A power of two is a number of the form where is an
integer, that is, the result of
exponentiation 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 cultu ...
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,
128 128 may refer to
* 128 (number), a natural number
* AD 128, a year in the 2nd century AD
* 128 BC, a year in the 2nd century BC
* 128 (New Jersey bus)
See also
* List of highways numbered
*
* 12/8 (disambiguation)
{{numberdis ...
,
256
Year 256 ( CCLVI) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Claudius and Glabrio (or, less frequently, year 1009 ''Ab urbe condi ...
,
512
__NOTOC__
Year 512 ( DXII) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Paulus and Moschianus (or, less frequently, year ...
, ...
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 notation ...
, 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
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 fi ...
in the
decimal 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 represented ...
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, 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 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 uni ...
, 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-release ...
'' 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 en ...
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 com ...
), 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 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 of ...
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, , and . Put another way, they have fairly regular bit patterns.
Mersenne and Fermat primes
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 (2
5). 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
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 number
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite... The impolite numbers are exactly the powers of two, an ...
s; 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 notation ...
, 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 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. 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 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 ord ...
of 2 modulo , which is (see
Multiplicative group of integers modulo n).
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 r ...
and
lower hyperoperations. For more about representing signed numbers see
two's complement.
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 2-powers''.
The numbers
form an
irrationality sequence: 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 represented ...
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 uni ...
, 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 com ...
. (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 uni ...
is often defined as a
collection of bits rather than the strict definition of an 8-bit quantity, as demonstrated by the term
kilobyte.)
; 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 wor ...
, 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 uni ...
s = 1
kilobyte (or
kibibyte).
; 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 ma ...
size of an
Intel x86-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 on a
16-bit
16-bit microcomputers are microcomputers that use 16-bit microprocessors.
A 16-bit register can store 216 different values. The range of integer values that can be stored in 16 bits depends on the integer representation used. With the two mos ...
processor, such as the original
x86 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 ar ...
variable in the
C#, and
Java programming languages. The maximum range of a Word or Smallint variable in the
Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Frenc ...
programming language.
: The number of
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
s on a 4-element set.
; 2
20 = 1,048,576
: The binary approximation of the
mega-
Mega is a unit prefix in metric systems of units denoting a factor of one million (106 or ). It has the unit symbol M. It was confirmed for use in the International System of Units (SI) in 1960. ''Mega'' comes from grc, μέγας, mégas, grea ...
, 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 uni ...
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 associa ...
s that can be displayed in
truecolor, which is used by common
computer monitor
A computer monitor is an output device that displays information in pictorial or textual form. A discrete monitor comprises a visual display, support electronics, power supply, housing, electrical connectors, and external user controls.
The d ...
s.
: 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 additi ...
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
Notable 24-bit machines include the CDC 924 – a 24-bit version of the CDC 1604, CDC lower 3000 series, SDS 930 and SDS 940, the ICT 1900 series, the Elliott 4100 series, and the Datacraft minicomputers/ Harris H series.
The term SWORD is ...
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-, 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 uni ...
s = 1
gigabyte
The gigabyte () is a multiple of the unit byte for digital information. The prefix '' giga'' means 109 in the International System of Units (SI). Therefore, one gigabyte is one billion bytes. The unit symbol for the gigabyte is GB.
This definit ...
(or
gibibyte).
; 2
31 = 2,147,483,648
: The number of non-negative values for a ''signed'' 32-bit integer. Since
Unix time 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.
; 2
32 = 4,294,967,296
: The number of distinct values representable in a single
word on a
32-bit
In computer architecture, 32-bit computing refers to computer systems with a processor, memory, and other major system components that operate on data in 32-bit units. Compared to smaller bit widths, 32-bit computers can perform large calculati ...
processor. Or, the number of values representable in a
doubleword
In computer science, an integer is a data, datum of integral data type, a data type that represents some interval (mathematics), range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain ...
on a
16-bit
16-bit microcomputers are microcomputers that use 16-bit microprocessors.
A 16-bit register can store 216 different values. The range of integer values that can be stored in 16 bits depends on the integer representation used. With the two mos ...
processor, such as the original
x86 processors.
: The range of an
int
variable in the
Java and
C# programming languages.
: The range of a
Cardinal
or
Integer
variable in the
Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Frenc ...
programming language.
: The minimum range of a
long integer variable in the
C and
C++
C, or c, is the third letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''.
History
"C" ...
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 addresses). See:
IPv4 address exhaustion.
: 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-, 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 uni ...
s = 1
terabyte 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 uni ...
s = 1
petabyte 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-
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 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 uni ...
s = 1
exabyte 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 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 comput ...
processor. Or, the number of values representable in a
doubleword
In computer science, an integer is a data, datum of integral data type, a data type that represents some interval (mathematics), range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain ...
on a
32-bit
In computer architecture, 32-bit computing refers to computer systems with a processor, memory, and other major system components that operate on data in 32-bit units. Compared to smaller bit widths, 32-bit computers can perform large calculati ...
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 ar ...
on a
16-bit
16-bit microcomputers are microcomputers that use 16-bit microprocessors.
A 16-bit register can store 216 different values. The range of integer values that can be stored in 16 bits depends on the integer representation used. With the two mos ...
processor, such as the original
x86 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 mensu ...
variable in the
Java and
C# programming languages.
: The range of a Int64 or QWord variable in the
Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Frenc ...
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 pac ...
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.
; 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-, 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 uni ...
s = 1
zettabyte (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-, 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 uni ...
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
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
d 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 pac ...
es generally given to a
local Internet registry. In
CIDR 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
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 identi ...
available under
IPv6. 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, and hence the maximum number that can be represented by many programs, for example
Microsoft Excel.
; 2
82,589,933 =
: One more than the
largest known prime number . It has 24,862,047 digits.
Other properties
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 te ...
s is equal to . Consider the set of all -digit binary integers. Its
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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'' � ...
s.
The number of
vertices of an -dimensional
hypercube is . Similarly, the number of -faces of an -dimensional
cross-polytope
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
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, ''a''
''n'' + ''b''
''n'' is
irreducible, 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 value
In music notation, a note value indicates the relative duration of a note, using the texture or shape of the '' notehead'', the presence or absence of a ''stem'', and the presence or absence of ''flags/ beams/hooks/tails''. Unmodified note value ...
s 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
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
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.
See also
*
2048 (video game)
''2048'' is a single-player sliding tile puzzle video game written by Italian web developer Gabriele Cirulli and published on GitHub. The objective of the game is to slide numbered tiles on a grid to combine them to create a tile with the numbe ...
*
Binary number
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 notation ...
*
Fermi–Dirac prime
*
Geometric progression
*
Gould's sequence
*
Inchworm Song
*
Integer binary logarithm
*
Octave (electronics)
In electronics, an octave (symbol: oct) is a logarithmic unit for ratios between frequencies, with one octave corresponding to a doubling of frequency. For example, the frequency one octave above 40 Hz is 80 Hz. The term is derived fro ...
*
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 fi ...
*
Power of three
In mathematics, a power of three is a number of the form where is an integer – that is, the result of exponentiation with number three as the base and integer as the exponent.
Applications
The powers of three give the place values in ...
*
Sum-free sequence In mathematics, a sum-free sequence is an increasing sequence of positive integers,
:a_1, a_2, a_3, \ldots,
such that no term a_n can be represented as a sum of any subset of the preceding elements of the sequence.
This differs from a sum-free ...
References
{{DEFAULTSORT:Power Of Two
Binary arithmetic
Integer sequences
Integers