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?
Can I say "fingers" when referring to toes?
Plot of a tornado-shaped surface
How to say when an application is taking the half of your screen on a computer
Why did the EU agree to delay the Brexit deadline?
Store Credit Card Information in Password Manager?
Can I still be respawned if I die by falling off the map?
What are the advantages of simplicial model categories over non-simplicial ones?
Strong empirical falsification of quantum mechanics based on vacuum energy density?
Non-trope happy ending?
Does Doodling or Improvising on the Piano Have Any Benefits?
What is the highest possible scrabble score for placing a single tile
Calculate sum of polynomial roots
Temporarily disable WLAN internet access for children, but allow it for adults
Quasinilpotent , non-compact operators
Can a stoichiometric mixture of oxygen and methane exist as a liquid at standard pressure and some (low) temperature?
Why "had" in "[something] we would have made had we used [something]"?
Why should universal income be universal?
Does an advisor owe his/her student anything? Will an advisor keep a PhD student only out of pity?
How to explain what's wrong with this application of the chain rule?
It grows, but water kills it
How could a planet have erratic days?
What features enable the Su-25 Frogfoot to operate with such a wide variety of fuels?
How do you respond to a colleague from another team when they're wrongly expecting that you'll help them?
Biological Blimps: Propulsion
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?
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
$begingroup$
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
5 hours ago
add a comment |
$begingroup$
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
$endgroup$
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
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
5 hours ago
add a comment |
$begingroup$
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
5 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
5 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
5 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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 "$sqrt2$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt2)^2=2$. So then you could compute $$(sqrt2-1)(sqrt2+1) = (sqrt2)^2 + sqrt2 - sqrt2 - 1 = 2 - 1 = 1$$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
$begingroup$
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 "$sqrt2$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt2)^2=2$. So then you could compute $$(sqrt2-1)(sqrt2+1) = (sqrt2)^2 + sqrt2 - sqrt2 - 1 = 2 - 1 = 1$$
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.
$endgroup$
add a comment |
$begingroup$
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 "$sqrt2$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt2)^2=2$. So then you could compute $$(sqrt2-1)(sqrt2+1) = (sqrt2)^2 + sqrt2 - sqrt2 - 1 = 2 - 1 = 1$$
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.
$endgroup$
add a comment |
$begingroup$
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 "$sqrt2$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt2)^2=2$. So then you could compute $$(sqrt2-1)(sqrt2+1) = (sqrt2)^2 + sqrt2 - sqrt2 - 1 = 2 - 1 = 1$$
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.
$endgroup$
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 "$sqrt2$" without particularly worrying about what it is, except that it obeys the usual arithmetic rules and that $(sqrt2)^2=2$. So then you could compute $$(sqrt2-1)(sqrt2+1) = (sqrt2)^2 + sqrt2 - sqrt2 - 1 = 2 - 1 = 1$$
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.
answered 5 hours ago
David RicherbyDavid Richerby
68.9k15105194
68.9k15105194
add a comment |
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered 2 hours ago
gnasher729gnasher729
11.3k1217
11.3k1217
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
$begingroup$
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
5 hours ago