Zero to the power of zero, denoted by , is a
mathematical expression
In mathematics, an expression or mathematical expression is a finite combination of Glossary of mathematical symbols, symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can desi ...
that is either defined as 1 or left
undefined
Undefined may refer to:
Mathematics
* Undefined (mathematics), with several related meanings
** Indeterminate form, in calculus
Computing
* Undefined behavior, computer code whose behavior is not specified under certain conditions
* Undefined ...
, depending on context. In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
and
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, one typically defines . In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the expression is sometimes left undefined.
Computer programming languages and software also have
differing ways of handling this expression.
Discrete exponents
Many widely used formulas involving
natural-number exponents require to be defined as . For example, the following three interpretations of make just as much sense for as they do for positive integers :
* The interpretation of as an
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
assigns it the value .
* The
combinatorial interpretation of is the number of
0-tuples of elements from a -element set; there is exactly one 0-tuple.
* The
set-theoretic interpretation of is the number of
functions from the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
to a -element set; there is exactly one such function, namely, the
empty function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
.
All three of these specialize to give .
Polynomials and power series
When evaluating
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 exa ...
s, it is convenient to define as . A (real) polynomial is an expression of the form , where is an indeterminate, and the coefficients are
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. Polynomials are added termwise, and multiplied by applying the
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
and the usual rules for exponents. With these operations, polynomials form a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
. The
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of is the polynomial ; that is, times any polynomial is just .
Also, polynomials can be evaluated by specializing to a real number. More precisely, for any given real number , there is a unique unital
-algebra homomorphism such that . Because is unital, . That is, for each real number , including 0. The same argument applies with replaced by any
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
Defining is necessary for many polynomial identities. For example, the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
holds for only if .
Similarly, rings of
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
require to be defined as 1 for all specializations of . For example, identities like and hold for only if .
In order for the polynomial to define a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
, one must define .
In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, the
power rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated usin ...
is valid for at only if .
Continuous exponents
Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an
indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this su ...
.
The expression is an indeterminate form: Given real-valued functions and approaching (as approaches a real number or ) with , the limit of can be any non-negative real number or , or it can
diverge, depending on and . For example, each limit below involves a function with as (a
one-sided limit
In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right.
The limit as x decreases in value approaching a (x approaches ...
), but their values are different:
Thus, the two-variable function , though continuous on the set ,
cannot be extended to a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
on , no matter how one chooses to define .
On the other hand, if and are
analytic functions
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
on an open neighborhood of a number , then as approaches from any side on which is positive.
This and more general results can be obtained by studying the limiting behavior of the function .
Complex exponents
In the
complex domain
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 function may be defined for nonzero by choosing a
branch
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk (botany), trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' ...
of and defining as . This does not define since there is no branch of defined at , let alone in a neighborhood of .
History
As a value
In 1752,
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in ''
Introductio in analysin infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...
'' wrote that
and explicitly mentioned that .
An annotation attributed
to
Mascheroni in a 1787 edition of Euler's book ''
Institutiones calculi differentialis
''Institutiones calculi differentialis'' (''Foundations of differential calculus'') is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus. It consists of a single volu ...
''
offered the "justification"
as well as another more involved justification. In the 1830s,
Libri published several further arguments attempting to justify the claim , though these were far from convincing, even by standards of rigor at the time.
As a limiting form
Euler, when setting , mentioned that consequently the values of the function take a "huge jump", from for , to at , to for .
In 1814,
Pfaff
PFAFF (german: PFAFF Industriesysteme und Maschinen AG, PFAFF Industrial) is a German manufacturer of sewing machines and is now owned by the SGSB Co. Ltd.
History
PFAFF was founded in Kaiserslautern, Germany, in 1862 by instrument maker Geo ...
used a
squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions.
The squeeze theorem is used in calculus and mathematical anal ...
argument to prove that as .
On the other hand, in 1821
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
explained why the limit of as positive numbers and approach ''while being constrained by some fixed relation'' could be made to assume any value between and by choosing the relation appropriately. He deduced that the limit of the full ''two-variable'' function without a specified constraint is "indeterminate". With this justification, he listed along with expressions like in a
table of indeterminate forms.
Apparently unaware of Cauchy's work,
Möbius in 1834, building on Pfaff's argument, claimed incorrectly that whenever as approaches a number (presumably is assumed positive away from ). Möbius reduced to the case , but then made the mistake of assuming that each of and could be expressed in the form for some continuous function not vanishing at and some nonnegative integer , which is true for analytic functions, but not in general. An anonymous commentator pointed out the unjustified step;
then another commentator who signed his name simply as "S" provided the explicit counterexamples and as and expressed the situation by writing that " can have many different values".
Current situation
* Some authors define as because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define is based on convenience, not on correctness. If we refrain from defining , then certain assertions become unnecessarily awkward. ... The consensus is to use the definition , although there are textbooks that refrain from defining ."
Knuth (1992) contends more strongly that "''has'' to be "; he draws a distinction between the ''value'' , which should equal , and the ''limiting form'' (an abbreviation for a limit of where ), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."
* Other authors leave undefined because is an indeterminate form: does not imply .
There do not seem to be any authors assigning a specific value other than 1.
Treatment on computers
IEEE floating-point standard
The
IEEE 754-2008
The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operation ...
floating-point standard is used in the design of most floating-point libraries. It recommends a number of operations for computing a power:
*
pown
(whose exponent is an integer) treats as ; see .
*
pow
(whose intent is to return a non-
NaN
Nan or NAN may refer to:
Places China
* Nan County, Yiyang, Hunan, China
* Nan Commandery, historical commandery in Hubei, China
Thailand
* Nan Province
** Nan, Thailand, the administrative capital of Nan Province
* Nan River
People Given name ...
result when the exponent is an integer, like
pown
) treats as .
*
powr
treats as NaN (Not-a-Number) due to the indeterminate form; see .
The
pow
variant is inspired by the
pow
function from
C99
C99 (previously known as C9X) is an informal name for ISO/IEC 9899:1999, a past version of the C programming language standard. It extends the previous version ( C90) with new features for the language and the standard library, and helps impl ...
, mainly for compatibility.
It is useful mostly for languages with a single power function. The
pown
and
powr
variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above).
Programming languages
The C and C++ standards do not specify the result of (a domain error may occur). But for C, as of
C99
C99 (previously known as C9X) is an informal name for ISO/IEC 9899:1999, a past version of the C programming language standard. It extends the previous version ( C90) with new features for the language and the standard library, and helps impl ...
, if the
normative
Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good, desirable, or permissible, and others as bad, undesirable, or impermissible. A norm in ...
annex F is supported, the result for real floating-point types is required to be because there are significant applications for which this value is more useful than
NaN
Nan or NAN may refer to:
Places China
* Nan County, Yiyang, Hunan, China
* Nan Commandery, historical commandery in Hubei, China
Thailand
* Nan Province
** Nan, Thailand, the administrative capital of Nan Province
* Nan River
People Given name ...
(for instance, with
discrete exponents); the result on complex types is not specified, even if the informative annex G is supported. 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 ...
standard,
the
.NET Framework
The .NET Framework (pronounced as "''dot net"'') is a proprietary software framework developed by Microsoft that runs primarily on Microsoft Windows. It was the predominant implementation of the Common Language Infrastructure (CLI) until bein ...
method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to:
*Scien ...
System.Math.Pow
,
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g. ...
, and
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
also treat as . Some languages document that their exponentiation operation corresponds to the
pow
function from the
C mathematical library; this is the case with
Lua
Lua or LUA may refer to:
Science and technology
* Lua (programming language)
* Latvia University of Agriculture
* Last universal ancestor, in evolution
Ethnicity and language
* Lua people, of Laos
* Lawa people, of Thailand sometimes referred t ...
and
Perl
Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...
's
**
operator
(where it is explicitly mentioned that the result of
0**0
is platform-dependent).
Mathematical and scientific software
APL,
R,
Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...
,
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, numbe ...
,
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
,
Magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
,
GAP,
Singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
,
PARI/GP
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems.
System overview
The P ...
,
and
GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a langu ...
evaluate to .
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
and
Macsyma
Macsyma (; "Project MAC's SYmbolic MAnipulator") is one of the oldest general-purpose computer algebra systems still in wide use. It was originally developed from 1968 to 1982 at MIT's Project MAC.
In 1982, Macsyma was licensed to Symbolics and ...
simplify to even if no constraints are placed on ; however, if is entered directly, it is treated as an error or indeterminate.
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, numbe ...
does not simplify .
Maple
''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
,
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
and
PARI/GP
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems.
System overview
The P ...
further distinguish between integer and floating-point values: If the exponent is a zero of integer type, they return a of the type of the base; exponentiation with a floating-point exponent of value zero is treated as undefined, indeterminate or error.
References
External links
sci.math FAQ: What is ?What does {{math, 00 (zero to the zeroth power) equal?on AskAMathematician.com
Exponentials
Mathematical analysis
Computer arithmetic
Computer errors
0 (number)
Software anomalies
1 (number)