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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
asked 8 hours ago
evil_potatoevil_potato
698189
$endgroup$
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
asked 8 hours ago
evil_potatoevil_potato
698189
$endgroup$
$begingroup$
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
7 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
7 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
asked 8 hours ago
evil_potatoevil_potato
698189
$endgroup$
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 8 hours ago
evil_potatoevil_potato
698189
asked 8 hours ago
evil_potatoevil_potato
698189
asked 8 hours ago
evil_potatoevil_potato
698189
asked 8 hours ago
evil_potatoevil_potato
698189
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
7 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
7 hours ago
add a comment |
$begingroup$
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
7 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
7 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
7 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
7 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
7 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
7 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.
answered 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
$endgroup$
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
51 mins ago
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.
answered 5 hours ago
gnasher729gnasher729
11.3k1217
$endgroup$
add a comment |
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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.
answered 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
$endgroup$
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
51 mins ago
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.
answered 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
$endgroup$
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
51 mins ago
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.
answered 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
$endgroup$
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 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
answered 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
answered 7 hours ago
David RicherbyDavid Richerby
68.9k15105194
answered 7 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
51 mins ago
add a comment |
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
51 mins 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
51 mins 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
51 mins ago
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.
answered 5 hours ago
gnasher729gnasher729
11.3k1217
$endgroup$
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.
answered 5 hours ago
gnasher729gnasher729
11.3k1217
$endgroup$
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.
answered 5 hours ago
gnasher729gnasher729
11.3k1217
$endgroup$
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 5 hours ago
gnasher729gnasher729
11.3k1217
answered 5 hours ago
gnasher729gnasher729
11.3k1217
answered 5 hours ago
gnasher729gnasher729
11.3k1217
answered 5 hours ago
gnasher729gnasher729
11.3k1217
11.3k1217
add a comment |
add a comment |
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$begingroup$
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
7 hours ago
$begingroup$
IEEE 754 is the answer.
$endgroup$
– Jasper
7 hours ago