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In this context, real numbers are represented as arbitrary infinite sequences of symbols. These sequences could for instance represent the digits of a real number. Such sequences '''need not be computable''' — this freedom is both an important one and philosophically unproblematic.<ref>An uncomputable real number can be generated with near certainty by sampling each digit at random in an infinite unending process.</ref> Note that the programs that act on these sequences ''do'' need to be computable in a reasonable sense.
In the case of real numbers, the usual decimal or binary representations are not appropriate. Instead a signed digit representation first suggested by Brouwer often gets used: The number system is base 2, but the digits are <math>\overline 1</math> (representing <math>-1</math>), 0 and 1. In particular, this means <math>1/2</math> can be represented both as <math>0.1</math> and <math>
To understand why decimal notation is inappropriate, consider the problem of computing <math>z=x+y</math> where <math>x = 0.(3)</math> and <math>y=0.(6)</math>, and giving the result <math>z</math> in decimal notation. The value of <math>z</math> is either <math>0.(9)</math> or <math>1.(0)</math>. If the latter result were given for instance, then a finite number <math>n</math> of digits of <math>x</math> would be read before choosing the digit <math>1</math> before the decimal point in <math>z</math> — but then if the <math>n+1</math>th digit of <math>x</math> were decreased to 2, then the result for <math>z</math> would be wrong. Similarly, the former choice <math>0.(9)</math> for <math>z</math> would be wrong sometimes. This is essentially the [[Table-maker's dilemma|tablemaker's dilemma]].
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