How does a computer interpret real numbers?












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
    8 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
    8 hours ago










  • $begingroup$
    IEEE 754 is the answer.
    $endgroup$
    – Jasper
    8 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
    8 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
    8 hours ago










  • $begingroup$
    IEEE 754 is the answer.
    $endgroup$
    – Jasper
    8 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 8 hours ago









evil_potatoevil_potato

698189




698189












  • $begingroup$
    Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
    $endgroup$
    – John Dvorak
    8 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
    8 hours ago










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


















  • $begingroup$
    Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
    $endgroup$
    – John Dvorak
    8 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
    8 hours ago










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
















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




$begingroup$
Possible duplicate of this Wikipedia article: en.wikipedia.org/wiki/Floating-point_arithmetic
$endgroup$
– John Dvorak
8 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
8 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
8 hours ago












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




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










2 Answers
2






active

oldest

votes


















5












$begingroup$

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









$endgroup$









  • 1




    $begingroup$
    +1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
    $endgroup$
    – ruakh
    1 hour ago



















1












$begingroup$

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









$endgroup$













  • $begingroup$
    In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
    $endgroup$
    – Graham
    36 mins ago












  • $begingroup$
    @Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
    $endgroup$
    – ruakh
    16 mins ago











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









5












$begingroup$

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









$endgroup$









  • 1




    $begingroup$
    +1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
    $endgroup$
    – ruakh
    1 hour ago
















5












$begingroup$

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









$endgroup$









  • 1




    $begingroup$
    +1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
    $endgroup$
    – ruakh
    1 hour ago














5












5








5





$begingroup$

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 8 hours ago









David RicherbyDavid Richerby

68.9k15105194




68.9k15105194








  • 1




    $begingroup$
    +1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
    $endgroup$
    – ruakh
    1 hour ago














  • 1




    $begingroup$
    +1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
    $endgroup$
    – ruakh
    1 hour ago








1




1




$begingroup$
+1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
$endgroup$
– ruakh
1 hour ago




$begingroup$
+1. For folks interested in more information about your second paragraph, some good search terms are symbolic computation and computer algebra.
$endgroup$
– ruakh
1 hour ago











1












$begingroup$

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









$endgroup$













  • $begingroup$
    In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
    $endgroup$
    – Graham
    36 mins ago












  • $begingroup$
    @Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
    $endgroup$
    – ruakh
    16 mins ago
















1












$begingroup$

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









$endgroup$













  • $begingroup$
    In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
    $endgroup$
    – Graham
    36 mins ago












  • $begingroup$
    @Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
    $endgroup$
    – ruakh
    16 mins ago














1












1








1





$begingroup$

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 5 hours ago









gnasher729gnasher729

11.3k1217




11.3k1217












  • $begingroup$
    In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
    $endgroup$
    – Graham
    36 mins ago












  • $begingroup$
    @Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
    $endgroup$
    – ruakh
    16 mins ago


















  • $begingroup$
    In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
    $endgroup$
    – Graham
    36 mins ago












  • $begingroup$
    @Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
    $endgroup$
    – ruakh
    16 mins ago
















$begingroup$
In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
$endgroup$
– Graham
36 mins ago






$begingroup$
In fact, infinitely many real numbers can't be represented. It's just a slightly smaller infinity than the infinite number of all real numbers. :)
$endgroup$
– Graham
36 mins ago














$begingroup$
@Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
$endgroup$
– ruakh
16 mins ago




$begingroup$
@Graham: It's actually the same infinite cardinality. gnasher729 is using "almost all" in a precise sense; see en.wikipedia.org/wiki/Almost_all for details.
$endgroup$
– ruakh
16 mins ago


















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