A power of two is a number of the form where is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, that is, the result of
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
with number
two
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultur ...
as the
base and integer as the
exponent.
In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2
multiplied by itself a certain number of times.
The first ten powers of 2 for non-negative values of are:
:
1,
2,
4,
8,
16,
32,
64,
128,
256,
512, ...
Because two is the base of the
binary numeral system
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
, powers of two are common in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a
power of 10
A power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The f ...
in the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system.
Computer science
Two to the exponent of , written as , is the number of ways the
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s in a
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
word of length can be arranged. A word, interpreted as an unsigned
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, can represent values from 0 () to () inclusively. Corresponding
signed integer values can be positive, negative and zero; see
signed number representations
In computing, signed number representations are required to encode negative numbers in binary number systems.
In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU regis ...
. 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
Video games, also known as computer games, are electronic games that involves interaction with a user interface or input device such as a joystick, controller, keyboard, or motion sensing device to generate visual feedback. This fee ...
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
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
, which is
8 bits long, to store the number, giving a maximum value of . For example, in the original ''
Legend of Zelda
''The Legend of Zelda'' is an action-adventure game franchise created by the Japanese game designers Shigeru Miyamoto and Takashi Tezuka. It is primarily developed and published by Nintendo, although some portable installments and re-rele ...
'' 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
''Kill Screen'' (stylized as ''KILL SCREEN'') was a print and online magazine founded in 2009 by Jamin Warren and Chris Dahlen and owned by Kill Screen Media, Inc. It focused on video games and culture, but also included articles based on e ...
at level 256.
Powers of two are often used to measure computer memory. A byte is now considered eight bits (an
octet
Octet may refer to:
Music
* Octet (music), ensemble consisting of eight instruments or voices, or composition written for such an ensemble
** String octet, a piece of music written for eight string instruments
*** Octet (Mendelssohn), 1825 compos ...
), resulting in the possibility of 256 values (2
8). (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 (2
10). However, in general, the term ''kilo'' has been used in the
International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
to mean 1,000 (10
3).
Binary prefixes
A binary prefix is a unit prefix for multiples of units. It is most often used in data processing, data transmission, and digital information, principally in association with the bit and the byte, to indicate multiplication by a power o ...
have been standardized, such as ''kibi'' (Ki) meaning 1,024. Nearly all
processor register
A processor register is a quickly accessible location available to a computer's processor. Registers usually consist of a small amount of fast storage, although some registers have specific hardware functions, and may be read-only or write-only. ...
s 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 drive
Disk storage (also sometimes called drive storage) is a general category of storage mechanisms where data is recorded by various electronic, magnetic, optical, or mechanical changes to a surface layer of one or more rotating disks. A disk drive is ...
s, 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, , and . Put another way, they have fairly regular bit patterns.
Mersenne and Fermat primes
A
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
that is one less than a power of two is called a
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
. For example, the prime number
31 is a Mersenne prime because it is 1 less than 32 (2
5). Similarly, a prime number (like
257) that is one more than a positive power of two is called a
Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 429496 ...
—the exponent itself is a power of two. A
fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
that has a power of two as its
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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
The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the
binary numeral system
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
, 1, 10, 100, 1000, 10000, 100000, ... ) is important in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
. Book IX, Proposition 36 of
''Elements'' proves that if the sum of the first terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the th term is a
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
. 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 divides 496 and it is not amongst these numbers. Assume is equal to , or 31 is to as is to 16. Now cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or 16.
Therefore, 31 cannot divide . And since 31 does not divide and measures 496, the
fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
implies that must divide 16 and be amongst the numbers 1, 2, 4, 8 or 16. Let be 4, then must be 124, which is impossible since by hypothesis is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248.
Table of values
Starting with 2 the last digit is periodic with period 4, with the cycle 2–4–8–6–, and starting with 4 the last two digits are periodic with period 20. These patterns are generally true of any power, with respect to any
base. The pattern continues where each pattern has starting point , and the period is the
multiplicative order In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n.
In other words, the multiplicative order ...
of 2 modulo , which is (see
Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to ''n'' from the set \ of ''n'' non-negative integers form a group under multiplication modulo ''n'', called the multiplicative group of integers modulo ''n''. Equivalently, the ele ...
).
Powers of 1024
The first few powers of 2
10 are slightly larger than those same powers of 1000 (10
3):
Powers of two whose exponents are powers of two
Because data (specifically integers) and the addresses of data are stored using the same hardware, and the data is stored in one or more octets (),
double exponentials of two are common. For example,
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 distinct values, which can either be regarded as mere bit-patterns, or are more commonly interpreted as the unsigned numbers from 0 to , or as the range of signed numbers between and . Also see
tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
and
lower hyperoperations. For more about representing signed numbers see
two's complement
Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
.
In a connection with
nimber
In mathematics, the nimbers, also called ''Grundy numbers'', are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ' ...
s, these numbers are often called ''
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...
2-powers''.
The numbers
form an
irrationality sequence
In mathematics, a sequence of positive integers ''a'n'' is called an irrationality sequence if it has the property that for every sequence ''x'n'' of positive integers, the sum of the series
: \sum_^\infty \frac
exists (that is, it conve ...
: for every sequence
of
positive integers, the
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
:
converges to an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
. Despite the rapid growth of this sequence, it is the slowest-growing irrationality sequence known.
Selected powers of two
; 2
8 = 256
:The number of values represented by the 8
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s in a
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
, more specifically termed as an
octet
Octet may refer to:
Music
* Octet (music), ensemble consisting of eight instruments or voices, or composition written for such an ensemble
** String octet, a piece of music written for eight string instruments
*** Octet (Mendelssohn), 1825 compos ...
. (The term
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
is often defined as a
collection of bits rather than the strict definition of an 8-bit quantity, as demonstrated by the term
kilobyte
The kilobyte is a multiple of the unit byte for digital information.
The International System of Units (SI) defines the prefix '' kilo'' as 1000 (103); per this definition, one kilobyte is 1000 bytes.International Standard IEC 80000-13 Quant ...
.)
; 2
10 = 1,024
: The binary approximation of the
kilo-
Kilo is a decimal unit prefix in the metric system denoting multiplication by one thousand (103). It is used in the International System of Units, where it has the symbol k, in lowercase.
The prefix ''kilo'' is derived from the Greek word () ...
, or 1,000 multiplier, which causes a change of prefix. For example: 1,024
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
kilobyte
The kilobyte is a multiple of the unit byte for digital information.
The International System of Units (SI) defines the prefix '' kilo'' as 1000 (103); per this definition, one kilobyte is 1000 bytes.International Standard IEC 80000-13 Quant ...
(or
kibibyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
).
; 2
12 = 4,096
: The hardware
page
Page most commonly refers to:
* Page (paper), one side of a leaf of paper, as in a book
Page, PAGE, pages, or paging may also refer to:
Roles
* Page (assistance occupation), a professional occupation
* Page (servant), traditionally a young m ...
size of an
Intel x86
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was intr ...
-compatible processor.
; 2
15 = 32,768
: The number of non-negative values for a ''signed'' 16-bit integer.
; 2
16 = 65,536
: The number of distinct values representable in a single
word
A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
on a
16-bit processor, such as the original
x86
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was intr ...
processors.
[Though they vary in word size, all x86 processors use the term "word" to mean 16 bits; thus, a 32-bit x86 processor refers to its native wordsize as a dword]
: The maximum range of a
short integer
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are ...
variable in the
C#, and
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
programming languages. The maximum range of a Word or Smallint variable in the
Pascal programming language.
: The number of
binary relations on a 4-element set.
; 2
20 = 1,048,576
: The binary approximation of the
mega-
Mega is a metric prefix, unit prefix in metric systems of units denoting a factor of one million (106 or 1000000 (number), ). It has the unit symbol M. It was confirmed for use in the International System of Units (SI) in 1960. ''Mega'' comes fro ...
, or 1,000,000 multiplier, which causes a change of prefix. For example: 1,048,576
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
megabyte (or
mebibyte).
; 2
24 = 16,777,216
: The number of unique
color
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...
s that can be displayed in
truecolor, which is used by common
computer monitors.
: This number is the result of using the three-channel
RGB
The RGB color model is an additive color model in which the red, green and blue primary colors of light are added together in various ways to reproduce a broad array of colors. The name of the model comes from the initials of the three addi ...
system, with 8 bits for each channel, or 24 bits in total.
: The size of the largest unsigned integer or address in computers with
24-bit registers or data buses.
; 2
29 = 536,870,912
: The largest power of two with distinct digits in base ten.
; 2
30 = 1,073,741,824
: The binary approximation of the
giga-
Giga ( or ) is a unit prefix in the metric system denoting a factor of a short-scale billion or long-scale milliard (109 or ). It has the symbol G.
''Giga'' is derived from the Greek word (''gÃgas''), meaning "giant". The ''Oxford English Dic ...
, or 1,000,000,000 multiplier, which causes a change of prefix. For example, 1,073,741,824
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
gigabyte (or
gibibyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
).
; 2
31 = 2,147,483,648
: The number of non-negative values for a ''signed'' 32-bit integer. Since
Unix time
Current Unix time ()
Unix time is a date and time representation widely used in computing. It measures time by the number of seconds that have elapsed since 00:00:00 UTC on 1 January 1970, the beginning of the Unix epoch, less adjustments m ...
is measured in seconds since January 1, 1970, it will run out at 2,147,483,647 seconds or 03:14:07 UTC on Tuesday, 19 January 2038 on 32-bit computers running Unix, a problem known as the
year 2038 problem
The year 2038 problem (also known as Y2038, Y2K38, or the Epochalypse) is a time formatting bug in computer systems with representing times after 03:14:07 UTC on 19 January 2038.
The problem exists in systems which measure Unix time â ...
.
; 2
32 = 4,294,967,296
: The number of distinct values representable in a single
word
A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
on a
32-bit processor. Or, the number of values representable in a
doubleword
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are ...
on a
16-bit processor, such as the original
x86
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was intr ...
processors.
: The range of an
int
variable in the
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
and
C# programming languages.
: The range of a
Cardinal
or
Integer
variable in the
Pascal programming language.
: The minimum range of a
long integer variable in the
C and
C++
C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
programming languages.
: The total number of
IP address
An Internet Protocol address (IP address) is a numerical label such as that is connected to a computer network that uses the Internet Protocol for communication.. Updated by . An IP address serves two main functions: network interface ident ...
es under
IPv4. Although this is a seemingly large number, we've already run out of numbers for IPv4 addresses (but not for
IPv6
Internet Protocol version 6 (IPv6) is the most recent version of the Internet Protocol (IP), the communication protocol, communications protocol that provides an identification and location system for computers on networks and routes traffic ...
addresses). See:
IPv4 address exhaustion
IPv4 address exhaustion is the depletion of the pool of unallocated IPv4 addresses. Because the original Internet architecture had fewer than 4.3 billion addresses available, depletion has been anticipated since the late 1980s, when the Interne ...
.
: The number of
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s with domain equal to any 4-element set, such as
GF(4).
; 2
40 = 1,099,511,627,776
: The binary approximation of the
tera-
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pr ...
, or 1,000,000,000,000 multiplier, which causes a change of prefix. For example, 1,099,511,627,776
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
terabyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
or tebibyte.
; 2
50 = 1,125,899,906,842,624
: The binary approximation of the
peta-
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...
, or 1,000,000,000,000,000 multiplier. 1,125,899,906,842,624
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
petabyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
or pebibyte.
; 2
53 = 9,007,199,254,740,992
: The number until which all integer values can exactly be represented in IEEE
double precision floating-point format
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
Flo ...
. Also the first power of 2 to start with the digit 9 in decimal.
; 2
56 = 72,057,594,037,927,936
: The number of different possible keys in the obsolete 56 bit
DES
Des is a masculine given name, mostly a short form (hypocorism) of Desmond. People named Des include:
People
* Des Buckingham, English football manager
* Des Corcoran, (1928–2004), Australian politician
* Des Dillon (disambiguation), sever ...
symmetric cipher.
; 2
60 = 1,152,921,504,606,846,976
: The binary approximation of the
exa-, or 1,000,000,000,000,000,000 multiplier. 1,152,921,504,606,846,976
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
exabyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
or exbibyte.
; 2
63 = 9,223,372,036,854,775,808
: The number of non-negative values for a signed 64-bit integer.
: 2
63 − 1, a common maximum value (equivalently the number of positive values) for a signed 64-bit integer in programming languages.
; 2
64 = 18,446,744,073,709,551,616
: The number of distinct values representable in a single
word
A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
on a
64-bit
In computer architecture, 64-bit integers, memory addresses, or other data units are those that are 64 bits wide. Also, 64-bit CPUs and ALUs are those that are based on processor registers, address buses, or data buses of that size. A compu ...
processor. Or, the number of values representable in a
doubleword
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are ...
on a
32-bit processor. Or, the number of values representable in a
quadword
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are ...
on a
16-bit processor, such as the original
x86
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was intr ...
processors.
: The range of a
long
Long may refer to:
Measurement
* Long, characteristic of something of great duration
* Long, characteristic of something of great length
* Longitude (abbreviation: long.), a geographic coordinate
* Longa (music), note value in early music mens ...
variable in the
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
and
C# programming languages.
: The range of a Int64 or QWord variable in the
Pascal programming language.
: The total number of
IPv6 address
An Internet Protocol Version 6 address (IPv6 address) is a numeric label that is used to identify and locate a network interface of a computer or a network node participating in a computer network using IPv6. IP addresses are included in the ...
es generally given to a single LAN or subnet.
: 2
64 − 1, the number of grains of rice on a chessboard,
according to the old story, where the first square contains one grain of rice and each succeeding square twice as many as the previous square. For this reason the number is sometimes known as the "chess number".
: 2
64 − 1 is also the number of moves required to complete the legendary 64-disk version of the
Tower of Hanoi
The Tower of Hanoi (also called The problem of Benares Temple or Tower of Brahma or Lucas' Tower and sometimes pluralized as Towers, or simply pyramid puzzle) is a mathematical game or puzzle consisting of three rods and a number of disks of v ...
.
; 2
68 = 295,147,905,179,352,825,856
: The first power of 2 to contain all decimal digits.
; 2
70 = 1,180,591,620,717,411,303,424
: The binary approximation of the
zetta-
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...
, or 1,000,000,000,000,000,000,000 multiplier. 1,180,591,620,717,411,303,424
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
zettabyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
(or
zebibyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
).
; 2
80 = 1,208,925,819,614,629,174,706,176
: The binary approximation of the
yotta-
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...
, or 1,000,000,000,000,000,000,000,000 multiplier. 1,208,925,819,614,629,174,706,176
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s = 1
yottabyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
(or
yobibyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
).
; 2
86 = 77,371,252,455,336,267,181,195,264
: 2
86 is
conjectured to be the largest power of two not containing a zero in decimal.
; 2
96 = 79,228,162,514,264,337,593,543,950,336
: The total number of
IPv6 address
An Internet Protocol Version 6 address (IPv6 address) is a numeric label that is used to identify and locate a network interface of a computer or a network node participating in a computer network using IPv6. IP addresses are included in the ...
es generally given to a
local Internet registry
A regional Internet registry (RIR) is an organization that manages the allocation and registration of Internet number resources within a region of the world. Internet number resources include IP addresses and autonomous system (AS) numbers.
...
. In
CIDR
Classless Inter-Domain Routing (CIDR ) is a method for allocating IP addresses and for IP routing. The Internet Engineering Task Force introduced CIDR in 1993 to replace the previous classful network addressing architecture on the Internet. Its g ...
notation, ISPs are given a , which means that 128-32=96 bits are available for addresses (as opposed to network designation). Thus, 2
96 addresses.
; 2
108 = 324,518,553,658,426,726,783,156,020,576,256
: The largest known power of 2 not containing a 9 in decimal.
; 2
126 = 85,070,591,730,234,615,865,843,651,857,942,052,864
: The largest known power of 2 not containing a pair of consecutive equal digits.
; 2
128 =
: The total number of
IP addresses available under
IPv6
Internet Protocol version 6 (IPv6) is the most recent version of the Internet Protocol (IP), the communication protocol, communications protocol that provides an identification and location system for computers on networks and routes traffic ...
. Also the number of distinct
universally unique identifiers (UUIDs).
; 2
168 =
: The largest known power of 2 not containing all decimal digits (the digit 2 is missing in this case).
; 2
192 =
: The total number of different possible keys in the
AES 192-bit
key space (symmetric cipher).
; 2
229 = 862,718,293,348,820,473,429,344,482,784,628,181,556,388,621,521,298,319,395,315,527,974,912
: 2
229 is the largest known power of two containing the least number of zeros relative to its power. It is conjectured by Metin Sariyar that every digit 0 to 9 is inclined to appear an equal number of times in the decimal expansion of power of two as the power increases.
; 2
256 =
: The total number of different possible keys in the
AES 256-bit
key space (symmetric cipher).
; 2
333 =
: The smallest power of 2 greater than a
googol (10
100).
; 2
1024 =
: The maximum number that can fit in an IEEE
double-precision floating-point format
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
Fl ...
, and hence the maximum number that can be represented by many programs, for example
Microsoft Excel
Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
.
; 2
82,589,933 =
: One more than the
largest known prime number
The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.
A prime number is a posi ...
. It has 24,862,047 digits.
Other properties
![sum_of_powers_of_two_including_fractions](https://upload.wikimedia.org/wikipedia/commons/2/20/Sum_of_powers_of_two_including_fractions.svg)
The sum of all -choose
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s is equal to . Consider the set of all -digit binary integers. Its
cardinality is . It is also the sums of the cardinalities of certain subsets: the subset of integers with no 1s (consisting of a single number, written as 0s), the subset with a single 1, the subset with two 1s, and so on up to the subset with 1s (consisting of the number written as 1s). Each of these is in turn equal to the binomial coefficient indexed by and the number of 1s being considered (for example, there are 10-choose-3 binary numbers with ten digits that include exactly three 1s).
Currently, powers of two are the only known
almost perfect number
In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ''n'' such that the sum of all divisors of ''n'' (the sum-of-divisors function ''σ''(''n'')) is equal to 2''n'' â ...
s.
The number of
vertices of an -dimensional
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
is . Similarly, the number of -faces of an -dimensional
cross-polytope is also and the formula for the number of -faces an -dimensional cross-polytope has is
The
sum of the reciprocals of the powers of two is
1. The
sum of the reciprocals of the squared powers of two (powers of four) is 1/3.
The smallest natural power of two whose
decimal representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
begins with 7 is
:
Every power of 2 (excluding 1) can be written as the
sum of four square numbers in 24 ways. The powers of 2 are the natural numbers greater than 1 that can be written as the sum of four square numbers in the fewest ways.
As a real
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
, ''a''
''n'' + ''b''
''n'' is
irreducible
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
, if and only if ''n'' is a power of two. (If ''n'' is odd, then ''a''
''n'' + ''b''
''n'' is divisible by ''a''+''n'', and if ''n'' is even but not a power of 2, then ''n'' can be written as ''n''=''mp'', where ''m'' is odd, and thus
, which is divisible by ''a''
''p'' + ''b''
''p''.)
But in the domain of
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, the polynomial
(where ''n''>=1) can always be factorized as
,
even if ''n'' is a power of two.
Powers of two in music theory
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 note, Dotted or otherwise modified notes have other durations. In time signatures the lower numeral, the ''beat unit'', which can be seen as the
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
of a fraction, is almost always a power of two.
If the ratio of frequency, frequencies of two pitches is a power of two, then the interval (music), interval between those pitches is full octaves. In this case, the corresponding notes have the same name.
See also
* 2048 (video game)
* Binary number
* Fermi–Dirac prime
* Geometric progression
* Gould's sequence
* Inchworm Song
* Binary logarithm#Integer rounding, Integer binary logarithm
* Octave (electronics)
* Power of 10
* Power of three
* Sum-free sequence
References
{{DEFAULTSORT:Power Of Two
Binary arithmetic
Integer sequences
Integers