-size: 11.5pt;" aria-label="mathematics self-study (20 items)">mathematics self-study<span class="tag-link-count"> (20)</span></a> <a href="https://www.physicsforums.com/insights/tag/physicist/" class="tag-cloud-link tag-link-11 tag-link-position-12" style="font-size: 15.777777777778pt;" aria-label="Physicist (26 items)">Physicist<span class="tag-link-count"> (26)</span></a> <a href="https://www.physicsforums.com/insights/tag/programming/" class="tag-cloud-link tag-link-42 tag-link-position-13" style="font-size: 9.9444444444444pt;" aria-label="programming (18 items)">programming<span class="tag-link-count"> (18)</span></a> <a href="https://www.physicsforums.com/insights/tag/quantum-field-theory/" class="tag-cloud-link tag-link-87 tag-link-position-14" style="font-size: 18.888888888889pt;" aria-label="Quantum Field Theory (31 items)">Quantum Field Theory<span class="tag-link-count"> (31)</span></a> <a href="https://www.physicsforums.com/insights/tag/quantum-mechanics/" class="tag-cloud-link tag-link-62 tag-link-position-15" style="font-size: 21.222222222222pt;" aria-label="quantum mechanics (36 items)">quantum mechanics<span class="tag-link-count"> (36)</span></a> <a href="https://www.physicsforums.com/insights/tag/quantum-physics/" class="tag-cloud-link tag-link-314 tag-link-position-16" style="font-size: 13.055555555556pt;" aria-label="quantum physics (22 items)">quantum physics<span class="tag-link-count"> (22)</span></a> <a href="https://www.physicsforums.com/insights/tag/relativity/" class="tag-cloud-link tag-link-114 tag-link-position-17" style="font-size: 22pt;" aria-label="relativity (38 items)">relativity<span class="tag-link-count"> (38)</span></a> <a href="https://www.physicsforums.com/insights/tag/special-relativity/" class="tag-cloud-link tag-link-89 tag-link-position-18" style="font-size: 8pt;" aria-label="Special Relativity (16 items)">Special Relativity<span class="tag-link-count"> (16)</span></a> <a href="https://www.physicsforums.com/insights/tag/technology/" class="tag-cloud-link tag-link-861 tag-link-position-19" style="font-size: 10.722222222222pt;" aria-label="technology (19 items)">technology<span class="tag-link-count"> (19)</span></a> <a href="https://www.physicsforums.com/insights/tag/universe/" class="tag-cloud-link tag-link-265 tag-link-position-20" style="font-size: 12.277777777778pt;" aria-label="universe (21 items)">universe<span class="tag-link-count"> (21)</span></a></div> <span class="seperator extralight-border"></span></section></div></aside></div></div><footer class='container_wrap socket_color' id='socket' role="contentinfo" itemscope="itemscope" itemtype="https://schema.org/WPFooter" aria-label="Copyright and company info" ><div class='container'><span class='copyright'> 2024 © Physics Forums, ALL RIGHTS RESERVED - <a href="https://www.physicsforums.com/misc/contact">Contact Us</a> - <a href="https://www.physicsforums.com/threads/physics-forums-privacy-policy.797829/">Privacy Policy</a> - <a href="https://www.physicsforums.com/insights/about-pf-insights/">About PF Insights</a></span><ul class='noLightbox social_bookmarks icon_count_4'><li class='social_bookmarks_twitter av-social-link-twitter social_icon_1'><a target="_blank" aria-label="Link to X" href="https://twitter.com/physicsforums" aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='X' rel="noopener"><span class='avia_hidden_link_text'>X</span></a></li><li class='social_bookmarks_facebook av-social-link-facebook social_icon_2'><a target="_blank" aria-label="Link to Facebook" href='https://www.facebook.com/physicsforums' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='Facebook' rel="noopener"><span class='avia_hidden_link_text'>Facebook</span></a></li><li class='social_bookmarks_linkedin av-social-link-linkedin social_icon_3'><a target="_blank" aria-label="Link to LinkedIn" href='https://www.linkedin.com/company/physics-forums/' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='LinkedIn' rel="noopener"><span class='avia_hidden_link_text'>LinkedIn</span></a></li><li class='social_bookmarks_youtube av-social-link-youtube social_icon_4'><a target="_blank" aria-label="Link to Youtube" href='https://www.youtube.com/@physicsforum' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='Youtube' rel="noopener"><span class='avia_hidden_link_text'>Youtube</span></a></li></ul></div></footer></div><a class='avia-post-nav avia-post-prev with-image' href='https://www.physicsforums.com/insights/an-accurate-hookes-law-laboratory/' ><span class="label iconfont" aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span><span class="entry-info-wrap"><span class="entry-info"><span class='entry-title'>An Example of An Accurate Hooke’s Law Laboratory</span><span class='entry-image'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4MCIgaGVpZ2h0PSI0NSIgdmlld0JveD0iMCAwIDgwIDQ1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" loading="lazy" width="80" height="45" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/hookes_law_experiment.png" class="wp-image-23286 avia-img-lazy-loading-23286 attachment-thumbnail size-thumbnail wp-post-image" alt="Hooke's Law Experiement" decoding="async" /><noscript><img loading="lazy" width="80" height="45" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/hookes_law_experiment.png" class="wp-image-23286 avia-img-lazy-loading-23286 attachment-thumbnail size-thumbnail wp-post-image" alt="Hooke's Law Experiement" decoding="async" /></noscript></span></span></span></a><a class='avia-post-nav avia-post-next with-image' href='https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-representations/' ><span class="label iconfont" aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello'></span><span class="entry-info-wrap"><span class="entry-info"><span class='entry-image'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4MCIgaGVpZ2h0PSI0NSIgdmlld0JveD0iMCAwIDgwIDQ1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" loading="lazy" width="80" height="45" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/lie_algebra_representations.png" class="wp-image-23401 avia-img-lazy-loading-23401 attachment-thumbnail size-thumbnail wp-post-image" alt="lie algebra representations" decoding="async" /><noscript><img loading="lazy" width="80" height="45" src="https://www.physicsforums.com/insights/wp-content/uploads/2019/01/lie_algebra_representations.png" class="wp-image-23401 avia-img-lazy-loading-23401 attachment-thumbnail size-thumbnail wp-post-image" alt="lie algebra representations" decoding="async" /></noscript></span><span class='entry-title'>Lie Algebras: A Walkthrough The Representations</span></span></span></a></div><a href='#top' title='Scroll to top' id='scroll-top-link' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello' tabindex='-1'><span class="avia_hidden_link_text">Scroll to top</span></a><div id="fb-root"></div> <script data-optimized="1" type="text/javascript" src="https://www.physicsforums.com/insights/wp-content/litespeed/js/ffe2445567cebbe851874e864ab606aa.js?ver=606aa" defer></script> <script type='text/javascript'>/* */ var avia_framework_globals = avia_framework_globals || {}; avia_framework_globals.frameworkUrl = 'https://www.physicsforums.com/insights/wp-content/themes/enfold/framework/'; avia_framework_globals.installedAt = 'https://www.physicsforums.com/insights/wp-content/themes/enfold/'; avia_framework_globals.ajaxurl = 'https://www.physicsforums.com/insights/wp-admin/admin-ajax.php'; /* */</script> <script type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-includes/js/jquery/jquery.min.js?ver=3.7.1" id="jquery-core-js"></script> <script type="text/javascript" id="wpil-frontend-script-js-extra">/* */ var wpilFrontend = {"ajaxUrl":"\/insights\/wp-admin\/admin-ajax.php","postId":"23309","postType":"post","openInternalInNewTab":"0","openExternalInNewTab":"1","disableClicks":"0","openLinksWithJS":"0","trackAllElementClicks":"0","clicksI18n":{"imageNoText":"Image in link: No Text","imageText":"Image Title: ","noText":"No Anchor Text Found"}}; /* */</script> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/32fbbb44e22394d8c0f1ada21cfadab5.js?ver=ef02b" id="wpil-frontend-script-js"></script> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/28eaaf920db7121b0c5763a98afd6217.js?ver=f2d22" id="comment-reply-js" async="async" data-wp-strategy="async"></script> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/966b486962f7a18865706d52697208b2.js?ver=208b2" id="avia-footer-scripts-js"></script> <script data-no-optimize="1">!function(t,e){"object"==typeof exports&&"undefined"!=typeof module?module.exports=e():"function"==typeof define&&define.amd?define(e):(t="undefined"!=typeof globalThis?globalThis:t||self).LazyLoad=e()}(this,function(){"use strict";function e(){return(e=Object.assign||function(t){for(var e=1;e<arguments.length;e++){var n,a=arguments[e];for(n in a)Object.prototype.hasOwnProperty.call(a,n)&&(t[n]=a[n])}return t}).apply(this,arguments)}function i(t){return e({},it,t)}function o(t,e){var n,a="LazyLoad::Initialized",i=new t(e);try{n=new CustomEvent(a,{detail:{instance:i}})}catch(t){(n=document.createEvent("CustomEvent")).initCustomEvent(a,!1,!1,{instance:i})}window.dispatchEvent(n)}function l(t,e){return t.getAttribute(gt+e)}function c(t){return l(t,bt)}function s(t,e){return function(t,e,n){e=gt+e;null!==n?t.setAttribute(e,n):t.removeAttribute(e)}(t,bt,e)}function r(t){return s(t,null),0}function u(t){return null===c(t)}function d(t){return c(t)===vt}function f(t,e,n,a){t&&(void 0===a?void 0===n?t(e):t(e,n):t(e,n,a))}function _(t,e){nt?t.classList.add(e):t.className+=(t.className?" ":"")+e}function v(t,e){nt?t.classList.remove(e):t.className=t.className.replace(new RegExp("(^|\\s+)"+e+"(\\s+|$)")," ").replace(/^\s+/,"").replace(/\s+$/,"")}function g(t){return t.llTempImage}function b(t,e){!e||(e=e._observer)&&e.unobserve(t)}function p(t,e){t&&(t.loadingCount+=e)}function h(t,e){t&&(t.toLoadCount=e)}function n(t){for(var e,n=[],a=0;e=t.children[a];a+=1)"SOURCE"===e.tagName&&n.push(e);return n}function m(t,e){(t=t.parentNode)&&"PICTURE"===t.tagName&&n(t).forEach(e)}function a(t,e){n(t).forEach(e)}function E(t){return!!t[st]}function I(t){return t[st]}function y(t){return delete t[st]}function A(e,t){var n;E(e)||(n={},t.forEach(function(t){n[t]=e.getAttribute(t)}),e[st]=n)}function k(a,t){var i;E(a)&&(i=I(a),t.forEach(function(t){var e,n;e=a,(t=i[n=t])?e.setAttribute(n,t):e.removeAttribute(n)}))}function L(t,e,n){_(t,e.class_loading),s(t,ut),n&&(p(n,1),f(e.callback_loading,t,n))}function w(t,e,n){n&&t.setAttribute(e,n)}function x(t,e){w(t,ct,l(t,e.data_sizes)),w(t,rt,l(t,e.data_srcset)),w(t,ot,l(t,e.data_src))}function O(t,e,n){var a=l(t,e.data_bg_multi),i=l(t,e.data_bg_multi_hidpi);(a=at&&i?i:a)&&(t.style.backgroundImage=a,n=n,_(t=t,(e=e).class_applied),s(t,ft),n&&(e.unobserve_completed&&b(t,e),f(e.callback_applied,t,n)))}function N(t,e){!e||0<e.loadingCount||0<e.toLoadCount||f(t.callback_finish,e)}function C(t,e,n){t.addEventListener(e,n),t.llEvLisnrs[e]=n}function M(t){return!!t.llEvLisnrs}function z(t){if(M(t)){var e,n,a=t.llEvLisnrs;for(e in a){var i=a[e];n=e,i=i,t.removeEventListener(n,i)}delete t.llEvLisnrs}}function R(t,e,n){var a;delete t.llTempImage,p(n,-1),(a=n)&&--a.toLoadCount,v(t,e.class_loading),e.unobserve_completed&&b(t,n)}function T(o,r,c){var l=g(o)||o;M(l)||function(t,e,n){M(t)||(t.llEvLisnrs={});var a="VIDEO"===t.tagName?"loadeddata":"load";C(t,a,e),C(t,"error",n)}(l,function(t){var e,n,a,i;n=r,a=c,i=d(e=o),R(e,n,a),_(e,n.class_loaded),s(e,dt),f(n.callback_loaded,e,a),i||N(n,a),z(l)},function(t){var e,n,a,i;n=r,a=c,i=d(e=o),R(e,n,a),_(e,n.class_error),s(e,_t),f(n.callback_error,e,a),i||N(n,a),z(l)})}function G(t,e,n){var a,i,o,r,c;t.llTempImage=document.createElement("IMG"),T(t,e,n),E(c=t)||(c[st]={backgroundImage:c.style.backgroundImage}),o=n,r=l(a=t,(i=e).data_bg),c=l(a,i.data_bg_hidpi),(r=at&&c?c:r)&&(a.style.backgroundImage='url("'.concat(r,'")'),g(a).setAttribute(ot,r),L(a,i,o)),O(t,e,n)}function D(t,e,n){var a;T(t,e,n),a=e,e=n,(t=It[(n=t).tagName])&&(t(n,a),L(n,a,e))}function V(t,e,n){var a;a=t,(-1<yt.indexOf(a.tagName)?D:G)(t,e,n)}function F(t,e,n){var a;t.setAttribute("loading","lazy"),T(t,e,n),a=e,(e=It[(n=t).tagName])&&e(n,a),s(t,vt)}function j(t){t.removeAttribute(ot),t.removeAttribute(rt),t.removeAttribute(ct)}function P(t){m(t,function(t){k(t,Et)}),k(t,Et)}function S(t){var e;(e=At[t.tagName])?e(t):E(e=t)&&(t=I(e),e.style.backgroundImage=t.backgroundImage)}function U(t,e){var n;S(t),n=e,u(e=t)||d(e)||(v(e,n.class_entered),v(e,n.class_exited),v(e,n.class_applied),v(e,n.class_loading),v(e,n.class_loaded),v(e,n.class_error)),r(t),y(t)}function $(t,e,n,a){var i;n.cancel_on_exit&&(c(t)!==ut||"IMG"===t.tagName&&(z(t),m(i=t,function(t){j(t)}),j(i),P(t),v(t,n.class_loading),p(a,-1),r(t),f(n.callback_cancel,t,e,a)))}function q(t,e,n,a){var i,o,r=(o=t,0<=pt.indexOf(c(o)));s(t,"entered"),_(t,n.class_entered),v(t,n.class_exited),i=t,o=a,n.unobserve_entered&&b(i,o),f(n.callback_enter,t,e,a),r||V(t,n,a)}function H(t){return t.use_native&&"loading"in HTMLImageElement.prototype}function B(t,i,o){t.forEach(function(t){return(a=t).isIntersecting||0<a.intersectionRatio?q(t.target,t,i,o):(e=t.target,n=t,a=i,t=o,void(u(e)||(_(e,a.class_exited),$(e,n,a,t),f(a.callback_exit,e,n,t))));var e,n,a})}function J(e,n){var t;et&&!H(e)&&(n._observer=new IntersectionObserver(function(t){B(t,e,n)},{root:(t=e).container===document?null:t.container,rootMargin:t.thresholds||t.threshold+"px"}))}function K(t){return Array.prototype.slice.call(t)}function Q(t){return t.container.querySelectorAll(t.elements_selector)}function W(t){return c(t)===_t}function X(t,e){return e=t||Q(e),K(e).filter(u)}function Y(e,t){var n;(n=Q(e),K(n).filter(W)).forEach(function(t){v(t,e.class_error),r(t)}),t.update()}function t(t,e){var n,a,t=i(t);this._settings=t,this.loadingCount=0,J(t,this),n=t,a=this,Z&&window.addEventListener("online",function(){Y(n,a)}),this.update(e)}var Z="undefined"!=typeof window,tt=Z&&!("onscroll"in window)||"undefined"!=typeof navigator&&/(gle|ing|ro)bot|crawl|spider/i.test(navigator.userAgent),et=Z&&"IntersectionObserver"in window,nt=Z&&"classList"in document.createElement("p"),at=Z&&1<window.devicePixelRatio,it={elements_selector:".lazy",container:tt||Z?document:null,threshold:300,thresholds:null,data_src:"src",data_srcset:"srcset",data_sizes:"sizes",data_bg:"bg",data_bg_hidpi:"bg-hidpi",data_bg_multi:"bg-multi",data_bg_multi_hidpi:"bg-multi-hidpi",data_poster:"poster",class_applied:"applied",class_loading:"litespeed-loading",class_loaded:"litespeed-loaded",class_error:"error",class_entered:"entered",class_exited:"exited",unobserve_completed:!0,unobserve_entered:!1,cancel_on_exit:!0,callback_enter:null,callback_exit:null,callback_applied:null,callback_loading:null,callback_loaded:null,callback_error:null,callback_finish:null,callback_cancel:null,use_native:!1},ot="src",rt="srcset",ct="sizes",lt="poster",st="llOriginalAttrs",ut="loading",dt="loaded",ft="applied",_t="error",vt="native",gt="data-",bt="ll-status",pt=[ut,dt,ft,_t],ht=[ot],mt=[ot,lt],Et=[ot,rt,ct],It={IMG:function(t,e){m(t,function(t){A(t,Et),x(t,e)}),A(t,Et),x(t,e)},IFRAME:function(t,e){A(t,ht),w(t,ot,l(t,e.data_src))},VIDEO:function(t,e){a(t,function(t){A(t,ht),w(t,ot,l(t,e.data_src))}),A(t,mt),w(t,lt,l(t,e.data_poster)),w(t,ot,l(t,e.data_src)),t.load()}},yt=["IMG","IFRAME","VIDEO"],At={IMG:P,IFRAME:function(t){k(t,ht)},VIDEO:function(t){a(t,function(t){k(t,ht)}),k(t,mt),t.load()}},kt=["IMG","IFRAME","VIDEO"];return t.prototype={update:function(t){var e,n,a,i=this._settings,o=X(t,i);{if(h(this,o.length),!tt&&et)return H(i)?(e=i,n=this,o.forEach(function(t){-1!==kt.indexOf(t.tagName)&&F(t,e,n)}),void h(n,0)):(t=this._observer,i=o,t.disconnect(),a=t,void i.forEach(function(t){a.observe(t)}));this.loadAll(o)}},destroy:function(){this._observer&&this._observer.disconnect(),Q(this._settings).forEach(function(t){y(t)}),delete this._observer,delete this._settings,delete this.loadingCount,delete this.toLoadCount},loadAll:function(t){var e=this,n=this._settings;X(t,n).forEach(function(t){b(t,e),V(t,n,e)})},restoreAll:function(){var e=this._settings;Q(e).forEach(function(t){U(t,e)})}},t.load=function(t,e){e=i(e);V(t,e)},t.resetStatus=function(t){r(t)},Z&&function(t,e){if(e)if(e.length)for(var n,a=0;n=e[a];a+=1)o(t,n);else o(t,e)}(t,window.lazyLoadOptions),t});!function(e,t){"use strict";function a(){t.body.classList.add("litespeed_lazyloaded")}function n(){console.log("[LiteSpeed] Start Lazy Load Images"),d=new LazyLoad({elements_selector:"[data-lazyloaded]",callback_finish:a}),o=function(){d.update()},e.MutationObserver&&new MutationObserver(o).observe(t.documentElement,{childList:!0,subtree:!0,attributes:!0})}var d,o;e.addEventListener?e.addEventListener("load",n,!1):e.attachEvent("onload",n)}(window,document);</script></body></html>