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.
Read more about this topic: Binary Numeral System
Famous quotes containing the words representing, real and/or numbers:
“He who has learned what is commonly considered the whole art of painting, that is, the art of representing any natural object faithfully, has as yet only learned the language by which his thoughts are to be expressed.”
—John Ruskin (1819–1900)
“A people’s literature is the great textbook for real knowledge of them. The writings of the day show the quality of the people as no historical reconstruction can.”
—Edith Hamilton (1867–1963)
“Old age equalizes—we are aware that what is happening to us has happened to untold numbers from the beginning of time. When we are young we act as if we were the first young people in the world.”
—Eric Hoffer (1902–1983)