[Axiom-developer] [RealNumbers]

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Real Numbers

  In mathematics, the real numbers are intuitively defined as numbers
that are in one-to-one correspondence with the points on an infinite
line—the number line. The term "real number" is a retronym coined in
response to "imaginary number".

Real numbers may be rational or irrational; algebraic or transcendental;
and positive, negative, or zero. ...

A real number is said to be computable if there exists an algorithm that
yields its digits. Because there are only countably many algorithms, but
an uncountable number of reals, most real numbers are not computable. Some
constructivists accept the existence of only those reals that are computable.
The set of definable numbers is broader, but still only countable.

Computers can only approximate most real numbers with rational numbers;
these approximations are known as floating point numbers or fixed-point
numbers; see real data type. Computer algebra systems are able to treat
some real numbers exactly by storing an algebraic description (such as 
$\sqrt{2}$)
rather than their decimal approximation. 

See: http://encyclopedia.laborlawtalk.com/Real_number


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