Binary Numeral System - Representing Real Numbers

Representing Real Numbers

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 11.01₂ thus means:

1 × 21	(1 × 2 = 2)	plus
1 × 20	(1 × 1 = 1)	plus
0 × 2−1	(0 × ½ = 0)	plus
1 × 2−2	(1 × ¼ = 0.25)

For a total of 3.25 decimal.

All dyadic rational numbers 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

= = 0.0101010101…₂

= = 0.10110100 10110100 10110100...₂

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 0.111111… is the sum of the geometric series 2−1 + 2−2 + 2−3 + ... which is 1.

Binary numerals which neither terminate nor recur represent irrational numbers. 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, the square root of 2, another irrational. It has no discernible pattern. See irrational number.