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How does a computer interpret real numbers?

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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

asked 5 hours ago

evil_potatoevil_potato

698189

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

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

asked 5 hours ago

evil_potatoevil_potato

698189

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

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

share|cite|improve this question

asked 5 hours ago

evil_potatoevil_potato

698189

$endgroup$

share|cite|improve this question

asked 5 hours ago

evil_potatoevil_potato

698189

share|cite|improve this question

asked 5 hours ago

evil_potatoevil_potato

698189

asked 5 hours ago

evil_potatoevil_potato

698189

asked 5 hours ago

evil_potatoevil_potato

698189

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 "$sqrt2$" 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.

share|cite|improve this answer

answered 5 hours ago

David RicherbyDavid Richerby

68.9k15105194

$endgroup$

add a comment | 

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

answered 2 hours ago

gnasher729gnasher729

11.3k1217

$endgroup$

add a comment | 

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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 "$sqrt2$" 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.

share|cite|improve this answer

answered 5 hours ago

David RicherbyDavid Richerby

68.9k15105194

$endgroup$

add a comment | 

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 "$sqrt2$" 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.

share|cite|improve this answer

answered 5 hours ago

David RicherbyDavid Richerby

68.9k15105194

$endgroup$

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.

share|cite|improve this answer

answered 5 hours ago

David RicherbyDavid Richerby

68.9k15105194

$endgroup$

share|cite|improve this answer

answered 5 hours ago

David RicherbyDavid Richerby

68.9k15105194

answered 5 hours ago

David RicherbyDavid Richerby

68.9k15105194

answered 5 hours ago

David RicherbyDavid Richerby

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

answered 2 hours ago

gnasher729gnasher729

11.3k1217

$endgroup$

add a comment | 

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

answered 2 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.

share|cite|improve this answer

answered 2 hours ago

gnasher729gnasher729

11.3k1217

$endgroup$

share|cite|improve this answer

answered 2 hours ago

gnasher729gnasher729

11.3k1217

answered 2 hours ago

gnasher729gnasher729

11.3k1217

answered 2 hours ago

gnasher729gnasher729

11.3k1217