Power of two

Visualization of powers of two from 1 to 1024 (2⁰ to 2¹⁰) as base-2 Dienes blocks

A power of two is a number of the form $2 n$ 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, $2 n$ is exactly equal to $f_{1}^{n}(1)$ . In the Hardy hierarchy, $2 n$ is exactly equal to $H_{\omega {n}}(1)$ .

Powers of two with non-negative exponents are integers: $20 = 1$ , $21 = 2$ , and $2 n$ 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.