How does a computer interpret real numbers?Permuting natural numbersWhat advanced math topics are recommended...

Is there a way to get `mathscr' with lower case letters in pdfLaTeX?

Do the primes contain an infinite almost arithmetic progression?

Can I say "fingers" when referring to toes?

Temporarily disable WLAN internet access for children, but allow it for adults

Why "had" in "[something] we would have made had we used [something]"?

How does a computer interpret real numbers?

User Story breakdown - Technical Task + User Feature

Yosemite Fire Rings - What to Expect?

Quasinilpotent , non-compact operators

Mixing PEX brands

How do apertures which seem too large to physically fit work?

What is the evidence for the "tyranny of the majority problem" in a direct democracy context?

Fear of getting stuck on one programming language / technology that is not used in my country

The IT department bottlenecks progress, how should I handle this?

What if you are holding an Iron Flask with a demon inside and walk into Antimagic Field?

Why Shazam when there is already Superman?

How to explain what's wrong with this application of the chain rule?

Why is short-wave infrared portion of electromagnetic spectrum so sensitive to fire?

Recommended PCB layout understanding - ADM2572 datasheet

How to say when an application is taking the half of your screen on a computer

Why does a simple loop result in ASYNC_NETWORK_IO waits?

Why did the EU agree to delay the Brexit deadline?

What is the highest possible scrabble score for placing a single tile

Does the UK parliament need to pass secondary legislation to accept the Article 50 extension



How does a computer interpret real numbers?


Permuting natural numbersWhat advanced math topics are recommended for computer science?Can a computer count to infinity?Representing Computations on Transcendental NumbersHow to find a subset of numbers such that its average is close to the average of the full set?What precisely differentiates Computer Science from Mathematics in theoretical context?Mathematics topics or fields that increase computer programming proficiency?How does maths consistency impacts on computer science?“Mathematical Physics”'s equivalent for Computer ScienceHow to generate evenly distributed random numbers from the tics of a Geiger Counter?













1












$begingroup$


I understand that the modern day digital computer works on the binary number system. I can also get, that the binary representation can be converted to rational numbers.



But I want to know how does the present day computational model interpret real numbers.



For eg:




On a daily basis we can see that a computer can plot graphs. But here, graphs may be continuous entities. What is the
mathematical basis, that transforms a discrete (or countable, at most)
like the binary system to something mathematically continuous like a
say, the graph of $f(x) = x$.







mathematical-programming mathematical-foundations







share|cite|improve this question









$endgroup$












  • $begingroup$
    Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
    $endgroup$
    – John Dvorak
    5 hours ago










  • $begingroup$
    Aside from floating-point approximations, there are systems that deal with the reals as mathematical entities, most notably computer algebra systems and theorem provers. Representing a particular real number "directly" (for a certain interpretation of "directly") is restricted to the computable numbers.
    $endgroup$
    – Jeroen Mostert
    5 hours ago










  • $begingroup$
    IEEE 754 is the answer.
    $endgroup$
    – Jasper
    4 hours ago
















1












$begingroup$


I understand that the modern day digital computer works on the binary number system. I can also get, that the binary representation can be converted to rational numbers.



But I want to know how does the present day computational model interpret real numbers.



For eg:




On a daily basis we can see that a computer can plot graphs. But here, graphs may be continuous entities. What is the
mathematical basis, that transforms a discrete (or countable, at most)
like the binary system to something mathematically continuous like a
say, the graph of $f(x) = x$.







mathematical-programming mathematical-foundations







share|cite|improve this question









$endgroup$












  • $begingroup$
    Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
    $endgroup$
    – John Dvorak
    5 hours ago










  • $begingroup$
    Aside from floating-point approximations, there are systems that deal with the reals as mathematical entities, most notably computer algebra systems and theorem provers. Representing a particular real number "directly" (for a certain interpretation of "directly") is restricted to the computable numbers.
    $endgroup$
    – Jeroen Mostert
    5 hours ago










  • $begingroup$
    IEEE 754 is the answer.
    $endgroup$
    – Jasper
    4 hours ago














1












1








1





$begingroup$


I understand that the modern day digital computer works on the binary number system. I can also get, that the binary representation can be converted to rational numbers.



But I want to know how does the present day computational model interpret real numbers.



For eg:




On a daily basis we can see that a computer can plot graphs. But here, graphs may be continuous entities. What is the
mathematical basis, that transforms a discrete (or countable, at most)
like the binary system to something mathematically continuous like a
say, the graph of $f(x) = x$.







mathematical-programming mathematical-foundations







share|cite|improve this question









$endgroup$




I understand that the modern day digital computer works on the binary number system. I can also get, that the binary representation can be converted to rational numbers.



But I want to know how does the present day computational model interpret real numbers.



For eg:




On a daily basis we can see that a computer can plot graphs. But here, graphs may be continuous entities. What is the
mathematical basis, that transforms a discrete (or countable, at most)
like the binary system to something mathematically continuous like a
say, the graph of $f(x) = x$.







mathematical-programming mathematical-foundations




mathematical-programming mathematical-foundations






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 5 hours ago









evil_potatoevil_potato

698189




698189












  • $begingroup$
    Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
    $endgroup$
    – John Dvorak
    5 hours ago










  • $begingroup$
    Aside from floating-point approximations, there are systems that deal with the reals as mathematical entities, most notably computer algebra systems and theorem provers. Representing a particular real number "directly" (for a certain interpretation of "directly") is restricted to the computable numbers.
    $endgroup$
    – Jeroen Mostert
    5 hours ago










  • $begingroup$
    IEEE 754 is the answer.
    $endgroup$
    – Jasper
    4 hours ago


















  • $begingroup$
    Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
    $endgroup$
    – John Dvorak
    5 hours ago










  • $begingroup$
    Aside from floating-point approximations, there are systems that deal with the reals as mathematical entities, most notably computer algebra systems and theorem provers. Representing a particular real number "directly" (for a certain interpretation of "directly") is restricted to the computable numbers.
    $endgroup$
    – Jeroen Mostert
    5 hours ago










  • $begingroup$
    IEEE 754 is the answer.
    $endgroup$
    – Jasper
    4 hours ago
















$begingroup$
Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
$endgroup$
– John Dvorak
5 hours ago




$begingroup$
Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
$endgroup$
– John Dvorak
5 hours ago












$begingroup$
Aside from floating-point approximations, there are systems that deal with the reals as mathematical entities, most notably computer algebra systems and theorem provers. Representing a particular real number "directly" (for a certain interpretation of "directly") is restricted to the computable numbers.
$endgroup$
– Jeroen Mostert
5 hours ago




$begingroup$
Aside from floating-point approximations, there are systems that deal with the reals as mathematical entities, most notably computer algebra systems and theorem provers. Representing a particular real number "directly" (for a certain interpretation of "directly") is restricted to the computable numbers.
$endgroup$
– Jeroen Mostert
5 hours ago












$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
4 hours ago




$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
4 hours ago










2 Answers
2






active

oldest

votes


















4












$begingroup$

They represent continuous quantities with discrete approximations. Mostly, this is done with floating point, which is analogous to scientific notation. Essentially, they work with something like $1.xyztimes 10^k$, with some appropriate number of decimal places (and in binary, rather than decimal).



It's also possible to work with some irrational numbers directly. For example, you could create yourself an object called "$sqrt{2}$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt{2})^2=2$. So then you could compute $$(sqrt{2}-1)(sqrt{2}+1) = (sqrt{2})^2 + sqrt{2} - sqrt{2} - 1 = 2 - 1 = 1$$
exactly, as an algebraic fact that's not susceptible to rounding errors.



Note that, if you're plotting a graph on screen, regardless of how far you zoom in, you're plotting it as discrete pixels so using an appropriate number of significant figures basically gets you everything you need.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    The real numbers are uncountable. The set of real numbers that can be represented in any way is countable. Therefore, almost all real numbers cannot be represented by a computer at all.



    The most common method is to store floating point numbers, which are reasonably precise approximations to real numbers that are not excessively large or small.






    share|cite|improve this answer









    $endgroup$













      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "419"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: false,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: null,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f105923%2fhow-does-a-computer-interpret-real-numbers%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      They represent continuous quantities with discrete approximations. Mostly, this is done with floating point, which is analogous to scientific notation. Essentially, they work with something like $1.xyztimes 10^k$, with some appropriate number of decimal places (and in binary, rather than decimal).



      It's also possible to work with some irrational numbers directly. For example, you could create yourself an object called "$sqrt{2}$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt{2})^2=2$. So then you could compute $$(sqrt{2}-1)(sqrt{2}+1) = (sqrt{2})^2 + sqrt{2} - sqrt{2} - 1 = 2 - 1 = 1$$
      exactly, as an algebraic fact that's not susceptible to rounding errors.



      Note that, if you're plotting a graph on screen, regardless of how far you zoom in, you're plotting it as discrete pixels so using an appropriate number of significant figures basically gets you everything you need.






      share|cite|improve this answer









      $endgroup$


















        4












        $begingroup$

        They represent continuous quantities with discrete approximations. Mostly, this is done with floating point, which is analogous to scientific notation. Essentially, they work with something like $1.xyztimes 10^k$, with some appropriate number of decimal places (and in binary, rather than decimal).



        It's also possible to work with some irrational numbers directly. For example, you could create yourself an object called "$sqrt{2}$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt{2})^2=2$. So then you could compute $$(sqrt{2}-1)(sqrt{2}+1) = (sqrt{2})^2 + sqrt{2} - sqrt{2} - 1 = 2 - 1 = 1$$
        exactly, as an algebraic fact that's not susceptible to rounding errors.



        Note that, if you're plotting a graph on screen, regardless of how far you zoom in, you're plotting it as discrete pixels so using an appropriate number of significant figures basically gets you everything you need.






        share|cite|improve this answer









        $endgroup$
















          4












          4








          4





          $begingroup$

          They represent continuous quantities with discrete approximations. Mostly, this is done with floating point, which is analogous to scientific notation. Essentially, they work with something like $1.xyztimes 10^k$, with some appropriate number of decimal places (and in binary, rather than decimal).



          It's also possible to work with some irrational numbers directly. For example, you could create yourself an object called "$sqrt{2}$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt{2})^2=2$. So then you could compute $$(sqrt{2}-1)(sqrt{2}+1) = (sqrt{2})^2 + sqrt{2} - sqrt{2} - 1 = 2 - 1 = 1$$
          exactly, as an algebraic fact that's not susceptible to rounding errors.



          Note that, if you're plotting a graph on screen, regardless of how far you zoom in, you're plotting it as discrete pixels so using an appropriate number of significant figures basically gets you everything you need.






          share|cite|improve this answer









          $endgroup$



          They represent continuous quantities with discrete approximations. Mostly, this is done with floating point, which is analogous to scientific notation. Essentially, they work with something like $1.xyztimes 10^k$, with some appropriate number of decimal places (and in binary, rather than decimal).



          It's also possible to work with some irrational numbers directly. For example, you could create yourself an object called "$sqrt{2}$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt{2})^2=2$. So then you could compute $$(sqrt{2}-1)(sqrt{2}+1) = (sqrt{2})^2 + sqrt{2} - sqrt{2} - 1 = 2 - 1 = 1$$
          exactly, as an algebraic fact that's not susceptible to rounding errors.



          Note that, if you're plotting a graph on screen, regardless of how far you zoom in, you're plotting it as discrete pixels so using an appropriate number of significant figures basically gets you everything you need.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 5 hours ago









          David RicherbyDavid Richerby

          68.9k15105194




          68.9k15105194























              0












              $begingroup$

              The real numbers are uncountable. The set of real numbers that can be represented in any way is countable. Therefore, almost all real numbers cannot be represented by a computer at all.



              The most common method is to store floating point numbers, which are reasonably precise approximations to real numbers that are not excessively large or small.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                The real numbers are uncountable. The set of real numbers that can be represented in any way is countable. Therefore, almost all real numbers cannot be represented by a computer at all.



                The most common method is to store floating point numbers, which are reasonably precise approximations to real numbers that are not excessively large or small.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  The real numbers are uncountable. The set of real numbers that can be represented in any way is countable. Therefore, almost all real numbers cannot be represented by a computer at all.



                  The most common method is to store floating point numbers, which are reasonably precise approximations to real numbers that are not excessively large or small.






                  share|cite|improve this answer









                  $endgroup$



                  The real numbers are uncountable. The set of real numbers that can be represented in any way is countable. Therefore, almost all real numbers cannot be represented by a computer at all.



                  The most common method is to store floating point numbers, which are reasonably precise approximations to real numbers that are not excessively large or small.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  gnasher729gnasher729

                  11.3k1217




                  11.3k1217






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Computer Science Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f105923%2fhow-does-a-computer-interpret-real-numbers%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      Gersau Kjelder | Navigasjonsmeny46°59′0″N 8°31′0″E46°59′0″N...

                      Image
                      Tidlegare republikkarSkipingar i 1433Kommunar i SchwyzDistrikt i SchwyzNedleggingar i 1798Stader ved Vierwaldstättersee kommunekantonenSchwyzSveitskm²VierwaldstätterseeLauerz Gersau Kommunevåpenet til Gersau Plassering Styresmakter Land   Sveits Kanton kantonen Schwyz Distrikt Gersau Språk tysk Geografi Flatevidd  - kommune 23,70 km² Innbyggjarar  - kommune (desember 2010)    - folketettleik 2 094   88,4 /km² Koordinatar 46°59′0″N 8°31′0″E  Koordinatar: 46°59′0″N 8°31′0″E  Høgd over havet 435 moh Diverse annan informasjon Postnummer 6442 Nettstad: www.gersau.ch Freistaat und Republik Gersau Fristaten og republikken Gersau By-republikk og alliert protektorat i Det gamle sveitsiske eidsforundet ←   ← 1433–1798 1814–1818 →   → Hovudstad Gersau Styreform Republikk Historisk periode Tidleg moderne tid  - Kjøpte fridom frå ...
                      Read more

                      Nässjö kommun Tettstader | Kjelder | NavigasjonsmenyeVIAFISNIGeoNamesMusicBrainz (area)

                      Image
                      Nässjö kommunKommunar i Jönköpings län svenskeJönköpings länAdministrasjonssenterNässjöAneby kommunEksjö kommunVetlanda kommunSävsjö kommuneVaggeryd kommuneJönköping kommunejernbaneknutepunktBackyard BabiesEidsbergtettstader31. desember2005 Nässjö kommun kommune Kommunevåpen Land   Sverige Len Jönköpings län Landskap Småland Areal 988,49 km²  - land 930,29 km²  - vatn 58,2 km² Folketal 29 375 Wikimedia Commons: Nässjö Municipality Nässjö kommune ligg i det svenske länet Jönköpings län. Administrasjonssenter i kommunen er Nässjö. Kommunen grensar til Aneby kommun i nord, Eksjö kommun og Vetlanda kommun i aust, Sävsjö kommune og Vaggeryd kommune i sør og Jönköping kommune i vest. Kommunen er kjend som jernbaneknutepunkt, og som heimstaden til rockebandet Backyard Babies. Nässjö er venskapskommune med Eidsberg. Nässjö sentrum sett frå vasstårnet. Tettstader | Nässjö kommune har tol...
                      Read more

                      Kvitkval Innhaldsliste Taksonomi og utvikling | Utsjånad og levevis | Utbreiing | Åtferd |...

                      Image
                      Nær truga artarPattedyr i NoregNarkvalfamilien tannkvalarktiskenarkvalenmiocenkvalendelfinanetannkvalarLatinbiosonarenekkolokaliseringisbjørnenspekkhoggarinuittarArktisNew YorkNord-AmerikaEuropaJapan Kvitkval Status i verda: Nær truga Status i Noreg: Datamangel Kvitkval Systematikk Domene: Eukaryota Rike: Animalia Rekkje: Chordata Underrekkje: Vertebrata Klasse: Mammalia Orden: Cetaceae Familie: Monodontidae Slekt: Delphinapterus Art: D. leucas Vitskapleg namn Delphinapterus leucas Pallas, 1776 Kvitkval er ein tannkval som lever i arktiske og subarktiske område. Innhaldsliste 1 Taksonomi og utvikling 2 Utsjånad og levevis 3 Utbreiing 4 Åtferd 5 Bestandsstorleik og menneskeleg påverknad 6 Kjelder Taksonomi og utvikling | Kvitkvalen vart første gongen skildra av Peter Simon Pallas i 1776. Han er medlem av Monodontidae -familien saman med narkvalen. D...
                      Read more