Power of two

A power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with the number two as the base and integer n as the exponent. In the fast-growing hierarchy, 2n is exactly equal to

f

1

n

(

1

)

{\displaystyle f_{1}^{n}(1)}

. In the Hardy hierarchy, 2n is exactly equal to

H

ω

n

(

1

)

{\displaystyle H_{\omega {n}}(1)}

.

Powers of two with non-negative exponents are integers: 20 = 1, 21 = 2, and 2n is n 2s multiplied together. The first ten powers of 2 for non-negative values of n are:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (sequence A000079 in the OEIS)

By comparison, powers of two with negative exponents are fractions: for positive integer n, 2−n is one half multiplied by itself n times. Thus the first few negative powers of 2 are ⁠1/2⁠, ⁠1/4⁠, ⁠1/8⁠, ⁠1/16⁠, etc. Sometimes these are called inverse powers of two because each is the multiplicative inverse of a positive power of two.

Computer science

Most computers use a binary (base two) representation of numbers, a number system in which each place represents a power of 2, and the only possible binary digits ("bits") are 0 and 1. Powers of two are thus ubiquitous in computing and computer science.