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Arithmetic operation
exponentiation, denoted bn, is an operation involving two numbers: the base, b, and the exponent, n. When n is a positive integer, exponentiation corresponds
Exponentiation
Algorithm for fast exponentiation
are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic
Exponentiation_by_squaring
Exponentation in modular arithmetic
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography
Modular_exponentiation
Performing order of mathematical operations
a property of exponentiation that (ab)c = abc, so it's unnecessary to use serial exponentiation for this. However, when exponentiation is represented
Order_of_operations
Arithmetic operation
tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no universal notation for tetration, though Knuth's up arrow
Tetration
Operations on ordinals that extend classical arithmetic
operations on ordinal numbers such as addition, multiplication, and exponentiation. Each can be defined in two different ways: either by constructing an
Ordinal_arithmetic
Matrix operation generalizing exponentiation of scalar numbers
multiplication, hence also exponentiation, of diagonal matrices is equivalent to element-wise addition and multiplication, and hence exponentiation; in particular
Matrix_exponential
Size of a possibly infinite set
if μ ≤ π. It will be unique (and equal to π) if and only if μ < π. Exponentiation is given by | X | | Y | = | X Y | , {\displaystyle |X|^{|Y|}=\left|X^{Y}\right|
Cardinal_number
Generalization of addition, multiplication, exponentiation, tetration, etc.
multiplication (n = 2), and exponentiation (n = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity
Hyperoperation
Method of exponentiation using a minimal number of multiplications
mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number
Addition-chain_exponentiation
Exponential function of an exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f ( x ) = a b x = a ( b x ) {\displaystyle
Double_exponential_function
Branch of elementary mathematics
subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can
Arithmetic
In general, exponentiation fails to be commutative
In general, exponentiation fails to be commutative. However, the equation x y = y x {\displaystyle x^{y}=y^{x}} has an infinity of solutions, consisting
Equation_xy_=_yx
Mathematical function, inverse of an exponential function
single-variable function, the logarithm to base b is the inverse of exponentiation with base b. The logarithm base 10 is called the decimal or common logarithm
Logarithm
Method of notation of very large integers
names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function
Knuth's_up-arrow_notation
Topics referred to by the same term
ciphers Modulo (mathematics) Modular arithmetic Modulo operation Modular exponentiation MOD., a science museum at the University of South Australia, Adelaide
Mod
Decimal unit prefix in the metric system
occur in exponentiation, such as in square and cubic forms, any multiplier prefix is part of the unit, and thus included in the exponentiation. 1 km2 means
Kilo-
Convex polytope, the n-dimensional analogue of a square and a cube
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract. It is
Hypercube
Mathematical fallacy
In mathematics, the freshman's dream, also known as freshman exponentiation, the child's binomial theorem, (rarely) the schoolboy binomial theorem, or
Freshman's_dream
Collection of mathematical objects
considered sets. These operations are Cartesian product, disjoint union, set exponentiation and power set. Given sets A {\displaystyle A} and B {\displaystyle
Set_(mathematics)
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 2x 2 4 8 16 32 64 128 256 512 1024 2048 4096 x2 1 4 9 16 25 36 49 64 81 100 121 144
2
Metric prefix
Tera- (/ˈtɛrə/; symbol T) is a metric prefix denoting a factor of a short-scale trillion or long-scale billion (1012 or 1000000000000). It was adopted
Tera-
Type of mathematical expression
involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An
Polynomial
Algorithm for public-key cryptography
private key. The modular exponentiation to the power of e is used in encryption and in verifying signatures, and exponentiation to the power of d is used
RSA_cryptosystem
Computation modulo a fixed integer
ak ≡ bk (mod m) for any non-negative integer k (compatibility with exponentiation) p(a) ≡ p(b) (mod m), for any polynomial p(x) with integer coefficients
Modular_arithmetic
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4x 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864 268435456 1073741824
4
Field of knowledge
2023-03-23. Retrieved 2022-11-19. Marker, Dave (Jul 1996). "Model theory and exponentiation". Notices of the American Mathematical Society. 43 (7): 753–759. Archived
Mathematics
Major unsolved problem in transcendental number theory
Wilkie, for example, proved that the theory of the real field with exponentiation, R {\displaystyle \mathbb {R} } exp, is decidable provided Schanuel's
Schanuel's_conjecture
Typographical mark (^)
The use of the caret for exponentiation can be traced back to ALGOL 60,[citation needed] which expressed the exponentiation operator as an upward-pointing
Caret
Metric prefix
in exponentiation, such as in square and cubic forms, any multiples-prefix is considered part of the unit, and thus included in the exponentiation. 1 Mm2
Mega-
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 10x 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000 x10 1 1024 59049 1048576 9765625 60466176
10
General-purpose programming language
//, and floating-point division /. Python uses the ** operator for exponentiation. Python uses the + operator for string concatenation. The language uses
Python_(programming_language)
Method for representing or encoding numbers
allowed digits for the given base.) Positional numeral systems work using exponentiation of the base. A digit's value is the digit multiplied by the value of
Positional_notation
Concept in the philosophy of mathematics
their objection to the totality of number theoretic functions like exponentiation over natural numbers. Like other finitists, ultrafinitists deny the
Ultrafinitism
Theorem on modular exponentiation
Theorem on modular exponentiation
Euler's_theorem
Hardware description language
Addition - Subtraction - 2's complement * Multiplication / Division ** Exponentiation (*Verilog-2001) Relational > Greater than < Less than >= Greater than
Verilog
Natural number
{\displaystyle \scriptstyle {{\text{10,000,000,000,000 }}\div {\text{ x}}}} Exponentiation 10,000,000,000,000 x {\displaystyle \scriptstyle {{\text{10,000,000
10,000,000,000,000
Theorem in axiomatic set theory
function is used for studying the continuum function and the cardinal exponentiation function. The symbol ℷ {\displaystyle \gimel } is a serif form of the
Gimel_function
Method of exchanging cryptographic keys
logarithm problem. The computation of ga mod p is known as modular exponentiation and can be done efficiently even for large numbers. Note that g need
Diffie–Hellman_key_exchange
Digital verification standard
for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem. In a digital signature system, there
Digital_Signature_Algorithm
Integer
complex numbers, the equation x2 = −1 has infinitely many solutions. Exponentiation of a non‐zero real number can be extended to negative integers, where
−1
Problem of inverting exponentiation in groups
Regardless of the specific algorithm used, this operation is called modular exponentiation. For example, consider Z17×. To compute 3 4 {\displaystyle 3^{4}} in
Discrete_logarithm
Measurement unit derived from basic metric value
more of the base units, possibly scaled by an appropriate power of exponentiation (see: Buckingham π theorem). Some are dimensionless, as when the units
SI_derived_unit
Mathematical field with an extra operation
group to its multiplicative group. This generalizes the usual idea of exponentiation on the real numbers, where the base is a chosen positive real number
Exponential_field
Used to count, measure, and label
familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer
Number
Property of a mathematical operation
operations are non-associative; some examples include subtraction, exponentiation, and the vector cross product. In contrast to the theoretical properties
Associative_property
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 11x 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937424601 285311670611 x11 1 2048 177147 4194304
11_(number)
Three raised to an integer power
number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent. The first
Power_of_three
Inequality about exponentiations of ''1+x''
inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x {\displaystyle 1+x} . It is often employed in real analysis
Bernoulli's_inequality
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 9x 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 x9 1 512 19683 262144 1953125 10077696 40353607
9
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 6x 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016 x6 1 64 729
6
Graphical symbol or pictogram used to point or indicate direction
notation uses multiple up arrows, such as ⇈, for iterated, or repeated, exponentiation (tetration). The quantum theory of electron spin uses either upward
Arrow_(symbol)
Mathematical problem
whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms
Tarski's high school algebra problem
Tarski's_high_school_algebra_problem
Mathematical function, denoted exp(x) or e^x
of the form f ( x ) = b x {\displaystyle f(x)=b^{x}} , which is exponentiation with a fixed base b {\displaystyle b} . More generally, and especially
Exponential_function
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3x 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 4782969 14348907 43046721
3
One of the four basic arithmetic operations
{\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.} Exponentiation base exponent base power } = {\displaystyle \scriptstyle \left
Subtraction
subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms, and the use of such operations
Outline_of_arithmetic
Numbers obtained by adding the two previous ones
matrix can be computed in O(log n) arithmetic operations, using the exponentiation by squaring method. Taking the determinant of both sides of this equation
Fibonacci_sequence
Quotient of two integers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction p q {\displaystyle {\tfrac {p}{q}}} of two integers
Rational_number
Study of matrices and their algebraic properties
operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices),
Matrix_analysis
Integer factorization algorithm
+ 1 contains only small factors. It uses Lucas sequences to perform exponentiation in a quadratic field. It is analogous to Pollard's p − 1 algorithm.
Williams's_p_+_1_algorithm
Whole number
number multiplied by 0 produces 1), a consequence of the previous rule. Exponentiation: x0 = x/x = 1, except that the case x = 0 is considered undefined
0
Topics referred to by the same term
object known as "the" hypercube Exponentiation for powers above 3 Fourth power, more narrowly for the specific exponentiation to the power of 4, also known
Hypercube_(disambiguation)
Quantum algorithm for integer factorization
j {\displaystyle U^{2^{j}}} . This can be accomplished via modular exponentiation, which is the slowest part of the algorithm. The gate thus defined satisfies
Shor's_algorithm
Type of transfinite numbers
numbers are those of the form ωωβ. The standard definition of ordinal exponentiation with base α is: α 0 = 1 , {\displaystyle \alpha ^{0}=1\,,} α β = α β
Epsilon_number
Hypercomplex number system
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented
Octonion
Digital circuit that produces sums from inputs
XOR Bit shifts Bit manipulation See also Kochanski multiplication (exponentiation) Multiply–accumulate operation Categories Category:Binary arithmetic
Adder_(electronics)
Recursive mathematical formula
function of n {\displaystyle n} defined by iterating the process of exponentiation, analogously to the way that the factorial can be defined by iterated
Exponential_factorial
Two raised to an integer power
number of the form 2n where n is an integer, that is, the result of exponentiation with the number two as the base and integer n as the exponent. In the
Power_of_two
Typographical symbol (™)
baseline TM, the letters written as superscripts, as in mathematical exponentiation ᵀᴹ, using symbols from the Phonetic Extensions block in Unicode Look
Trademark_symbol
Random process in probability theory
A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of
Compound_Poisson_process
Proposition in mathematical logic
directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation ℵ α ℵ β {\displaystyle \aleph
Continuum_hypothesis
Number ten to the power of a googol
written as 1010100 using the conventional interpretation for serial exponentiation. A typical book can be printed with one million zeros (around 400 pages
Googolplex
Arithmetic logic circuit
XOR Bit shifts Bit manipulation See also Kochanski multiplication (exponentiation) Multiply–accumulate operation Categories Category:Binary arithmetic
Carry-lookahead_adder
Decomposition of a number into a product
algorithms Chakravala Cornacchia Exponentiation by squaring Integer square root Integer relation (LLL; KZ) Modular exponentiation Montgomery reduction Schoof
Integer_factorization
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5x 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 244140625 1220703125 6103515625 30517578125
5
Natural number
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 12x 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256 x12
12_(number)
Arithmetical operation
all factors are identical, a product of n factors is equivalent to exponentiation: ∏ i = 1 n x = x ⋅ x ⋅ … ⋅ x = x n . {\displaystyle \prod _{i=1}^{n}x=x\cdot
Multiplication
3-volume treatise on mathematics, 1910–1913
cardinals of different types. PM define addition, multiplication and exponentiation of cardinals, and compare different definitions of finite and infinite
Principia_Mathematica
Algorithm for factoring polynomials over finite fields
fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by David G. Cantor
Cantor–Zassenhaus_algorithm
Modern form of the metric system
'm' and the 's' are lowercase because neither the metre nor the second are named after people, and exponentiation is represented with a superscript '2'.
International_System_of_Units
Combinational digital circuit
XOR Bit shifts Bit manipulation See also Kochanski multiplication (exponentiation) Multiply–accumulate operation Categories Category:Binary arithmetic
Arithmetic_logic_unit
Arithmetic notation system
Cutler in 2004. The idea is based on iterated exponentiation in much the same way that exponentiation is iterated multiplication. A regular exponential
Cutler's_bar_notation
Elementary functions and their finitely iterated integrals
algebraic operations ( addition, subtraction, multiplication, division, exponentiation, and root extraction) and antiderivatives. The logarithmic function
Liouvillian_function
Complex exponential in terms of sine and cosine
also use the definition of the logarithm (as the inverse operator of exponentiation): a = e ln a , {\displaystyle a=e^{\ln a},} and that e a e b = e a
Euler's_formula
Mathematical symbols (+ and −)
the rules for the order of operations mean that −52 is equal to −25: Exponentiation binds more strongly than the unary minus, which binds more strongly
Plus_and_minus_signs
Type of mathematical equation
matrix exponential of a diagonal matrix is the same as element-wise exponentiation of its elements) exp ( [ 3 − 4 4 − 7 ] t ) = [ 4 e t / 3 − e − 5 t
Matrix_differential_equation
Probability of shared birthdays
In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday
Birthday_problem
Greek mathematician and physicist (c. 287 – 212 BC)
and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply
Archimedes
Set of rules defining correctly structured programs
x : y Math.min(1, -2) −2 Minimum: (x < y) ? x : y Math.pow(-3, 2) 9 Exponentiation (raised to the power of): Math.pow(x, y) gives xy Math.random() e.g
JavaScript_syntax
Mathematical concept
3, …>. 0.999... Absolute infinite Aleph number Ananta Apeirophobia Exponentiation Indeterminate form Infinite monkey theorem Names of large numbers Paradoxes
Infinity
Topics referred to by the same term
Legio XX Valeria Victrix, a legion of the Roman Empire xx, to express exponentiation in the initial version of the FORTRAN programming language .xx ("dot
XX
Approach to public-key cryptography
provide equivalent security, compared to cryptosystems based on modular exponentiation in finite fields, such as the RSA cryptosystem and ElGamal cryptosystem
Elliptic-curve_cryptography
Large number coined by Ronald Graham
enormous size of Graham's number, it may be helpful to express—in terms of exponentiation alone—just the first term (g1) of the rapidly growing 64-term sequence
Graham's_number
Topics referred to by the same term
Exponential may refer to any of several mathematical topics related to exponentiation, including: Exponential function, also: Matrix exponential, the matrix
Exponential
x} . In particular, tetration can be interpreted as superfunction of exponentiation for some real base b {\displaystyle b} ; in this case, f = exp b . {\displaystyle
Superfunction
Topics referred to by the same term
** may refer to: **, to express exponentiation in some programming languages **, a pointer to a pointer (or double pointer) in C syntax **, interpolation
**
Certain large countable ordinal
{\displaystyle \operatorname {LVO} ^{\operatorname {LVO} }} (ordinal exponentiation) and Ω {\displaystyle \Omega } is used as a place holder for LVO. Notice
Large_Veblen_ordinal
Computing circuit
XOR Bit shifts Bit manipulation See also Kochanski multiplication (exponentiation) Multiply–accumulate operation Categories Category:Binary arithmetic
Adder–subtractor
Probabilistic primality test
1{\pmod {p}}} , because the congruence relation is compatible with exponentiation. It also holds trivially for a ≡ − 1 ( mod p ) {\displaystyle a\equiv
Fermat_primality_test
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