Power of two

In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times 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 10000...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 one example, in the video game The Legend of Zelda for the 8-bit Nintendo, one can only hold 255 rupeess at one time - 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 (some prefer the word kibibyte) is 1 024 (2¹⁰) bytes. Nearly all processor registers have sizes that are powers of two (32 being currently used in most personal computers).

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).

The first forty powers of two

2 1	=	2		211	=	2 048		221	=	2 097 152		231	=	2 147 483 648
2 2	=	4		212	=	4 096		222	=	4 194 304		232	=	4 294 967 296
2 3	=	8		213	=	8 192		223	=	8 388 608		233	=	8 589 934 592
2 4	=	16		214	=	16 384		224	=	16 777 216		234	=	17 179 869 184
2 5	=	32		215	=	32 768		225	=	33 554 432		235	=	34 359 738 368
2 6	=	64		216	=	65 536		226	=	67 108 864		236	=	68 719 476 736
2 7	=	128		217	=	131 072		227	=	134 217 728		237	=	137 438 953 472
2 8	=	256		218	=	262 144		228	=	268 435 456		238	=	274 877 906 944
2 9	=	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 often hold a number of bits which is a power of two, the most frequent
powers of two to appear are those whose exponent is also a power of two.  –   A short list of some of these follows :

2	=	21
4	=	22
16	=	24
256	=	28
65 536	=	216
4 294 967 296	=	232
18 446 744 073 709 551 616	=	264
340 282 366 920 938 463 463 374 607 431 768 211 456	=	2128
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	=	2256

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 recognizable powers of two

• 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.

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