Power of two

A power of two is a number of the form $2 n$ where $n$ is an integer, that is, the result of exponentiation with number two as the base and integer $n$ as the exponent. If $n$ is negative, $2 n$ is called an negative power of two or inverse power of two.

"Power of 2" redirects here. For other uses of this and of "Power of two", see Power of two (disambiguation)

In a context where only integers are considered, $n$ is restricted to non-negative values,^[1] so there are 1, 2, and 2 multiplied by itself a certain number of times.^[2]

The first ten powers of 2 for non-negative values of $n$ are:

Base of the binary numeral system[edit]

Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system.

Computer science[edit]

Two to the exponent of $n$ , written as $2 n$ , is the number of ways the bits in a binary word of length $n$ can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 ( $000...000 2$ ) to $2 n - 1$ ( $111...111 2$ ) inclusively. Corresponding signed integer values can be positive, negative and zero; see signed number representations. Either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this form show up frequently in computer software. As an example, a video game running on an 8-bit system might limit the score or the number of items the player can hold to 255—the result of using a byte, which is 8 bits long, to store the number, giving a maximum value of $28 - 1 = 255$ . For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees (the currency of the game) at any given time, and the video game Pac-Man famously has a kill screen at level 256.

Powers of two are often used to measure computer memory. A byte is now considered eight bits (an octet), resulting in the possibility of 256 values (2⁸). (The term byte once meant (and in some cases, still means) a collection of bits, typically of 5 to 32 bits, rather than only an 8-bit unit.) The prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024 (2¹⁰). However, in general, the term kilo has been used in the International System of Units to mean 1,000 (10³). Binary prefixes have been standardized, such as kibi (Ki) meaning 1,024. Nearly all processor registers have sizes that are powers of two, 32 or 64 being very common.

Powers of two occur in a range of other places as well. For many disk drives, at least one of the sector size, number of sectors per track, and number of tracks per surface is a power of two. The logical block size is almost always a power of two.

Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of only two or three powers of two, or powers of two minus one. For example, $640 = 32 \times 20$ , and $480 = 32 \times 15$ . Put another way, they have fairly regular bit patterns.

Mersenne and Fermat primes[edit]

A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it is 1 less than 32 (2⁵). Similarly, a prime number (like 257) that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational. The numbers that can be represented as sums of consecutive positive integers are called polite numbers; they are exactly the numbers that are not powers of two.

Euclid's Elements, Book IX[edit]

The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, ... ) is important in number theory. Book IX, Proposition 36 of Elements proves that if the sum of the first $n$ terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the $n$ th term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number.

Book IX, Proposition 35, proves that in a geometric series if the first term is subtracted from the second and last term in the sequence, then as the excess of the second is to the first—so is the excess of the last to all those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31, 62, 124, 248, 496 (which results from 1, 2, 4, 8, 16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248. Therefore, the numbers 1, 2, 4, 8, 16, 31, 62, 124 and 248 add up to 496 and further these are all the numbers that divide 496. For suppose that $p$ divides 496 and it is not amongst these numbers. Assume $p q$ is equal to $16 \times 31$ , or 31 is to $q$ as $p$ is to 16. Now $p$ cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or 16. Therefore, 31 cannot divide $q$ . And since 31 does not divide $q$ and $q$ measures 496, the fundamental theorem of arithmetic implies that $q$ must divide 16 and be among the numbers 1, 2, 4, 8 or 16. Let $q$ be 4, then $p$ must be 124, which is impossible since by hypothesis $p$ is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248.

Powers of two in music theory[edit]

In musical notation, all unmodified note values have a duration equal to a whole note divided by a power of two; for example a half note (1/2), a quarter note (1/4), an eighth note (1/8) and a sixteenth note (1/16). Dotted or otherwise modified notes have other durations. In time signatures the lower numeral, the beat unit, which can be seen as the denominator of a fraction, is almost always a power of two.

If the ratio of frequencies of two pitches is a power of two, then the interval between those pitches is full octaves. In this case, the corresponding notes have the same name.

The mathematical coincidence $2^{7}\approx ({\tfrac {3}{2}})^{12}$ , from ${\frac {\log 3}{\log 2}}=1.5849\ldots \approx {\frac {19}{12}}$ , closely relates the interval of 7 semitones in equal temperament to a perfect fifth of just intonation: $2^{7/12}\approx 3/2$ , correct to about 0.1%. The just fifth is the basis of Pythagorean tuning; the difference between twelve just fifths and seven octaves is the Pythagorean comma.^[10]

Negative powers of two[edit]

Huffman codes deliver optimal lossless data compression when probabilities of the source symbols are all negative powers of two.^[12]

Power of two

Base of the binary numeral system[edit]

Computer science[edit]

Mersenne and Fermat primes[edit]

Euclid's Elements, Book IX[edit]

Powers of two in music theory[edit]

Negative powers of two[edit]

Fermi–Dirac prime

Gould's sequence

Binary logarithm

Power of three

Power of 10