Scientific Notation | Math in Science

<!DOCTYPE html> <html lang="en" dir="ltr"> <head>  <meta http-equiv="X-UA-Compatible" content="ie=edge"> <meta charset="utf-8"> <base href="https://www.visionlearning.com"> <title>Scientific Notation | Math in Science | Visionlearning</title> <link rel="canonical" href="https://www.visionlearning.com/en/library/math-in-science/62/scientific-notation/250"> <meta name="description" content="The blue whale weighs approximately 190,000 kilograms, while a plankton weighs just 0.5 milligrams—a difference of 11 orders of magnitude. Scientific notation and order of magnitude are fundamental concepts in all branches of science. They are especially useful when expressing and comparing very large and very small measurements. This module traces the history of our base-ten numeration system and gives a step-by-step explanation of how to write numbers in scientific notation. Sample problems demonstrate how to divide numbers in scientific notation to determine orders of magnitude."> <meta name="keywords" content="scientific notation, order of magnitude, exponents, numbers, multiplication"> <meta name="viewport" content="width=device-width, initial-scale=1.0, shrink-to-fit=no"> <meta name="msvalidate.01" content="D8E20F39AD48052260032E56DE409970"> <script type="application/ld+json"> { "@context": "http://schema.org/", "@type": "Article", "mainEntityOfPage": { "@type": "WebPage", "@id": "https://visionlearning.com/en/library/math-in-science/62/scientific-notation/250" }, "name": "Scientific Notation", "headline": "Scientific Notation: Working with orders of magnitude", "author": [ { "@type": "Person", "name": "Janet Shiver, Ph.D." } , { "@type": "Person", "name": "Teri Willard, Ed.D." }], "datePublished": "2016-10-10 04:19:54", "dateModified": "2017-02-12T08:30:00+05:00", "image": { "@type": "ImageObject", "url": "/img/library/moduleImages/featured_image_250-23061210060637.jpeg", "width": 696, "height": 464 }, "publisher": { "@type": "Organization", "name": "Visionlearning, Inc.", "logo": { "@type": "ImageObject", "url": "http://visionlearning.com/images/logo.png", "width": 278, "height": 60 } }, "description": "The blue whale weighs approximately 190,000 kilograms, while a plankton weighs just 0.5 milligrams—a difference of 11 orders of magnitude. Scientific notation and order of magnitude are fundamental concepts in all branches of science. They are especially useful when expressing and comparing very large and very small measurements. This module traces the history of our base-ten numeration system and gives a step-by-step explanation of how to write numbers in scientific notation. Sample problems demonstrate how to divide numbers in scientific notation to determine orders of magnitude.", "keywords": "scientific notation, order of magnitude, exponents, numbers, multiplication", "inLanguage": { "@type": "Language", "name": "English", "alternateName": "en" }, "copyrightHolder": { "@type": "Organization", "name": "Visionlearning, Inc." }, "copyrightYear": "2016"} </script> <meta property="og:url" content="https://visionlearning.com/en/library/math-in-science/62/scientific-notation/250"> <meta property="og:title" content="Scientific Notation | Math in Science | Visionlearning" /> <meta property="og:type" content="website"> <meta property="og:site_name" content="Visionlearning"> <meta property="og:description" content="The blue whale weighs approximately 190,000 kilograms, while a plankton weighs just 0.5 milligrams—a difference of 11 orders of magnitude. Scientific notation and order of magnitude are fundamental concepts in all branches of science. They are especially useful when expressing and comparing very large and very small measurements. This module traces the history of our base-ten numeration system and gives a step-by-step explanation of how to write numbers in scientific notation. Sample problems demonstrate how to divide numbers in scientific notation to determine orders of magnitude."> <meta property="og:image" content="https://visionlearning.com/images/logo.png"> <meta property="fb:admins" content="100000299664514"> <link rel="stylesheet" type="text/css" href="/css/visionlearning.css">  <link rel="stylesheet" type="text/css" href="/css/visionlearning-icons.css">  <link rel="preconnect" as="font" href="https://fonts.gstatic.com" crossorigin> <link href="https://fonts.googleapis.com/css2?family=Open+Sans:ital,wght@0,400;0,700;1,400;1,700&family=Schoolbell&display=swap" rel="stylesheet"> <style> textarea.myEditor { width: 90%; height: 350px; } </style> <script type="text/x-mathjax-config" src="/js/mathjax-config.js"></script> <script id="MathJax-script" async src="/js/mathjax/tex-svg.js"></script> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-9561344156007092" crossorigin="anonymous"></script> </head> <body>  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aria-controls="acc-panel-ecology" aria-expanded="false"> <span class="accordion__button__label"> Ecology </span> </button> <div class="accordion__panel" id="acc-panel-ecology" data-accordion="panel" aria-labelledby="acc-button-ecology" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/environmental-science/61/biodiversity-i/276">Biodiversity I</a></li> <li><a href="/en/library/environmental-science/61/biodiversity-ii/281">Biodiversity II</a></li> <li><a href="/en/library/environmental-science/61/ecosystem-services/279">Ecosystem Services</a></li> <li><a href="/en/library/environmental-science/61/population-biology/287">Population Biology</a></li> </ul> </div> <button class="accordion__button" id="acc-button-earth-cycles" data-accordion="button" aria-controls="acc-panel-earth-cycles" aria-expanded="false"> <span class="accordion__button__label"> Earth Cycles </span> </button> <div class="accordion__panel" id="acc-panel-earth-cycles" data-accordion="panel" aria-labelledby="acc-button-earth-cycles" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/environmental-science/61/the-nitrogen-cycle/98">The Nitrogen Cycle</a></li> <li><a href="/en/library/environmental-science/61/the-carbon-cycle/95">The Carbon Cycle</a></li> <li><a href="/en/library/environmental-science/61/the-phosphorus-cycle/197">The Phosphorus Cycle</a></li> </ul> </div> <button class="accordion__button" id="acc-button-scientific-research" data-accordion="button" aria-controls="acc-panel-scientific-research" aria-expanded="false"> <span class="accordion__button__label"> Scientific Research </span> </button> <div class="accordion__panel" id="acc-panel-scientific-research" data-accordion="panel" aria-labelledby="acc-button-scientific-research" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/environmental-science/61/collaborative-research-in-the-arctic-towards-understanding-climate-change/183">Collaborative Research in the Arctic Towards Understanding Climate Change</a></li> <li><a href="/en/library/environmental-science/61/atmospheric-chemistry-research-that-changed-global-policy/211">Atmospheric Chemistry Research that Changed Global Policy</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-general-science" data-accordion="button" aria-controls="acc-panel-general-science" aria-expanded="false"> <span class="accordion__button__label"> General Science </span> </button> <div class="accordion__panel" id="acc-panel-general-science" data-accordion="panel" aria-labelledby="acc-button-general-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-methods" data-accordion="button" aria-controls="acc-panel-methods" aria-expanded="false"> <span class="accordion__button__label"> Methods </span> </button> <div class="accordion__panel" id="acc-panel-methods" data-accordion="panel" aria-labelledby="acc-button-methods" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/general-science/3/the-scientific-method/45">The Scientific Method</a></li> </ul> </div> <button class="accordion__button" id="acc-button-measurement" data-accordion="button" aria-controls="acc-panel-measurement" aria-expanded="false"> <span class="accordion__button__label"> Measurement </span> </button> <div class="accordion__panel" id="acc-panel-measurement" data-accordion="panel" aria-labelledby="acc-button-measurement" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/general-science/3/the-metric-system/47">The Metric System</a></li> </ul> </div> <button class="accordion__button" id="acc-button-physical-properties" data-accordion="button" aria-controls="acc-panel-physical-properties" aria-expanded="false"> <span class="accordion__button__label"> Physical Properties </span> </button> <div class="accordion__panel" id="acc-panel-physical-properties" data-accordion="panel" aria-labelledby="acc-button-physical-properties" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/general-science/3/temperature/48">Temperature</a></li> <li><a href="/en/library/general-science/3/density-and-buoyancy/37">Density and Buoyancy</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-math-in-science" data-accordion="button" aria-controls="acc-panel-math-in-science" aria-expanded="false"> <span class="accordion__button__label"> Math in Science </span> </button> <div class="accordion__panel" id="acc-panel-math-in-science" data-accordion="panel" aria-labelledby="acc-button-math-in-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-equations" data-accordion="button" aria-controls="acc-panel-equations" aria-expanded="false"> <span class="accordion__button__label"> Equations </span> </button> <div class="accordion__panel" id="acc-panel-equations" 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class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/introduction-to-descriptive-statistics/218">Introduction to Descriptive Statistics</a></li> <li><a href="/en/library/math-in-science/62/introduction-to-inferential-statistics/224">Introduction to Inferential Statistics</a></li> <li><a href="/en/library/math-in-science/62/statistical-techniques/239">Statistical Techniques</a></li> </ul> </div> <button class="accordion__button" id="acc-button-trigonometric-functions" data-accordion="button" aria-controls="acc-panel-trigonometric-functions" aria-expanded="false"> <span class="accordion__button__label"> Trigonometric Functions </span> </button> <div class="accordion__panel" id="acc-panel-trigonometric-functions" data-accordion="panel" aria-labelledby="acc-button-trigonometric-functions" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/wave-mathematics/131">Wave Mathematics</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-physics" data-accordion="button" aria-controls="acc-panel-physics" aria-expanded="false"> <span class="accordion__button__label"> Physics </span> </button> <div class="accordion__panel" id="acc-panel-physics" data-accordion="panel" aria-labelledby="acc-button-physics" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-light-and-optics" data-accordion="button" aria-controls="acc-panel-light-and-optics" aria-expanded="false"> <span class="accordion__button__label"> Light and Optics </span> </button> <div class="accordion__panel" id="acc-panel-light-and-optics" data-accordion="panel" aria-labelledby="acc-button-light-and-optics" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/physics/24/the-nature-of-light/132">The Nature of Light</a></li> <li><a href="/en/library/physics/24/electromagnetism-and-light/138">Electromagnetism and Light</a></li> </ul> </div> <button 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id="acc-panel-process-of-science" data-accordion="panel" aria-labelledby="acc-button-process-of-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-introduction" data-accordion="button" aria-controls="acc-panel-introduction" aria-expanded="false"> <span class="accordion__button__label"> Introduction </span> </button> <div class="accordion__panel" id="acc-panel-introduction" data-accordion="panel" aria-labelledby="acc-button-introduction" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/the-process-of-science/176">The Process of Science</a></li> </ul> </div> <button class="accordion__button" id="acc-button-the-culture-of-science" data-accordion="button" aria-controls="acc-panel-the-culture-of-science" aria-expanded="false"> <span class="accordion__button__label"> The Culture of Science </span> </button> <div class="accordion__panel" id="acc-panel-the-culture-of-science" data-accordion="panel" aria-labelledby="acc-button-the-culture-of-science" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/the-nature-of-scientific-knowledge/185">The Nature of Scientific Knowledge</a></li> <li><a href="/en/library/process-of-science/49/scientists-and-the-scientific-community/172">Scientists and the Scientific Community</a></li> <li><a href="/en/library/process-of-science/49/scientific-ethics/161">Scientific Ethics</a></li> <li><a href="/en/library/process-of-science/49/scientific-institutions-and-societies/162">Scientific Institutions and Societies</a></li> </ul> </div> <button class="accordion__button" id="acc-button-ideas-in-science" data-accordion="button" aria-controls="acc-panel-ideas-in-science" aria-expanded="false"> <span class="accordion__button__label"> Ideas in Science </span> </button> <div class="accordion__panel" id="acc-panel-ideas-in-science" data-accordion="panel" aria-labelledby="acc-button-ideas-in-science" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/theories-hypotheses-and-laws/177">Theories, Hypotheses, and Laws</a></li> <li><a href="/en/library/process-of-science/49/scientific-controversy/181">Scientific Controversy</a></li> <li><a href="/en/library/process-of-science/49/creativity-in-science/182">Creativity in Science</a></li> </ul> </div> <button class="accordion__button" id="acc-button-research-methods" data-accordion="button" aria-controls="acc-panel-research-methods" aria-expanded="false"> <span class="accordion__button__label"> Research Methods </span> </button> <div class="accordion__panel" id="acc-panel-research-methods" data-accordion="panel" aria-labelledby="acc-button-research-methods" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/the-practice-of-science/148">The Practice of Science</a></li> <li><a href="/en/library/process-of-science/49/experimentation-in-scientific-research/150">Experimentation in Scientific Research</a></li> <li><a href="/en/library/process-of-science/49/description-in-scientific-research/151">Description in Scientific Research</a></li> <li><a href="/en/library/process-of-science/49/comparison-in-scientific-research/152">Comparison in Scientific Research</a></li> <li><a href="/en/library/process-of-science/49/modeling-in-scientific-research/153">Modeling in Scientific Research</a></li> </ul> </div> <button class="accordion__button" id="acc-button-data" data-accordion="button" aria-controls="acc-panel-data" aria-expanded="false"> <span class="accordion__button__label"> Data </span> </button> <div class="accordion__panel" id="acc-panel-data" data-accordion="panel" aria-labelledby="acc-button-data" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/data-analysis-and-interpretation/154">Data Analysis and Interpretation</a></li> <li><a href="/en/library/process-of-science/49/uncertainty-error-and-confidence/157">Uncertainty, Error, and Confidence</a></li> <li><a href="/en/library/process-of-science/49/statistics-in-science/155">Statistics in Science</a></li> <li><a href="/en/library/process-of-science/49/using-graphs-and-visual-data-in-science/156">Using Graphs and Visual Data in Science</a></li> </ul> </div> <button class="accordion__button" id="acc-button-scientific-communication" data-accordion="button" aria-controls="acc-panel-scientific-communication" aria-expanded="false"> <span class="accordion__button__label"> Scientific Communication </span> </button> <div class="accordion__panel" id="acc-panel-scientific-communication" data-accordion="panel" aria-labelledby="acc-button-scientific-communication" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/understanding-scientific-journals-and-articles/158">Understanding Scientific Journals and Articles</a></li> <li><a href="/en/library/process-of-science/49/utilizing-the-scientific-literature/173">Utilizing the Scientific Literature</a></li> <li><a href="/en/library/process-of-science/49/peer-review-in-scientific-publishing/159">Peer Review in Scientific Publishing</a></li> <li><a href="/en/library/process-of-science/49/the-how-and-why-of-scientific-meetings/186">The How and Why of Scientific Meetings</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-scientists-and-research" data-accordion="button" aria-controls="acc-panel-scientists-and-research" aria-expanded="false"> <span class="accordion__button__label"> Scientists and Research </span> </button> <div class="accordion__panel" id="acc-panel-scientists-and-research" data-accordion="panel" aria-labelledby="acc-button-scientists-and-research" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-scientific-research" data-accordion="button" aria-controls="acc-panel-scientific-research" aria-expanded="false"> <span class="accordion__button__label"> Scientific Research </span> </button> <div class="accordion__panel" id="acc-panel-scientific-research" data-accordion="panel" aria-labelledby="acc-button-scientific-research" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/scientists-and-research/58/collaborative-research-in-the-arctic-towards-understanding-climate-change/183">Collaborative Research in the Arctic Towards Understanding Climate Change</a></li> <li><a href="/en/library/scientists-and-research/58/from-stable-chromosomes-to-jumping-genes/184">From Stable Chromosomes to Jumping Genes</a></li> <li><a href="/en/library/scientists-and-research/58/an-elegant-experiment-to-test-the-process-of-dna-replication/187">An Elegant Experiment to Test the Process of DNA Replication</a></li> <li><a href="/en/library/scientists-and-research/58/the-founding-of-neuroscience/233">The Founding of Neuroscience</a></li> <li><a href="/en/library/scientists-and-research/58/tracking-endangered-jaguars-across-the-border/189">Tracking Endangered Jaguars across the Border</a></li> <li><a href="/en/library/scientists-and-research/58/atmospheric-chemistry-research-that-changed-global-policy/211">Atmospheric Chemistry Research that Changed Global Policy</a></li> <li><a href="/en/library/scientists-and-research/58/revolutionizing-medicine-with-monoclonal-antibodies/220">Revolutionizing Medicine with Monoclonal Antibodies</a></li> <li><a href="/en/library/scientists-and-research/58/uncovering-the-mysteries-of-chronic-mountain-sickness/238">Uncovering the Mysteries of Chronic Mountain Sickness</a></li> </ul> </div> <button class="accordion__button" id="acc-button-profiles-in-science" data-accordion="button" aria-controls="acc-panel-profiles-in-science" aria-expanded="false"> <span class="accordion__button__label"> Profiles in Science </span> </button> <div class="accordion__panel" id="acc-panel-profiles-in-science" data-accordion="panel" aria-labelledby="acc-button-profiles-in-science" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/scientists-and-research/58/luis-e.-miramontes/232">Luis E. Miramontes</a></li> <li><a href="/en/library/scientists-and-research/58/bernardo-houssay/237">Bernardo Houssay</a></li> <li><a href="/en/library/scientists-and-research/58/craig-lee/256">Craig Lee</a></li> <li><a href="/en/library/scientists-and-research/58/david-ho/241">David Ho</a></li> <li><a href="/en/library/scientists-and-research/58/louis-tompkins-wright/244">Louis Tompkins Wright</a></li> <li><a href="/en/library/scientists-and-research/58/carlos-j.-finlay/217">Carlos J. Finlay</a></li> <li><a href="/en/library/scientists-and-research/58/cecilia-payne/290">Cecilia Payne</a></li> <li><a href="/en/library/scientists-and-research/58/jazmin-scarlett/291">Jazmin Scarlett</a></li> <li><a href="/en/library/scientists-and-research/58/ramari-stewart/292">Ramari Stewart</a></li> <li><a href="/en/library/scientists-and-research/58/johnson-cerda/300">Johnson Cerda</a></li> <li><a href="/en/library/scientists-and-research/58/ellen-ochoa/201">Ellen Ochoa</a></li> <li><a href="/en/library/scientists-and-research/58/ruth-benerito/205">Ruth Benerito</a></li> <li><a href="/en/library/scientists-and-research/58/franklin-chang-díaz/219">Franklin Chang Díaz</a></li> <li><a href="/en/library/scientists-and-research/58/percy-lavon-julian/221">Percy Lavon Julian</a></li> <li><a href="/en/library/scientists-and-research/58/luis-walter-alvarez/229">Luis Walter Alvarez</a></li> <li><a href="/en/library/scientists-and-research/58/france-anne-dominic-córdova/230">France Anne-Dominic Córdova</a></li> </ul> </div> </div> </div> </div> </div> </li> <li>  <button class="button" data-toggle="dropdown">Math in Science </button> <div class="nav__dropdown box-shadow-1 padding-1"> <div class="accordion accordion--secondary font-size-sm"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-sub-button-equations" data-accordion="button" aria-controls="acc-sub-panel-equations" aria-expanded="false"> <span class="accordion__button__label"> Equations </span> </button> <div class="accordion__panel" id="acc-sub-panel-equations" data-accordion="panel" aria-labelledby="acc-sub-button-equations" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/unit-conversion/144">Unit Conversion</a></li> <li><a href="/en/library/math-in-science/62/linear-equations/194">Linear Equations</a></li> <li><a href="/en/library/math-in-science/62/exponential-equations-i/206">Exponential Equations I</a></li> <li><a href="/en/library/math-in-science/62/exponential-equations-ii/210">Exponential Equations II</a></li> <li class="current">Scientific Notation</li> <li><a href="/en/library/math-in-science/62/measurement/257">Measurement</a></li> </ul> </div> <button class="accordion__button" id="acc-sub-button-statistics" data-accordion="button" aria-controls="acc-sub-panel-statistics" aria-expanded="false"> <span class="accordion__button__label"> Statistics </span> </button> <div class="accordion__panel" id="acc-sub-panel-statistics" data-accordion="panel" aria-labelledby="acc-sub-button-statistics" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/introduction-to-descriptive-statistics/218">Introduction to Descriptive Statistics</a></li> <li><a href="/en/library/math-in-science/62/introduction-to-inferential-statistics/224">Introduction to Inferential Statistics</a></li> <li><a href="/en/library/math-in-science/62/statistical-techniques/239">Statistical Techniques</a></li> </ul> </div> <button class="accordion__button" id="acc-sub-button-trigonometric-functions" data-accordion="button" aria-controls="acc-sub-panel-trigonometric-functions" aria-expanded="false"> <span class="accordion__button__label"> Trigonometric Functions </span> </button> <div class="accordion__panel" id="acc-sub-panel-trigonometric-functions" data-accordion="panel" aria-labelledby="acc-sub-button-trigonometric-functions" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/wave-mathematics/131">Wave Mathematics</a></li> </ul> </div> </div> </div> </div> </li> </ul> </nav>  <div id="theTop"></div> <main id="skip-header-content"> <div class="margin-bottom-5"> <div class="container narrow wide--lg margin-y-4"> <article class="module"> <header class="module__header"> <span class="subcategory"> <strong><em>Equations</em></strong> </span> <h1>Scientific Notation: <sub><em>Working with orders of magnitude</em></sub></h1> <p class="byline">by Janet Shiver, Ph.D., Teri Willard, Ed.D.</p> </header> <nav class="module__tabs"> <ul class="tabs-nav tabs-nav--pill tabs-nav--horizontal--md library"> <li> <a href="/en/library/math-in-science/62/scientific-notation/250/reading" class="is-active" aria-current="page" >Reading</a> </li> <li> <a href="/en/library/math-in-science/62/scientific-notation/250/quiz" >Quiz</a> </li> <li> <a href="/en/library/math-in-science/62/scientific-notation/250/resources" >Teach with this</a> </li> </ul> </nav> <hr class="module__divider" />  <div class="module__tools"> <aside class="module__tools__container border-radius box-shadow-1"> <div class="tabs tabs--toggle-mobile--lg" role="tablist"> <ul class="tab__buttons"> <li> <button class="button button--icon-over-text" aria-label="In this module" aria-controls="tab-panel-module__tools" aria-selected="true" role="tab"> <span class="button__icon"> <span class="icon icon-list" aria-hidden="true"></span> </span> <span class="button__text">Contents</span> </button> </li> <li> <button class="button button--icon-over-text" aria-controls="tab-panel-toggle-terms" aria-selected="false" role="tab"> <span class="button__icon"> <span class="icon icon-glossary-highlight"></span> </span> <span class="button__text">Glossary Terms</span> </button> </li> </ul> <div class="tabs__panel shown" id="tab-panel-module__tools" aria-labelledby="tab-button-module__tools" role="tabpanel"> <div class="table-of-contents"> <p class="table-of-contents__title"> Table of Contents </p> <ul class="table-of-contents__nav"> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_1">Numbers and numeration systems</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_2">History of our numeration system</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_3">Scientific notation</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_4">How to write a number in scientific notation</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_5">Scientific notation and the decimal place</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_6">Orders of magnitude</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_7">Dividing numbers in scientific notation</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_8">Example: All creatures great and small</a> </li> <li><a href="/en/library/math-in-science/62/scientific-notation/250#toc_9">Conclusion</a> </li> </ul> </div> </div> <div class="tabs__panel" id="tab-panel-toggle-terms" aria-labelledby="tab-button-toggle-terms" role="tabpanel"> <div class="reading-toggle"> <div class="reading-toggle__switch"> <div class="form-entry__option__switch"> <label> <input type="checkbox" name="termsToggleSwitch" id="terms-toggle-switch" /> <span class="switch__slider"></span> <span class="option__label text-decoration-none font-size-md"> Highlight Glossary Terms </span> </label> </div> </div> <div class="reading-toggle__help"> <p> <em> Activate glossary term highlighting to easily identify key terms within the module. Once highlighted, you can click on these terms to view their definitions. </em> </p> </div> </div> <div class="glossary-container"></div> </div> <div class="tabs__panel" id="tab-panel-toggle-ngss" aria-labelledby="tab-button-toggle-ngss" role="tabpanel"> <div class="reading-toggle"> <div class="reading-toggle__switch"> <div class="form-entry__option__switch"> <label> <input type="checkbox" name="ngssToggleSwitch" id="ngss-toggle-switch" /> <span class="switch__slider"></span> <span class="option__label text-decoration-none font-size-md"> Show NGSS Annotations </span> </label> </div> </div> <div class="reading-toggle__help"> <p> <em> Activate NGSS annotations to easily identify NGSS standards within the module. Once highlighted, you can click on them to view these standards. </em> </p> </div> </div> <div class="ngss-container"></div> </div> </div> </aside> <div class="margin-3"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-9561344156007092" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-9561344156007092" data-ad-slot="7634263342" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </div>     <div class="module__main"> <div class="module__main__container"> <div class="accordion">  <button class="accordion__button" id="acc-button-key-concepts" data-accordion="button" aria-controls="acc-panel-key-concepts" aria-expanded="true" tabindex="0"> Did you know? </button> <div class="accordion__panel shown show" id="acc-panel-key-concepts" data-accordion="panel" aria-labelledby="acc-button-key-concepts" role="region"> <div class="accordion__panel__content"> <p>Did you know that ancient Egyptians needed 18 digits to write the number 99? We use only two digits to write the same number because our modern system of writing numbers uses place values, where each place represents an order of magnitude. Orders of magnitude are a handy way to describe the size of an object and compare the sizes of different items.</p> </div> </div>  <button class="accordion__button" id="acc-button-table-of-contents" data-accordion="button" aria-controls="acc-panel-table-of-contents" aria-expanded="false" tabindex="0"> Key concepts </button> <div class="accordion__panel" id="acc-panel-table-of-contents" data-accordion="panel" aria-labelledby="acc-button-table-of-contents" role="region" aria-hidden="true"> <div class="accordion__panel__content"> <ul class="bulleted"> <li><p>Scientists often deal with very large and very small measurements, and so they think in terms of order of magnitude to effectively express these measurements and differences between them.</p></li> <li><p>Orders of magnitude differences are embedded in our base-ten measurement system, where one order of magnitude represents a ten-fold difference.</p></li> <li><p>Scientific notation is used to make it easier to express extremely large or extremely small numbers, and is rooted in multiplying a number by some power of ten (10<sup>x</sup>).</p></li> <li><p>Expressing numbers in scientific (base ten) notation can often make it easier to perform simple mathematical operations on that number.</p></li> </ul> </div> </div>  <button class="accordion__button" id="acc-button-terms-you-should-know" data-accordion="button" aria-controls="acc-panel-terms-you-should-know" aria-expanded="false" tabindex="0"> Terms you should know </button> <div class="accordion__panel" id="acc-panel-terms-you-should-know" data-accordion="panel" aria-labelledby="acc-button-terms-you-should-know" role="region" aria-hidden="true"> <div class="accordion__panel__content"> <dl> <dt><a href="/en/glossary/view/base">base </a></dt> <dd> a number raised to a power. For example, in the expression 10<sup>3</sup>, 10 is the base. </dd> <dt><a href="/en/glossary/view/exponent">exponent </a></dt> <dd> the raised numeral written behind the base indicating the number of times the base is used as a factor. For example, the expression 10<sup>3</sup> means 10 x 10 x 10; the exponent of 10<sup>3</sup> is 3. </dd> <dt><a href="/en/glossary/view/scientific+notation">scientific notation </a></dt> <dd> a method for expressing numbers using powers of 10. A number written in scientific notation has two factors, a number that is at least one, but less than ten, and a power of ten. For example, the age of the Earth can be expressed at 4.543 x 10<sup>9</sup>. </dd> <dt><a href="/en/glossary/view/order+of+magnitude">order of magnitude </a></dt> <dd> a classification that is a multiple of ten, expressed in scientific notation.</dd> </dl> </div> </div> </div> <section> <p>Measurement is the basis of science. Scientists measure and evaluate measurements to study and explain phenomena. Humans naturally want to compare the sizes of items. Say we wanted to compare how strong we are to how strong another animal is. We would see that the average human can comfortably lift one times her weight, while a gorilla can lift ten times his weight, an ant 100 times, and a dung beetle an astounding 1000 times its weight (that is a lot of dung!). To make comparisons like these scientists often think in terms of orders of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark>, so they might say that a dung beetle is 100 times as strong as a human.</p> <p>Order of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> is a <mark class="term" data-term="method" data-term-def="A procedure or process; a systematic way of performing a task or conducting research." data-term-url="/en/glossary/view/method/8238">method</mark> for describing the size of measures in terms of powers of ten. An <mark class="term" data-term="order of magnitude" data-term-def="A description of relative size given by the number of powers of 10 that there are in a specified quantity written&hellip;" data-term-url="/en/glossary/view/order+of+magnitude/11228">order of magnitude</mark> estimates the approximate <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark> or size of something in base-ten <mark class="term" data-term="unit" data-term-def="An accepted quantity used as a standard of measurement. For example, the meter, liter, and gram." data-term-url="/en/glossary/view/unit/848">units</mark> such as, “in the millions.” For example, our national debt is “in the ten-trillions” of dollars which means it can be anywhere from 10 trillion to 99 trillion. Order of magnitude is an especially important concept in science. It allows scientists to express a number in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark> and roughly determine how much larger or smaller one object is compared to another. To understand order of magnitude, we must first understand our base-ten <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark> and scientific notation.</p> <p><section id="toc_1" class=""> <h2>Numbers and numeration systems</h2></p> <p>A number is an abstract concept of an amount or measure, for example, two or three. A numeral, however, is a symbol representing a number, such as the symbol “2” or “3.” The set of symbols and rules that govern how numbers are represented is called a numeration <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark>. Many early numeration systems were based upon the <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> that symbols represented, and to make larger numbers, symbols were written in order and their values were added together. For example, in the Egyptian numeration system, the symbol | represents the number one, and || represents “1 + 1,” or two. ||| represents three, and so on (see Figure 1 for an example). The problem with this system is that numbers became extremely long. This was partially addressed by giving new symbols to each new power of ten, for example, ten was denoted as ∩. But this didn’t completely solve the problem as 99 looks like this in Egyptian symbols:</p><p> <div class="math-container"> <div class="mathml"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∩</mo> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo>∣</mo> <mspace width="mediummathspace" /> <mo stretchy="false">∣</mo> </math> </div> </div> </p> <div class="container margin-y-4 text-align-center"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-9561344156007092" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:inline-block;width:300px;height:250px" data-ad-client="ca-pub-9561344156007092" data-ad-slot="9090201191"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10859.jpg" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10859-161010031027.jpg" alt="Figure 1: Numerical hieroglyphs at the Temple of Karnak in north Luxor, Egypt." /> </button> <figcaption> <p><strong>Figure 1</strong>: Numerical hieroglyphs at the Temple of Karnak in north Luxor, Egypt.</p> <span class="credit">image ©Olaf Tausch</span> </figcaption> </figure> </div> <p>In contrast to the Egyptian <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark>, our modern numeration system is called a place <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark> system. Ten unique symbols are used to indicate amounts between zero and nine, and the position of a particular symbol represents the value of a particular power of ten. For example, the numeral “1” represents different values based on its position within the numeral. The numeral “11” represents the number eleven, not the number two as it would in the Egyptian system, because there is a “1” in the ten's place and a “1” in the one's place. So ten plus one equals eleven. One characteristic our system has in common with the Egyptian system is that we still add the values in each position to get our final number. Let’s take a look at the numeral 2,576. If we expand this numeral, we can show the value of each digit based on its position and how we would determine its value. The numeral 2,576 can be written in expanded notation as:</p><div class="figure"><figure> $$(2 \times 1000) + (5 \times 100) + ( 7 \times 10) + (6 \times 1)$$ </figure></div><p>Notice that each of the numerals is multiplied by a power of ten. For example, five is multiplied by the number 100. Since 100 is a multiple of ten, it can be written as 10 x 10 or 10<sup>2</sup>. 1000 is also a multiple of ten and can be written as 10 x 10 x 10 or 10<sup>3</sup>. Using these powers of ten, we can rewrite the expanded notation to highlight that feature:</p><div class="figure"><figure> $$(2 \times 10^3) + (5 \times 10^2) + (7 \times 10^1) + (6 \times 10^0)$$ </figure></div><p>Writing it like this, you can see that the <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> denotes the position of the numeral. And the position of each numeral is important as it tells us the <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark>, or power of ten, to be included in the total sum for the number. Being able to express a number using powers of ten is important to understanding order of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark>, for example, take a look at the table below.</p><div class='\"figure\"'><table class="table" aria-describedby="configDescription"> <caption id="configDescription"> <strong>Table 1:</strong> Powers of ten. </caption> <thead> <tr> <th scope="col">Powers of ten</th> <th scope="col">Numeral</th> </tr> </thead> <tbody> <tr> <td scope="row">$10^5$</td> <td>100,000</td> </tr> <tr> <td scope="row">$10^4$</td> <td>10,000</td> </tr> <tr> <td scope="row">$10^3$</td> <td>1,000</td> </tr> <tr> <td scope="row">$10^2$</td> <td>100</td> </tr> <tr> <td scope="row">$10^1$</td> <td>10</td> </tr> <tr> <td scope="row">$10^0$</td> <td>1</td> </tr> </tbody> </table></div> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form name="cc10325"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">In the number 50,607,920, the numeral "7" represents</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-10325-0-option-a" name="quiz-option-10325" type="radio" value="7 x 10<sup>3</sup>" > <span class="option__label"> <span class="screen-reader-only">a.</span> 7 x 10<sup>3</sup> </span> </label> <span class="quiz__response" id="response-10325-0"> <strong>Correct!</strong> </span> </div> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-10325-1-option-b" name="quiz-option-10325" type="radio" value="7 x 10<sup>4</sup>" > <span class="option__label"> <span class="screen-reader-only">b.</span> 7 x 10<sup>4</sup> </span> </label> <span class="quiz__response" id="response-10325-1"> <strong>Incorrect.</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_2"> <h2>History of our numeration system</h2><p>Where did our numeration <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark> originate? Our base-ten numeration system began with symbols originating in India. These symbols traveled with Indian traders to the Arabic and Islamic people, and finally migrated to Europe. The symbols evolved over time as they were used by various civilizations. Abu’l Hasan Ahmad ibu Ibrahim al-Uqlidisi was the very first mathematician to use this type of system in a document dated 953. In this document, he discussed and wrote numbers in the base-ten place <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark> form similar to what we use today. Because our system was influenced by <mark class="term" data-term="development" data-term-def="The gradual exposure to stimuli in the early-developmental stages that influences the size, shape, and function of animal once mature." data-term-url="/en/glossary/view/development/13147">developments</mark> of the Indians and Arabs, it is called the Hindu-Arabic system. (See Figure 2 for an example of the <mark class="term" data-term="evolution" data-term-def="Change in the gene pool of a population from generation to generation by such processes as mutation, natural selection, and genetic drift." data-term-url="/en/glossary/view/evolution/5284">evolution</mark> of numerals.)</p>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10310.jpg" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10310-160813110844.jpg" alt="Figure 2: A selection of numerals from various cultures in history. You can see the progression of numerical symbols from Brahmi to Hindu to Arabic to 15th century European to our commonly recognized modern numbers." /> </button> <figcaption> <p><strong>Figure 2</strong>: A selection of numerals from various cultures in history. You can see the progression of numerical symbols from Brahmi to Hindu to Arabic to 15th century European to our commonly recognized modern numbers.</p> <span class="credit">image ©I.Taylor (Brahmi and 15th C European)</span> </figcaption> </figure> </div> </section> <section id="toc_3"> <h2>Scientific notation</h2><p>While our modern numeration <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark> makes it easier to write large numbers than the Egyptian system, writing very large and very small numbers can still be cumbersome and so scientists use <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark> as a way to write very large or very small numbers in a much more concise way that makes working with these numbers easier. For example, the Earth has a <mark class="term" data-term="surface" data-term-def="The outside or external part; the topside face of something." data-term-url="/en/glossary/view/surface/8275">surface</mark> area of 169,900,000 miles squared and the world’s population is approximately 7,403,000,000. What if you needed to divide these two numbers to find out how many people per square mile were on the earth? These giant numbers can be cumbersome to divide. Instead, we could express them in scientific notation as 1.699 x 10<sup>8</sup> and 7.403 x 10<sup>9</sup>. How did we convert these numbers and why is it easier to work with them in scientific notation? Let’s look at where scientific notation came from towards understanding its use.</p><div class='\"figure\"'><table class="table" aria-describedby="configDescription2"> <caption id="configDescription2"> <strong>Table 2:</strong> Standard versus scientific notation. </caption> <thead> <tr> <th scope="col"></th> <th scope="col">Earth's Surface Area</th> <th scope="col">World's Population</th> <th scope="col">People per Square Mile</th> </tr> </thead> <tbody> <tr> <td scope="row">Standard Number:</td> <td>169,900,000</td> <td>7,403,000,000</td> <td>$\frac{7,403,000,000}{169,900,000}$</td> </tr> <tr> <td scope="row">Scientific notation:</td> <td>$1.699 \times 10^8$</td> <td>$7.403 \times 10^9$</td> <td>$\frac{7.403 \times 10^9}{1.699 \times 10^8}$</td> </tr> </tbody> </table></div> <p>Scientific notation follows naturally from our base-ten <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark> as a shorthand way to write very large or very small numbers. <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">Scientific notation</mark> has two parts. Looking at an example, 1.6 x 10<sup>8</sup>, we can see that the first part is a decimal number greater than or equal to one, but less than ten (in this case, 1.6). The second part is a multiple of ten expressed using an <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> (here, 10<sup>8</sup>).</p><p>The notation we use today to denote an <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> was first used by Scottish mathematician, James Hume in 1636. However, he used Roman numerals for the exponents. Using Roman numerals as exponents became problematic since many of the exponents became very large so Hume’s notation didn’t last long. A year later in 1637, <mark class="term" data-term="Rene Descartes" data-term-def="Mathematician and philosopher born in Indre-et-Loire, France (1596-1650). Descartes invented analytical geometry and developed what is now called the Cartesian coordinate&hellip;" data-term-url="/en/glossary/view/Descartes%2C+Rene/4501">Rene Descartes</mark> became the first mathematician to use the Hindu-Arabic numerals of today as exponents. The exponent is used as a shorthand way to state how many times a number should be multiplied by itself, so 10<sup>3</sup> is equal to 10 x 10 x 10, and 2<sup>4</sup> is equal to 2 x 2 x 2 x 2.</p><p>In <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark> we also use a decimal numeral. The Flemish mathematician Simon Stevin (Figure 3) first used a decimal point to represent a fraction with a denominator of ten in 1585. While decimals had been used by both the Arabs and Chinese long before this time, Stevin is credited with popularizing their use in Europe. An English translation of Stevin’s work was published in 1608 and titled <em>Disme, The Arts of Tenths or Decimal Arithmetike</em>, and it inspired President Thomas Jefferson to propose a decimal-based currency for the United States. (For example, one tenth of a dollar is called a dime.)</p>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10860.jpg" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10860-161010031047.jpg" alt="Figure 3: Statue of Simon Stevin (1548-1620) in Bruges. Stevin was a Flemish mathematician and engineer, credited with introducing decimal fractions." /> </button> <figcaption> <p><strong>Figure 3</strong>: Statue of Simon Stevin (1548-1620) in Bruges. Stevin was a Flemish mathematician and engineer, credited with introducing decimal fractions.</p> <span class="credit">image ©Dennis Jarvis</span> </figcaption> </figure> </div> <p>While we can trace the history of the components (decimals and exponents) of <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>, it is difficult to determine who actually first used the term scientific notation. In fact, it wasn’t until 1961 that the term could be found in a dictionary indicating that it was being widely used. Even though it is difficult to pinpoint the exact origins of the phrase, it is often thought to have begun with computer scientists. In the 1940’s, Konrad Zuse introduced a concept that he referred to as the "floating point." Zuse’s floating point was a <mark class="term" data-term="method" data-term-def="A procedure or process; a systematic way of performing a task or conducting research." data-term-url="/en/glossary/view/method/8238">method</mark> of representing any number as a decimal greater than or equal to one but less than ten, times a number raised to a power. This notation made it easier to represent and conduct calculations on large and small numbers in the binary code used by computers, even given their limited computing power at the time. Over the next two decades, the term scientific notation often referred to a number expressed as a decimal (described above) times any second number raised to a power. For example, 2.45 x 2<sup>3</sup> would have been described as scientific notation in the 1960s or earlier. Today, we only use the term scientific notation when the second number is the numeral 10 raised to a power, such as in 2.45 x 10<sup>3</sup>.</p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form name="cc10326"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">The expression 16.7 x 2<sup>5</sup> is an example of modern scientific notation.</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-10326-0-option-a" name="quiz-option-10326" type="radio" value="True" > <span class="option__label"> <span class="screen-reader-only">a.</span> True </span> </label> <span class="quiz__response" id="response-10326-0"> <strong>Incorrect.</strong> </span> </div> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-10326-1-option-b" name="quiz-option-10326" type="radio" value="False" > <span class="option__label"> <span class="screen-reader-only">b.</span> False </span> </label> <span class="quiz__response" id="response-10326-1"> <strong>Correct!</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_4"> <h2>How to write a number in scientific notation</h2><p>Did you know that the Earth is 4,543,000,000 years old or that a carbon <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> weighs just 0.0000000000000000000000199265 grams? Those are really large and small numbers! If you had to write them several times, or worse yet, use them to calculate something, you may have a difficult time keeping up with all of those zeros. To help simplify these numbers and make them easier to work with, we can express them using <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>.</p><p>Consider the age of the Earth, 4,543,000,000 years. To rewrite 4,543,000,000 years in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark> we must express it as a decimal between one and ten. To do that we divide it by 1,000,000,000 giving us 4.543 years. But the age of the earth is not 4.543 years. So to express this number correctly, we must show that it has to be multiplied by 1,000,000,000. So we can write this as 4.543 times 1,000,000,000. But this can be shortened further because 1,000,000,000 is equal to 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 which can be expressed as 10<sup>9</sup>. So we can write the age of the earth in scientific notation as 4.543 x 10<sup>9</sup> years.</p><p>Table 3 shows how some small and large numbers are expressed in terms of powers of ten.</p><div class='\"figure\"'><table class="table" aria-describedby="configDescription3"> <caption id="configDescription3"> <strong>Table 3:</strong> Values for powers of 10. </caption> <thead> <tr> <th scope="col">Power of 10</th> <th scope="col">Expanding Meaning</th> <th scope="col">Equivalent Value</th> </tr> </thead> <tbody> <tr> <td scope="row">$10^{-5}$</td> <td>$\frac{1}{10^5} \, or \, \frac{1}{10 \times 10 \times 10 \times 10 \times 10} = \frac{1}{100000}$</td> <td>0.00001</td> </tr> <tr> <td scope="row">$10^{-4}$</td> <td>$\frac{1}{10^4} \, or \, \frac{1}{10 \times 10 \times 10 \times 10} = \frac{1}{10000}$</td> <td>0.0001</td> </tr> <tr> <td scope="row">$10^{-3}$</td> <td>$\frac{1}{10^3} \, or \, \frac{1}{10 \times 10 \times 10} = \frac{1}{1000}$</td> <td>0.001</td> </tr> <tr> <td scope="row">$10^{-2}$</td> <td>$\frac{1}{10^2} \, or \, \frac{1}{10 \times 10} = \frac{1}{100}$</td> <td>0.01</td> </tr> <tr> <td scope="row">$10^{-1}$</td> <td>$\frac{1}{10^1}$</td> <td>0.1</td> </tr> <tr> <td scope="row">$10^{0}$</td> <td>$1$</td> <td>1</td> </tr> <tr> <td scope="row">$10^{1}$</td> <td>$10$</td> <td>10</td> </tr> <tr> <td scope="row">$10^{2}$</td> <td>$10 \times 10$</td> <td>100</td> </tr> <tr> <td scope="row">$10^{3}$</td> <td>$10 \times 10 \times 10$</td> <td>1,000</td> </tr> <tr> <td scope="row">$10^{4}$</td> <td>$10 \times 10 \times 10 \times 10$</td> <td>10,000</td> </tr> <tr> <td scope="row">$10^{5}$</td> <td>$10 \times 10 \times 10 \times 10 \times 10$</td> <td>100,000</td> </tr> </tbody> </table></div> <p>As you can see from the table, when the <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> on the number ten is a negative number, we can think of this as multiplying by 1/10. For example, 10<sup>-4</sup> is the same as multiplying by the following:</p><div class="figure"><figure> $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} = \frac{1}{10000}$$ </figure></div><p>This number can be written in different ways, such as:</p><div class="figure"><figure> $$\frac{1}{10000} = \frac{1}{10^4} = 10^{-4} = .0001$$ </figure></div><p>The takeaway is that negative <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponents</mark> are used to express very small numbers in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>.</p><p>Let’s think about the weight of our carbon <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark>, 0.0000000000000000000000199265 grams. To express the weight of the carbon atom as a decimal number greater than or equal to one and less than ten, we would have to multiply by 100,000,000,000,000,000,000,000, or 10<sup>23</sup>. This would give us a decimal number of 1.99265, but we then have to add in the factor to show how the number is truly expressed. To reverse this <mark class="term" data-term="process" data-term-def="Method, procedure; series of actions or steps." data-term-url="/en/glossary/view/process/8256">process</mark> and return the atom to its original weight we would have to divide by 10<sup>23</sup>. So the original weight of a carbon atom can be written as:</p><div class="figure"><figure> $$\frac{1.99265}{10^{23}}$$ </figure></div><p>This representation is very close to <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>, but scientific notation is always written using multiplication, not division. So we can rewrite it as:</p> <div class="container margin-y-4 text-align-center"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-9561344156007092" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:inline-block;width:300px;height:250px" data-ad-client="ca-pub-9561344156007092" data-ad-slot="3321739899"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> <div class="figure"><figure> $$1.99265 \times \frac{1}{10^{23}}$$ </figure></div><p>Finally, we express the power of ten with a negative <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> and place it in the numerator:</p><div class="figure"><figure> $$1.99265 \times 10^{-23} \, grams$$ </figure></div><p>We now have the weight of an <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> written correctly in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>.</p></section> <section id="toc_5"> <h2>Scientific notation and the decimal place</h2><p>A faster <mark class="term" data-term="method" data-term-def="A procedure or process; a systematic way of performing a task or conducting research." data-term-url="/en/glossary/view/method/8238">method</mark> for rewriting a number in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark> is to think of how many times the decimal point would have to be moved. For example, in our age of the Earth example, to express 4,543,000,000 as a number greater than one and less than ten we can simply move the decimal.</p>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10313.jpg" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10313-160813020802.jpg" alt=" Scientific notation 1 " /> </button> </figure> </div> <p>In this case, we moved our decimal nine places to the left to get the number 4.543. Each shift of the decimal represents division of the number by ten. To return the number back to its original form we must do the opposite, multiply by nine 10s. Therefore, to express the number in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>, we would use a positive nine <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark>, thus giving us the same answer as earlier: 4.543 x 10<sup>9</sup>. To convert a number with a positive exponent back, you would move the decimal to the right the same number of places as indicated by the exponent. For example, 3.79 x 10<sup>5</sup>:</p>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10314.jpg" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10314-160813020804.jpg" alt=" Scientific notation 2 " /> </button> </figure> </div> <p>We can use a similar but opposite <mark class="term" data-term="process" data-term-def="Method, procedure; series of actions or steps." data-term-url="/en/glossary/view/process/8256">process</mark> to write very small numbers in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>. Let’s return to the carbon <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> that weighed 0.0000000000000000000000199265 grams. To begin the process, we must move our decimal 23 times to the right.</p>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10841.png" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10841-161010121003.png" alt=" Scientific notation 3_rev " /> </button> </figure> </div> <p>Since we are moving to the right we are multiplying 0.0000000000000000000000199265 by 10 twenty-three times. To return the number back to its original form we must reverse the <mark class="term" data-term="process" data-term-def="Method, procedure; series of actions or steps." data-term-url="/en/glossary/view/process/8256">process</mark> and divide by 10 twenty-three times. Since multiplying by a negative <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> is the same as dividing a number, the original weight of a carbon <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> can be written as 1.99265 x 10<sup>-23</sup> grams.</p><p>Notice that when you moved the decimal to the left you made the number you started with smaller. To return it to its original size you had to multiply by a bunch of tens. This means that the <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponent</mark> on the ten had to be positive. When you moved the decimal to the right, the opposite happened. You made the original number much bigger, so to return it to its original tiny size you had to divide by a bunch of tens, making the exponent on ten negative.</p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form name="cc10327"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">The number 0.0036 written in scientific notation is</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-10327-0-option-a" name="quiz-option-10327" type="radio" value="3.6 x 10<sup>3</sup>" > <span class="option__label"> <span class="screen-reader-only">a.</span> 3.6 x 10<sup>3</sup> </span> </label> <span class="quiz__response" id="response-10327-0"> <strong>Incorrect.</strong> </span> </div> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-10327-1-option-b" name="quiz-option-10327" type="radio" value="3.6 x 10<sup>-3</sup>" > <span class="option__label"> <span class="screen-reader-only">b.</span> 3.6 x 10<sup>-3</sup> </span> </label> <span class="quiz__response" id="response-10327-1"> <strong>Correct!</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_6"> <h2>Orders of magnitude</h2><p>Scientists describe the <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> or size of numbers using something called the <mark class="term" data-term="order of magnitude" data-term-def="A description of relative size given by the number of powers of 10 that there are in a specified quantity written&hellip;" data-term-url="/en/glossary/view/order+of+magnitude/11228">order of magnitude</mark>. We can think of order of magnitude to mean the power of ten closest to a given quantity. For example, we often hear there are a billion people in China. This is not the exact number of people in China. It is just an order of magnitude approximation. Since 1,000,000,000 can be written as 10<sup>9</sup> we can say that the population of China has an order of magnitude of nine. We can find an order of magnitude by simply writing a number in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>.</p></section> <section id="toc_7"> <h2>Dividing numbers in scientific notation</h2><p>Scientists often use order of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> to compare the sizes or distances of items. Orders of magnitude give us a quick <mark class="term" data-term="method" data-term-def="A procedure or process; a systematic way of performing a task or conducting research." data-term-url="/en/glossary/view/method/8238">method</mark> for determining the relationship between two quantities. But to do this, we need to be able to divide numbers written in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>. For example, the distance from the Earth to the Sun is 93,000,000 miles = 9.3 x 10<sup>7</sup> miles. However, the distance to the next nearest star, Proxima Centauri, is about 2.522 x 10<sup>13</sup> miles. How many times greater is the distance from Proxima Centauri to Earth than the distance from the Earth to the Sun?</p><p>To solve this problem, we need to divide the two distances.</p><div class="figure"><figure> $$\frac{2.522 \times 10^{13} miles}{9.3 \times 10^7 miles} = \frac{2.522}{9.3} \times \frac{10^{13}}{10^7} \approx 0.2712 \times 10^6$$ </figure></div><p>Notice that when you divide powers of ten, the net result is that the <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponents</mark> get subtracted from one another:</p><div class="figure"><figure> $$10^{13} \div 10^7 = 10^{13 - 7} = 10^6$$ </figure></div><p>Similarly, when you multiply powers of ten the net result is to add the <mark class="term" data-term="exponent" data-term-def="A number or expression written superscript to another number or expression, called the base, and indicating the power to which the&hellip;" data-term-url="/en/glossary/view/exponent/8759">exponents</mark> together. Now back to our example. Since the exponent on ten is six, it is tempting to say that the two numbers differ by a <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> of six, but this is not correct. The convention is to report the answer in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark>. The answer 0.2712 x 10<sup>6</sup> is not in correct form, since 0.2712 is not expressed as a number at least one but less than ten. To convert this into scientific notation, you need to move the decimal one place to the right, which is the same as multiplying the number by ten. To compensate for this you would have to reduce the exponent by one digit, which is equivalent to dividing by ten, and arrive at the same answer 2.712 x 10<sup>5</sup>. We can now correctly report that the two distances differ by a magnitude of five. Since 1 x 10<sup>5</sup> is equal to 10,000, we can say that Proxima Centauri is 10,000 times further from Earth than the Sun.</p>  <div class="figure"> <figure> <button class="lightbox-button" data-lightbox-src="/img/library/large_images/image_10861.jpg" data-lightbox="image"> <img src="/img/library/modules/mid250/Image/VLObject-10861-161010041017.jpg" alt="Figure 4: This diagram illustrates the locations of the star systems closest to the sun. Proxima Centauri is shown just after the 4 light year ring." /> </button> <figcaption> <p><strong>Figure 4</strong>: This diagram illustrates the locations of the star systems closest to the sun. Proxima Centauri is shown just after the 4 light year ring.</p> <span class="credit">image ©NASA/Penn State University</span> </figcaption> </figure> </div> <p>We see order of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> comparisons often arising in chemistry when discussing the <mark class="term" data-term="concentration" data-term-def="The amount of one substance in relation to other components within a given area." data-term-url="/en/glossary/view/concentration/8733">concentrations</mark> of substances. For example, a scientist might say the concentration of arsenic in the hundreds of parts per million (ppm) range. This means, that for every one million “parts” of water you would find 0.01 parts of arsenic. For example, you could say that for every 1kg of water, you could find approximately 0.01mg of arsenic (1kg = 1,000,000mg). Let’s write the comparison as a <mark class="term" data-term="ratio" data-term-def="The relationship between two or more quantities; relative amounts of two or more values expressed as a proportion." data-term-url="/en/glossary/view/ratio/8556">ratio</mark>.</p><div class="figure"><figure> $$\frac{0.01 \, mg}{1 \, kg} = \frac{0.01 \, mg}{1000000 \, mg}$$ </figure></div><p>Now let’s express the numerator and denominator in scientific notation:</p><div class="figure"><figure> $$\frac{0.01 \, mg}{1000000 \, mg} = \frac{1 \times 10^{-2} \, mg}{1 \times 10^6 \, mg}$$ </figure></div><p>Dividing we get:</p><div class="figure"><figure> $$\frac{1 \times 10^{-2} mg}{1 \times 10^6 \, mg} = \frac{1}{1} \times \frac{10^{-2}}{10^6} = 1 \times 10^{-2 - 6} = 1 \times 10^{-8}$$ </figure></div><p>We can now say that the amount of arsenic is eight orders of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> less than the amount of water.</p></section> <section id="toc_8"> <h2>Example: All creatures great and small</h2><p>Let’s return to the blue whale and plankton mentioned at the beginning of this module. The blue whale is the largest creature ever to inhabit the planet and yet it shares the ocean with one of the smallest, plankton, a creature that barely represents a tiny speck in the ocean. An average blue whale weighs approximately 190,000 kilograms while a single plankton weighs a mere 0.5 milligrams. To express these weights in the same <mark class="term" data-term="unit" data-term-def="An accepted quantity used as a standard of measurement. For example, the meter, liter, and gram." data-term-url="/en/glossary/view/unit/848">units</mark> (if you need help with metric conversions, refer to our <a href="/library/module_viewer.php?mid=47">Metric System</a> module) we can say that a Blue Whale weighs 190,000,000,000mg or 1.9 x 10<sup>11</sup>mg and a Plankton weighs 0.5mg or 5 x 10<sup>-1</sup>mg.</p><p>Comparing these two ocean dwellers we get:</p><div class="figure"><figure> $$\frac{1.9 \times 10^{11}}{5 \times 10^{-1}} = 0.38 \times (10^{11} \times 10^1)$$ $$ = 0.38 \times 10^{12}$$ $$ = 3.8 \times 10^{-1} \times 10^{12}$$ $$ = 3.8 \times 10^{11}$$ </figure></div><p>Thus the whale is 10<sup>11</sup>, or 100,000,000,000 times larger than the plankton!</p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form name="cc10328"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">When you compare 4.52 x 10<sup>7</sup> to 3.60 x 10<sup>-3</sup>,</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-10328-0-option-a" name="quiz-option-10328" type="radio" value="4.52 x 10<sup>7</sup> is 10 orders of magnitude larger than 3.60 x 10<sup>-3</sup>." > <span class="option__label"> <span class="screen-reader-only">a.</span> 4.52 x 10<sup>7</sup> is 10 orders of magnitude larger than 3.60 x 10<sup>-3</sup>. </span> </label> <span class="quiz__response" id="response-10328-0"> <strong>Correct!</strong> </span> </div> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-10328-1-option-b" name="quiz-option-10328" type="radio" value="4.52 x 10<sup>7</sup> is 4 orders of magnitude larger than 3.60 x 10<sup>-3</sup>." > <span class="option__label"> <span class="screen-reader-only">b.</span> 4.52 x 10<sup>7</sup> is 4 orders of magnitude larger than 3.60 x 10<sup>-3</sup>. </span> </label> <span class="quiz__response" id="response-10328-1"> <strong>Incorrect.</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_9"> <h2>Conclusion</h2><p>Scientific notation and order of <mark class="term" data-term="magnitude" data-term-def="Magnitude is a number assigned to a quantity and refers to the size or extent of something. There are several differing&hellip;" data-term-url="/en/glossary/view/magnitude/11226">magnitude</mark> are fundamental concepts in all branches of science. <mark class="term" data-term="order of magnitude" data-term-def="A description of relative size given by the number of powers of 10 that there are in a specified quantity written&hellip;" data-term-url="/en/glossary/view/order+of+magnitude/11228">Order of magnitude</mark> allows us to quickly estimate the size of something or the difference in measure between two things by expressing it as a power of ten. With practice, expressing numbers in <mark class="term" data-term="scientific notation" data-term-def="A method of expressing numbers in terms of a decimal number greater than or equal to 1 and less than 10&hellip;" data-term-url="/en/glossary/view/scientific+notation/11233">scientific notation</mark> provides a quick and easy way to compare, multiply and divide numbers. These concepts are especially useful when comparing very large and very small measurements such as the weight of an <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark>, the distance to a star, or the length of a tiny, <mark class="term" data-term="microscopic" data-term-def="That which cannot be seen with the unaided eye and requires a tool (such as a microscope) to view. Many scientific&hellip;" data-term-url="/en/glossary/view/microscopic/10590">microscopic</mark> ocean dweller.</p></section> <footer class="module__main__footer"> <hr class="border-color-dark"> <p class="citation"> <em> Janet Shiver, Ph.D., Teri Willard, Ed.D. “Scientific Notation” Visionlearning Vol. MAT-3 (7), 2016. </em> </p>  <div class="title-list" id="refs" name="refs"> <p class="h6 title-list__title"> References </p> <ul class="title-list__list"> <li>Schombert, J. (n.d.). History of Astronomy. Retrieved from: <a href="http://abyss.uoregon.edu/~js/ast121/lectures/lec02.html">http://abyss.uoregon.edu/~js/ast121/lectures/lec02.html</a></li> </ul> </div>  <div class="title-list" name="further"> <p class="h6 title-list__title"> Further Reading </p> <ul class="grid grid--column-2--md grid--column-3--md gap-1"> <li> <a class="no-hover-focus height-100" href="/en/library/General-Science/3/The-Metric-System/47"> <article class="flex-row align-items-center flex-column--md align-items-start--md height-100 theme-light padding-2 gap-2"> <div class="width-30 width-auto--md"> <img class="border-radius box-shadow-1" src="/img/library/moduleImages/featured_image_47-23061210060326.jpg" alt="The Metric System"> </div> <div class="flex-grow-shrink"> <h2 class="h6 font-weight-normal"> The Metric System: <em>Metric and scientific notation</em> </h2> </div> </article> </a> </li> <li> <a class="no-hover-focus height-100" href="/en/library/Math-in-Science/62/Unit-Conversion/144"> <article class="flex-row align-items-center flex-column--md align-items-start--md height-100 theme-light padding-2 gap-2"> <div class="width-30 width-auto--md"> <img class="border-radius box-shadow-1" src="/img/library/moduleImages/featured_image_144-23061210060537.jpeg" alt="Unit Conversion"> </div> <div class="flex-grow-shrink"> <h2 class="h6 font-weight-normal"> Unit Conversion: <em>Dimensional analysis</em> </h2> </div> </article> </a> </li> </ul> </div> </footer> </div>    </article> </div> </div> </main>   <footer class="position-relative box-shadow-1 font-size-md" id="global-footer"> <h2 class="screen-reader-only">Page Footer</h2> <div class="back-to-top"> <div class="container wide"> <button class="button button--has-icon font-size-sm"> <span class="icon icon-arrow-up"></span> <span class="button__text">Back to top</span> </button> </div> </div> <div class="container wide padding-y-2"> <div class="grid grid--column-2--md grid--column-4--lg gap-4 grid--divider--fill-x"> <nav> <ul class="nav font-weight-bold"> <li> <a href="/en/library" title="Readings & quizzes"> Library </a> </li> <li> <a href="/en/glossary" title="Science terms"> Glossary </a> </li> <li> <a href="/en/classroom" title="Courses & bookmarks"> Classroom </a> </li> </ul> </nav> <nav> <ul class="nav"> <li><a href="/en/about">About</a></li> <li><a href="/en/help">Contact</a></li> <li><a href="/en/about/jobs">Jobs</a></li> <li><a href="/en/help/faq">FAQ</a></li> </ul> </nav> <div> <ul class="nav nav--horizontal margin-bottom-2"> <li> <a class="display-flex" href="https://www.nsf.gov" target="_blank" rel="noopener"> <img src="/images/sponsor-nsf.png" width="60" height="60" alt="US Education Department Logo" /> </a> </li> <li> <a class="display-flex" href="https://www.ed.gov/" target="_blank" rel="noopener"> <img src="/images/sponsor-doe.png" width="60" height="60" alt="US Education Department Logo" /> </a> </li> </ul> <p>Visionlearning is supported by the The National Science Foundation and the U.S. Department of Education. 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