Power of two

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For the Indigo Girls song, see Power of Two (song).

In mathematics, a power of two is any of the integer powers of the number two; in other words, two multiplied by itself a certain number of times. Note that one is a power (the zeroth power) of two. Written in binary, a power of two always has the form 100...0, just like a power of ten in the decimal system.

Because two is the base of the binary system, powers of two are important to computer science. Specifically, two to the power of n is the number of ways the bits in a binary integer of length n can be arranged, and thus numbers that are one less than a power of two denote the upper bounds of integers in binary computers (one less because 0, not 1, is used as the lower bound). 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 a byte, which is 8 bits long, being used to store the number, giving a maximum value of 2⁸−1 = 255.

Powers of two also measure computer memory. A byte is eight (2³) bits, and a kilobyte is 1,024 (2¹⁰) bytes (standards prefer the word kibibyte, as "kilobyte" also means 1,000 bytes). Nearly all processor registers have sizes that are powers of two, 32 or 64 being most common (see word size).

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 which 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 = 512 + 128, and 480 = 32 × 15. Put another way, they have fairly regular bit patterns.

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 power of two is called a Fermat prime. The exponent will itself be a power of two. See Fermat number.

The 0th through 40th powers of 2

2⁰	=	1
2¹	=	2	2¹¹	=	2,048	2²¹	=	2,097,152	2³¹	=	2,147,483,648
2²	=	4	2¹²	=	4,096	2²²	=	4,194,304	2³²	=	4,294,967,296
2³	=	8	2¹³	=	8,192	2²³	=	8,388,608	2³³	=	8,589,934,592
2⁴	=	16	2¹⁴	=	16,384	2²⁴	=	16,777,216	2³⁴	=	17,179,869,184
2⁵	=	32	2¹⁵	=	32,768	2²⁵	=	33,554,432	2³⁵	=	34,359,738,368
2⁶	=	64	2¹⁶	=	65,536	2²⁶	=	67,108,864	2³⁶	=	68,719,476,736
2⁷	=	128	2¹⁷	=	131,072	2²⁷	=	134,217,728	2³⁷	=	137,438,953,472
2⁸	=	256	2¹⁸	=	262,144	2²⁸	=	268,435,456	2³⁸	=	274,877,906,944
2⁹	=	512	2¹⁹	=	524,288	2²⁹	=	536,870,912	2³⁹	=	549,755,813,888
2¹⁰	=	1,024	2²⁰	=	1,048,576	2³⁰	=	1,073,741,824	2⁴⁰	=	1,099,511,627,776

Powers of two whose exponents are powers of two

Because modern memory cells and registers are accessed by a Computer bus whose width (number of bits) is also a power of two, the most frequent powers of two to appear are those whose exponent is also a power of two. For example:

2¹ = 2

2² = 4

2⁴ = 16

2⁸ = 256

2¹⁶ = 65,536

2³² = 4,294,967,296

2⁶⁴ = 18,446,744,073,709,551,616

2¹²⁸ = 340,282,366,920,938,463,463,374,607,431,768,211,456

2²⁵⁶ = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936

Several of these numbers represent the number of values representable using common computer data types. For example, a 32-bit word consisting of 4 bytes can represent 2³² distinct values, which can either be regarded as mere bit-patterns, or are more commonly interpreted as the unsigned numbers from 0 to 2³²−1, or as the range of signed numbers between −2³¹ and 2³¹−1.

Other recognisable powers of two

2¹⁰ = 1,024

* the digital approximation of the kilo-, or 1,000 multiplier, which causes a change of prefix. For example: 1,024 bytes = 1 kilobyte (or kibibyte).

* This number has no special significance to computers, but is important to humans because we make use of powers of ten. 2²⁴ = 16,777,216

* The number of unique colors that can be displayed in truecolor, which is used by common computer monitors.

* This number is the result of using the three-channel RGB system, with 8 bits for each channel, or 24 bits in total.

Fast algorithm to check if a non-negative number is a power of two

The binary representation of integers makes it possible to apply a very fast test to determine whether a given positive integer x is a power of two:

x is a power of two

Leftrightarrow

(x & (x − 1)) equals zero.

where & is a bitwise logical AND operator. Examples:

−1	=	1...111...1	−1	=	1...111...111...1
x	=	0...010...0	y	=	0...010...010...0
x−1	=	0...001...1	y−1	=	0...010...001...1
x & (x−1)	=	0...000...0	y & (y−1)	=	0...010...000...0

Note that zero is incorrectly considered a power of two by this test.

Algorithm to convert any number into nearest power of two number

The following formula finds the nearest power of two with respect to binary logarithm of a given value $x > 0$ :

2^{mathrm{round}(log_2 x)}

Computer Pseudocode:
POT:= 2^ round(Log2(NPOT)); It does not, however, find the nearest power of two with respect to the actual value. For example, 47 is nearer to 32 than it is to 64, but previous formula rounds it to 64.

If $x$ is an integer value, following steps can be taken to find the nearest value (with respect to actual value rather than the binary logarithm) in a computer program:

Find the most significant bit $k$ that is "1" from the binary representation of $x$ , when $k = 0$ means the least significant bit
Assume that all bits $k < 0$ are zero. Then, if bit $k-1$ is zero, the result is $2^k$ . Otherwise the result is $2^{k+1}$ .

Actually, ignore the assumption that all bits $k < 0$ are zero, it doesn't do anything, and would make bit $k-1$ is zero always true. Here's the real way to round a number to the nearest power of two:

2^{mathrm{floor}((log_2 x) - (log_2 1.5) + 1)}

Algorithm to find the next-highest power of two

The pseudocode for an algorithm to compute the next-highest power of two for a particular number "n" is as follows:

n = n - 1;
n = n | (n >> 1);
n = n | (n >> 2);
n = n | (n >> 4);
n = n | (n >> 8);
n = n | (n >> 16);
...
n = n | (n >> (bitspace / 2));
n = n + 1;

Where "|" is a binary or operator, ">>" is the binary right-shift operator, and bitspace is the size (in bits) of the integer space represented by n. For most computer architectures, this value is either 8, 16, 32, or 64. This operator works by setting all bits on the right-hand side of a the most significant flagged bit to "1", and then incrementing the entire value at the end so it "rolls over" to the nearest power of two. An example of each step of this algorithm for the number 2689 is as follows:

Binary representation	Decimal representation
0101010000001	2689
0101010000000	2688
0111111000000	4032
0111111110000	4080
0111111111111	4095
1000000000000	4096

As demonstrated above, the algorithm yieds the correct value of 4096. It should be noted that the nearest power to 2689 happens to be 2048; however, this algorithm is designed only to give the next highest power of two to a given number, not the nearest power of two.

A C version of this code for the integer type T would be:

T nexthigher(T k) {
        k--;
        for (int i=1; i> i;
        return k+1;
}

References

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Last updated on Thursday March 13, 2008 at 13:30:06 PDT (GMT -0700)
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