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can be contacted" class="form-control mr-2" type="checkbox"> <label for="include_email" style="font-weight: normal">Include my email address so I can be contacted</label> </form></div> </scrollable-region> <div data-view-component="true" class="Overlay-footer Overlay-footer--alignEnd"> <button data-close-dialog-id="feedback-dialog" type="button" data-view-component="true" class="btn"> Cancel </button> <button form="code-search-feedback-form" data-action="click:qbsearch-input#submitFeedback" type="submit" data-view-component="true" class="btn-primary btn"> Submit feedback </button> </div> </dialog></dialog-helper> <custom-scopes data-target="qbsearch-input.customScopesManager"> <dialog-helper> <dialog data-target="custom-scopes.customScopesModalDialog" data-action="close:qbsearch-input#handleDialogClose cancel:qbsearch-input#handleDialogClose" id="custom-scopes-dialog" aria-modal="true" aria-labelledby="custom-scopes-dialog-title" aria-describedby="custom-scopes-dialog-description" 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KB","deleteTooltip":"You must be signed in to make or propose changes","editTooltip":"You must be signed in to make or propose changes","ghDesktopPath":"https://desktop.github.com","isGitLfs":false,"onBranch":true,"shortPath":"41f0f4d","siteNavLoginPath":"/login?return_to=https%3A%2F%2Fgithub.com%2Fpython%2Fpeps%2Fblob%2Fmain%2Fpeps%2Fpep-0228.rst","isCSV":false,"isRichtext":true,"toc":[{"level":2,"text":"Withdrawal","anchor":"withdrawal","htmlText":"Withdrawal"},{"level":2,"text":"Abstract","anchor":"abstract","htmlText":"Abstract"},{"level":2,"text":"Rationale","anchor":"rationale","htmlText":"Rationale"},{"level":2,"text":"Other Numerical Models","anchor":"other-numerical-models","htmlText":"Other Numerical Models"},{"level":2,"text":"Suggested Interface For Python's Numerical Model","anchor":"suggested-interface-for-pythons-numerical-model","htmlText":"Suggested Interface For Python's Numerical Model"},{"level":2,"text":"Coercion","anchor":"coercion","htmlText":"Coercion"},{"level":2,"text":"Inexact Operations","anchor":"inexact-operations","htmlText":"Inexact Operations"},{"level":2,"text":"Numerical Python Issues","anchor":"numerical-python-issues","htmlText":"Numerical Python Issues"},{"level":2,"text":"Unresolved Issues","anchor":"unresolved-issues","htmlText":"Unresolved Issues"},{"level":2,"text":"Copyright","anchor":"copyright","htmlText":"Copyright"}],"lineInfo":{"truncatedLoc":"145","truncatedSloc":"102"},"mode":"file"},"image":false,"isCodeownersFile":null,"isPlain":false,"isValidLegacyIssueTemplate":false,"issueTemplate":null,"discussionTemplate":null,"language":"reStructuredText","languageID":419,"large":false,"planSupportInfo":{"repoIsFork":null,"repoOwnedByCurrentUser":null,"requestFullPath":"/python/peps/blob/main/peps/pep-0228.rst","showFreeOrgGatedFeatureMessage":null,"showPlanSupportBanner":null,"upgradeDataAttributes":null,"upgradePath":null},"publishBannersInfo":{"dismissActionNoticePath":"/settings/dismiss-notice/publish_action_from_dockerfile","releasePath":"/python/peps/releases/new?marketplace=true","showPublishActionBanner":false},"rawBlobUrl":"https://github.com/python/peps/raw/refs/heads/main/peps/pep-0228.rst","renderImageOrRaw":false,"richText":"\u003carticle class=\"markdown-body entry-content container-lg\" itemprop=\"text\"\u003e\u003cp dir=\"auto\"\u003ePEP: 228\nTitle: Reworking Python's Numeric Model\nAuthor: Moshe Zadka \u0026lt;\u003ca href=\"mailto:moshez@zadka.site.co.il\"\u003emoshez@zadka.site.co.il\u003c/a\u003e\u0026gt;, Guido van Rossum \u0026lt;\u003ca href=\"mailto:guido@python.org\"\u003eguido@python.org\u003c/a\u003e\u0026gt;\nStatus: Withdrawn\nType: Standards Track\nCreated: 04-Nov-2000\nPost-History:\u003c/p\u003e\n\u003ca name=\"user-content-withdrawal\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eWithdrawal\u003c/h2\u003e\u003ca id=\"user-content-withdrawal\" class=\"anchor\" aria-label=\"Permalink: Withdrawal\" href=\"#withdrawal\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis PEP has been withdrawn in favor of \u003ca href=\"http://www.python.org/dev/peps/pep-3141\" rel=\"nofollow\"\u003ePEP 3141\u003c/a\u003e.\u003c/p\u003e\n\u003ca name=\"user-content-abstract\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eAbstract\u003c/h2\u003e\u003ca id=\"user-content-abstract\" class=\"anchor\" aria-label=\"Permalink: Abstract\" href=\"#abstract\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eToday, Python's numerical model is similar to the C numeric model:\nthere are several unrelated numerical types, and when operations\nbetween numerical types are requested, coercions happen. While\nthe C rationale for the numerical model is that it is very similar\nto what happens at the hardware level, that rationale does not\napply to Python. So, while it is acceptable to C programmers that\n\u003ccode\u003e2/3 == 0\u003c/code\u003e, it is surprising to many Python programmers.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNOTE: in the light of recent discussions in the newsgroup, the\nmotivation in this PEP (and details) need to be extended.\u003c/p\u003e\n\u003ca name=\"user-content-rationale\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eRationale\u003c/h2\u003e\u003ca id=\"user-content-rationale\" class=\"anchor\" aria-label=\"Permalink: Rationale\" href=\"#rationale\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eIn usability studies, one of the least usable aspect of Python was\nthe fact that integer division returns the floor of the division.\nThis makes it hard to program correctly, requiring casts to\n\u003ccode\u003efloat()\u003c/code\u003e in various parts through the code. Python's numerical\nmodel stems from C, while a model that might be easier to work with\ncan be based on the mathematical understanding of numbers.\u003c/p\u003e\n\u003ca name=\"user-content-other-numerical-models\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eOther Numerical Models\u003c/h2\u003e\u003ca id=\"user-content-other-numerical-models\" class=\"anchor\" aria-label=\"Permalink: Other Numerical Models\" href=\"#other-numerical-models\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003ePerl's numerical model is that there is one type of numbers --\nfloating point numbers. While it is consistent and superficially\nnon-surprising, it tends to have subtle gotchas. One of these is\nthat printing numbers is very tricky, and requires correct\nrounding. In Perl, there is also a mode where all numbers are\nintegers. This mode also has its share of problems, which arise\nfrom the fact that there is not even an approximate way of\ndividing numbers and getting meaningful answers.\u003c/p\u003e\n\u003ca name=\"user-content-suggested-interface-for-python-s-numerical-model\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eSuggested Interface For Python's Numerical Model\u003c/h2\u003e\u003ca id=\"user-content-suggested-interface-for-pythons-numerical-model\" class=\"anchor\" aria-label=\"Permalink: Suggested Interface For Python's Numerical Model\" href=\"#suggested-interface-for-pythons-numerical-model\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWhile coercion rules will remain for add-on types and classes, the\nbuilt in type system will have exactly one Python type -- a\nnumber. There are several things which can be considered \"number\nmethods\":\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003e\u003ccode\u003eisnatural()\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eisintegral()\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eisrational()\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eisreal()\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eiscomplex()\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eisexact()\u003c/code\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp dir=\"auto\"\u003eObviously, a number which answers true to a question from 1 to 5, will\nalso answer true to any following question. If \u003ccode\u003eisexact()\u003c/code\u003e is not true,\nthen any answer might be wrong.\n(But not horribly wrong: it's close to the truth.)\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow, there is two thing the models promises for the field operations\n(\u003ccode\u003e+\u003c/code\u003e, \u003ccode\u003e-\u003c/code\u003e, \u003ccode\u003e/\u003c/code\u003e, \u003ccode\u003e*\u003c/code\u003e):\u003c/p\u003e\n\u003cul dir=\"auto\"\u003e\n\u003cli\u003eIf both operands satisfy \u003ccode\u003eisexact()\u003c/code\u003e, the result satisfies\n\u003ccode\u003eisexact()\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eAll field rules are true, except that for not-\u003ccode\u003eisexact()\u003c/code\u003e numbers,\nthey might be only approximately true.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp dir=\"auto\"\u003eOne consequence of these two rules is that all exact calculations\nare done as (complex) rationals: since the field laws must hold,\nthen\u003c/p\u003e\n\u003cpre\u003e(a/b)*b == a\n\u003c/pre\u003e\n\u003cp dir=\"auto\"\u003emust hold.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThere is built-in function, \u003ccode\u003einexact()\u003c/code\u003e which takes a number\nand returns an inexact number which is a good approximation.\nInexact numbers must be as least as accurate as if they were\nusing IEEE-754.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eSeveral of the classical Python functions will return exact numbers\neven when given inexact numbers: e.g, \u003ccode\u003eint()\u003c/code\u003e.\u003c/p\u003e\n\u003ca name=\"user-content-coercion\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eCoercion\u003c/h2\u003e\u003ca id=\"user-content-coercion\" class=\"anchor\" aria-label=\"Permalink: Coercion\" href=\"#coercion\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe number type does not define \u003ccode\u003enb_coerce\u003c/code\u003e\nAny numeric operation slot, when receiving something other then \u003ccode\u003ePyNumber\u003c/code\u003e,\nrefuses to implement it.\u003c/p\u003e\n\u003ca name=\"user-content-inexact-operations\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eInexact Operations\u003c/h2\u003e\u003ca id=\"user-content-inexact-operations\" class=\"anchor\" aria-label=\"Permalink: Inexact Operations\" href=\"#inexact-operations\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe functions in the \u003ccode\u003emath\u003c/code\u003e module will be allowed to return\ninexact results for exact values. However, they will never return\na non-real number. The functions in the \u003ccode\u003ecmath\u003c/code\u003e module are also\nallowed to return an inexact result for an exact argument, and are\nfurthermore allowed to return a complex result for a real\nargument.\u003c/p\u003e\n\u003ca name=\"user-content-numerical-python-issues\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eNumerical Python Issues\u003c/h2\u003e\u003ca id=\"user-content-numerical-python-issues\" class=\"anchor\" aria-label=\"Permalink: Numerical Python Issues\" href=\"#numerical-python-issues\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003ePeople who use Numerical Python do so for high-performance vector\noperations. Therefore, NumPy should keep its hardware based\nnumeric model.\u003c/p\u003e\n\u003ca name=\"user-content-unresolved-issues\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eUnresolved Issues\u003c/h2\u003e\u003ca id=\"user-content-unresolved-issues\" class=\"anchor\" aria-label=\"Permalink: Unresolved Issues\" href=\"#unresolved-issues\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eWhich number literals will be exact, and which inexact?\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eHow do we deal with IEEE 754 operations? (probably, isnan/isinf should\nbe methods)\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eOn 64-bit machines, comparisons between ints and floats may be\nbroken when the comparison involves conversion to float. Ditto\nfor comparisons between longs and floats. This can be dealt with\nby avoiding the conversion to float. (Due to Andrew Koenig.)\u003c/p\u003e\n\u003ca name=\"user-content-copyright\"\u003e\u003c/a\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eCopyright\u003c/h2\u003e\u003ca id=\"user-content-copyright\" class=\"anchor\" aria-label=\"Permalink: Copyright\" href=\"#copyright\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis document has been placed in the public domain.\u003c/p\u003e\n\n\u003c/article\u003e","renderedFileInfo":null,"shortPath":null,"symbolsEnabled":true,"tabSize":8,"topBannersInfo":{"overridingGlobalFundingFile":false,"globalPreferredFundingPath":"/python/.github/blob/5a0a7cb55767fb3667fe2391bcb66bda638edcb4/FUNDING.yml","showInvalidCitationWarning":false,"citationHelpUrl":"https://docs.github.com/github/creating-cloning-and-archiving-repositories/creating-a-repository-on-github/about-citation-files","actionsOnboardingTip":null},"truncated":false,"viewable":true,"workflowRedirectUrl":null,"symbols":{"timed_out":false,"not_analyzed":true,"symbols":[]}},"copilotInfo":null,"copilotAccessAllowed":false,"modelsAccessAllowed":false,"csrf_tokens":{"/python/peps/branches":{"post":"ICKMApIwIMNrUKtTO2Qq6ucdBhk6NzHiAEO0IyrfhJ45TM1F1lJT-F6XEEm2xgRrcsongGzXQii90xROTnaQwA"},"/repos/preferences":{"post":"Lch-9IsJJXUnFOKYB8Nixy9HmT0pyTlvZNMebaz_J26sgSJAFgCneqYGe9vD9KRvedFuy_yoYuoLlHak0bGI-A"}}},"title":"peps/peps/pep-0228.rst at main · 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Python's Numeric Model Author: Moshe Zadka <<a href="mailto:moshez@zadka.site.co.il">moshez@zadka.site.co.il</a>>, Guido van Rossum <<a href="mailto:guido@python.org">guido@python.org</a>> Status: Withdrawn Type: Standards Track Created: 04-Nov-2000 Post-History:</p> <a name="user-content-withdrawal"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Withdrawal</h2><a id="user-content-withdrawal" class="anchor" aria-label="Permalink: Withdrawal" href="#withdrawal"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This PEP has been withdrawn in favor of <a href="http://www.python.org/dev/peps/pep-3141" rel="nofollow">PEP 3141</a>.</p> <a name="user-content-abstract"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Abstract</h2><a id="user-content-abstract" class="anchor" aria-label="Permalink: Abstract" href="#abstract"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Today, Python's numerical model is similar to the C numeric model: there are several unrelated numerical types, and when operations between numerical types are requested, coercions happen. While the C rationale for the numerical model is that it is very similar to what happens at the hardware level, that rationale does not apply to Python. So, while it is acceptable to C programmers that <code>2/3 == 0</code>, it is surprising to many Python programmers.</p> <p dir="auto">NOTE: in the light of recent discussions in the newsgroup, the motivation in this PEP (and details) need to be extended.</p> <a name="user-content-rationale"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Rationale</h2><a id="user-content-rationale" class="anchor" aria-label="Permalink: Rationale" href="#rationale"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">In usability studies, one of the least usable aspect of Python was the fact that integer division returns the floor of the division. This makes it hard to program correctly, requiring casts to <code>float()</code> in various parts through the code. Python's numerical model stems from C, while a model that might be easier to work with can be based on the mathematical understanding of numbers.</p> <a name="user-content-other-numerical-models"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Other Numerical Models</h2><a id="user-content-other-numerical-models" class="anchor" aria-label="Permalink: Other Numerical Models" href="#other-numerical-models"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Perl's numerical model is that there is one type of numbers -- floating point numbers. While it is consistent and superficially non-surprising, it tends to have subtle gotchas. One of these is that printing numbers is very tricky, and requires correct rounding. In Perl, there is also a mode where all numbers are integers. This mode also has its share of problems, which arise from the fact that there is not even an approximate way of dividing numbers and getting meaningful answers.</p> <a name="user-content-suggested-interface-for-python-s-numerical-model"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Suggested Interface For Python's Numerical Model</h2><a id="user-content-suggested-interface-for-pythons-numerical-model" class="anchor" aria-label="Permalink: Suggested Interface For Python's Numerical Model" href="#suggested-interface-for-pythons-numerical-model"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">While coercion rules will remain for add-on types and classes, the built in type system will have exactly one Python type -- a number. There are several things which can be considered "number methods":</p> <ol dir="auto"> <li><code>isnatural()</code></li> <li><code>isintegral()</code></li> <li><code>isrational()</code></li> <li><code>isreal()</code></li> <li><code>iscomplex()</code></li> <li><code>isexact()</code></li> </ol> <p dir="auto">Obviously, a number which answers true to a question from 1 to 5, will also answer true to any following question. If <code>isexact()</code> is not true, then any answer might be wrong. (But not horribly wrong: it's close to the truth.)</p> <p dir="auto">Now, there is two thing the models promises for the field operations (<code>+</code>, <code>-</code>, <code>/</code>, <code>*</code>):</p> <ul dir="auto"> <li>If both operands satisfy <code>isexact()</code>, the result satisfies <code>isexact()</code>.</li> <li>All field rules are true, except that for not-<code>isexact()</code> numbers, they might be only approximately true.</li> </ul> <p dir="auto">One consequence of these two rules is that all exact calculations are done as (complex) rationals: since the field laws must hold, then</p> <pre>(a/b)*b == a </pre> <p dir="auto">must hold.</p> <p dir="auto">There is built-in function, <code>inexact()</code> which takes a number and returns an inexact number which is a good approximation. Inexact numbers must be as least as accurate as if they were using IEEE-754.</p> <p dir="auto">Several of the classical Python functions will return exact numbers even when given inexact numbers: e.g, <code>int()</code>.</p> <a name="user-content-coercion"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Coercion</h2><a id="user-content-coercion" class="anchor" aria-label="Permalink: Coercion" href="#coercion"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">The number type does not define <code>nb_coerce</code> Any numeric operation slot, when receiving something other then <code>PyNumber</code>, refuses to implement it.</p> <a name="user-content-inexact-operations"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Inexact Operations</h2><a id="user-content-inexact-operations" class="anchor" aria-label="Permalink: Inexact Operations" href="#inexact-operations"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">The functions in the <code>math</code> module will be allowed to return inexact results for exact values. However, they will never return a non-real number. The functions in the <code>cmath</code> module are also allowed to return an inexact result for an exact argument, and are furthermore allowed to return a complex result for a real argument.</p> <a name="user-content-numerical-python-issues"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Numerical Python Issues</h2><a id="user-content-numerical-python-issues" class="anchor" aria-label="Permalink: Numerical Python Issues" href="#numerical-python-issues"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">People who use Numerical Python do so for high-performance vector operations. Therefore, NumPy should keep its hardware based numeric model.</p> <a name="user-content-unresolved-issues"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Unresolved Issues</h2><a id="user-content-unresolved-issues" class="anchor" aria-label="Permalink: Unresolved Issues" href="#unresolved-issues"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">Which number literals will be exact, and which inexact?</p> <p dir="auto">How do we deal with IEEE 754 operations? (probably, isnan/isinf should be methods)</p> <p dir="auto">On 64-bit machines, comparisons between ints and floats may be broken when the comparison involves conversion to float. Ditto for comparisons between longs and floats. This can be dealt with by avoiding the conversion to float. (Due to Andrew Koenig.)</p> <a name="user-content-copyright"></a> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Copyright</h2><a id="user-content-copyright" class="anchor" aria-label="Permalink: Copyright" href="#copyright"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This document has been placed in the public 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