Editing Binary number (section)

==Representing real numbers==
<!-- This section is linked from [[Chaitin's constant]] -->
Non-integers can be represented by using negative powers, which are set off from the other digits by means of a [[radix point]] (called a [[decimal point]] in the decimal system). For example, the binary number 11.01<sub>2</sub> means:

{|
|'''1''' × 2<sup>1</sup>  || (1 × 2 = '''2''')           || plus
|-
|'''1''' × 2<sup>0</sup>  || (1 × 1 = '''1''')           || plus
|-
|'''0''' × 2<sup>−1</sup> || (0 × {{frac|2}} = '''0''')  || plus
|-
|'''1''' × 2<sup>−2</sup> || (1 × {{frac|4}} = '''0.25''')
|}

For a total of 3.25 decimal.

All [[dyadic fraction|dyadic rational numbers]] <math>\frac{p}{2^a}</math> have a ''terminating'' binary numeral—the binary representation has a finite number of terms after the radix point. Other [[rational numbers]] have binary representation, but instead of terminating, they ''recur'', with a finite sequence of digits repeating indefinitely. For instance

<math display="block">\frac{1_{10}}{3_{10}} = \frac{1_2}{11_2} = 0.01010101\overline{01}\ldots\,_2 </math>
<math display="block">\frac{12_{10}}{17_{10}} = \frac{1100_2}{10001_2} = 0.10110100 10110100\overline{10110100}\ldots\,_2 </math>

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in [[decimal]]. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that [[0.111... = 1 (binary)|0.111111...]] is the sum of the [[geometric series]] 2<sup>−1</sup> + 2<sup>−2</sup> + 2<sup>−3</sup> + ... which is 1.

Binary numerals that neither terminate nor recur represent [[irrational number]]s. For instance,
* 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
* 1.0110101000001001111001100110011111110... is the binary representation of <math>\sqrt{2}</math>, the [[square root of 2]], another irrational. It has no discernible pattern.