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Encyclopedia > 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. For other meanings of mathematics or math, see mathematics (disambiguation). ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... 2 (two) is a number, numeral, and glyph. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ...


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. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... BIT is an acronym for: Bangalore Institute of Technology Bilateral Investment Treaty Bhilai Institute of Technology - Durg Birla Institute of Technology - Mesra Battles in Time (Doctor Who magazine) Category: ... In computer science, the term integer is used to refer to any data type which can represent some subset of the mathematical integers. ... Namcos Pac-Man was a hit, and became a cultural phenomenon. ... A byte is commonly used as a unit of storage measurement in computers, regardless of the type of data being stored. ...


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). A kibibyte (a contraction of kilo binary byte) is a unit of information or computer storage, commonly abbreviated KiB (never kiB). 1 kibibyte = 210 bytes = 1,024 bytes The kibibyte is closely related to the kilobyte, which can be used either as a synonym for kibibyte or to refer to... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to commonly used values—typically, the values being in the midst of a calculation at a given point in time. ... In computing, word is a term for the natural unit of data used by a particular computer design. ...


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. Disk Drive is the afternoon show on CBC Radio 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). In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a Mersenne prime is a prime number that is one less than a prime power of two. ... 31 is the natural number following 30 and preceding 32. ...

Contents


The first forty-one powers of two

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

Look up one in Wiktionary, the free dictionary. ... 2 (two) is a number, numeral, and glyph. ... This article discusses the number Four. ... 8 (eight) is the natural number following 7 and preceding 9. ... 16 (sixteen) is the natural number following 15 and preceding 17. ... 32 (thirty-two) is the natural number following 31 and preceding 33. ... 64 (sixty-four) is the natural number following 63 and preceding 65. ... 128 is the natural number following 127 and preceding 129. ... 256 (two hundred [and] fifty-six, CCLVI) is the natural number following 255 and preceding 257. ... 1024 is the natural number following 1023 and preceding 1025. ...

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. 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

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. 2 (two) is a number, numeral, and glyph. ... This article discusses the number Four. ... 16 (sixteen) is the natural number following 15 and preceding 17. ... 256 (two hundred [and] fifty-six, CCLVI) is the natural number following 255 and preceding 257. ... In computer science, a datatype or data type (often simply a type) is a name or label for a set of values and some operations which one can perform on that set of values. ...


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.

Kilo (symbol: k) is a prefix in the SI system denoting 103 or 1,000. ... A kilobyte (derived from the SI prefix kilo-, meaning 1000) is a unit of information or computer storage equal to either 1024 or 1000 bytes. ... A kibibyte (a contraction of kilo binary byte) is a unit of information or computer storage, commonly abbreviated KiB (never kiB). 1 kibibyte = 210 bytes = 1,024 bytes The kibibyte is closely related to the kilobyte, which can be used either as a synonym for kibibyte or to refer to... Color is an important part of the visual arts. ... Truecolor (also spelled Truecolour; called Millions on a Macintosh) graphics is a method of storing image information in a computers memory such that each pixel is represented by three or more bytes. ... Nineteen inch (48 cm) CRT computer monitor A computer display, monitor or screen is a computer peripheral device capable of showing still or moving images generated by a computer and processed by a graphics card. ... The RGB color model utilizes the additive model in which red, green, and blue light are combined in various ways to create other colors. ...

Fast algorithm to check if a 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 integer x is a power of two: The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ...

x is a power of two (x & (x − 1)) equals zero.

where & is a bitwise logical AND operator. Note that zero is incorrectly considered a power of two by this test. Therefore a more thorough (but slightly slower) test would be: In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ...

x is a power of two (x > 0) and ((x & (x − 1)) == 0)

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

Fast 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: Plot of log2 x In mathematics, the binary logarithm (log2 n) is the logarithm for base 2. ...

It does not, however, find the nearest power of two with respect to the actual value. For example, 47 is nearer to 16 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:

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

  Results from FactBites:
 
Power of two - Wikipedia, the free encyclopedia (578 words)
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.
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.
Powers of two whose exponents are powers of two
Two Conceptions of Power (15565 words)
If power is roughly defined as the ability to make or establish a claim on life, then the range of the presence of power may be broadened to include the notion that power is coextensive with life itself.
When power is defined in a unilateral or linear fashion as a capacity to influence another, it follows factually as well as logically that the gain in power by the other is experienced as a loss of one’s own power and therefore of one’s status and sense of worth.
With respect to the problem of power in relation to human sexual differentiation, I am not concerned to defend either traditional or modern versions of the roles of men and women, or to deny or affirm their distinctive natures, regardless of whether these differences are understood to be inherent or culturally derived.
  More results at FactBites »

 
 

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