Zero to the Power of Zero

Zero to the power of zero , denoted by 0 ⁰ , is a mathematical expression with no agreed-uponvalue. The most common possibilities are1 or leaving the expression undefined, with justifications existing for each, depending on context. Inalgebra andcombinatorics, the generally agreed upon value is  0 ⁰ = 1 , whereas in mathematical analysis , the expression is sometimes left undefined.also have differing ways of handling this expression.

Discrete exponents [ edit ]

There are many widely used formulas having terms involvingnatural-number exponents that require 0 ⁰ to be evaluated to 1 . For example, regarding b ⁰ as anempty product assigns it the value 1 , even when b = 0 . Alternatively, the combinatorial interpretation of b ⁰ is the number ofempty tuples of elements from a set with b elements; there is exactly one empty tuple, even if b = 0 . Equivalently, the set-theoretic interpretation of 0 ⁰ is the number of functions from the empty set to the empty set; there is exactly one such function, theempty function.

Polynomials and power series [ edit ]

Likewise, when working withpolynomials, it is convenient to define 0 ⁰ as having the value 1. A polynomial is an expression of the form where x is an indeterminate, and the coefficients are real numbers (or, more generally, elements of somering). The set of all real polynomials in x is denoted by . Polynomials are added termwise, and multiplied by the applying the usual rules for exponents in the indeterminate x (seeCauchy product). With these algebraic rules for manipulation, polynomials form apolynomial ring. The polynomial is theidentity element of the polynomial ring, meaning that it is the (unique) element such that the product of with any polynomial is just .Polynomials can be evaluated by specializing the indeterminate x to be a real number. More precisely, for any given real number there is a unique unitalring homomorphism such that .This is called the evaluation homomorphism . Because it is a unital homomorphism, we have That is, for all specializations of x to a real number (including zero).

This perspective is significant for many polynomial identities appearing in combinatorics. For example, thebinomial theorem is not valid for x = 0 unless 0 ⁰ = 1 .Similarly, rings ofpower series require to be true for all specializations of x . Thus identities like and are only true as functional identities (including at x = 0 ) if 0 ⁰ = 1 .

In differential calculus , thepower rule is not valid for n = 1 at x = 0 unless 0 ⁰ = 1 .

Continuous exponents [ edit ]

Plot of z = x ^y . The red curves (with z constant) yield different limits as ( x , y ) approaches (0, 0) . The green curves (of finite constant slope, y = ax ) all yield a limit of 1.

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 anindeterminate form.In fact, when f ( t ) and g ( t ) are real-valued functions both approaching 0 (as t approaches a real number or ±∞ ), with f ( t ) > 0 , the function f ( t ) ^{g ( t )} need not approach 1 ; depending on f and g , the limit of f ( t ) ^{g ( t )} can be any non-negative real number or +∞ , or it candiverge. For example, the functions below are of the form f ( t ) ^{g ( t )} with f ( t ), g ( t ) → 0 as t → 0 ⁺ (aone-sided limit), but the limits are different:

.

Thus, the two-variable function x ^y , though continuous on the set {( x , y ) : x > 0}, cannot be extended to acontinuous function on , no matter how one chooses to define 0 ⁰ .However, under certain conditions, such as when f and g are bothanalytic functions and f is positive on the open interval (0, b ) for some positive b , the limit approaching from the right is always 1 .

Complex exponents [ edit ]

In thecomplex domain, the function z ^w may be defined for nonzero z by choosing abranch of log z and defining z ^w as e ^{w log z} . This does not define 0 ^w since there is no branch of log z defined at z = 0 , let alone in a neighborhood of 0 .

History of differing points of view [ edit ]

The debate over the definition of has been going on at least since the early 19th century. At that time, most mathematicians agreed that , until in 1821Cauchy listed along with expressions like in a table of indeterminate forms . In the 1830s Guglielmo Libri Carucci dalla Sommaja published an unconvincing argument for , andMöbius sided with him, erroneously claiming that whenever . A commentator who signed his name simply as "S" provided the counterexample of , and this quieted the debate for some time. More historical details can be found in Knuth (1992).

More recent authors interpret the situation above in different ways:

• Some argue that the best value for depends on context, and hence thatdefining it once and for all is problematic.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 ."
• Others argue that should be defined as 1.Knuth (1992) contends strongly that " has to be 1", drawing a distinction between the value , which should equal 1 as advocated by Libri, and the limiting form (an abbreviation for a limit of where ), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."Vaughn gives several other examples of theorems whose (simplest) statements require 0 ⁰ = 1 as a convention.

Treatment on computers [ edit ]

IEEE floating-point standard [ edit ]

TheIEEE 754-2008 floating-point standard is used in the design of most floating-point libraries. It recommends a number of operations for computing a power:

• pow treats 0 ⁰ as 1. If the power is an exact integer the result is the same as for pown , otherwise the result is as for powr (except for some exceptional cases).
• pown treats 0 ⁰ as 1. The power must be an exact integer. The value is defined for negative bases; e.g., pown(−3,5) is −243.
• powr treats 0 ⁰ asNaN (Not-a-Number – undefined). The value is also NaN for cases like powr(−3,2) where the base is less than zero. The value is defined by e

  power×log(base)

  .

The pow variant is inspired by the pow function fromC99, 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 [ edit ]

The C and C++ standards do not specify the result of 0 ⁰ (a domain error may occur), but as ofC99, if thenormative annex F is supported, the result is required to be 1 because there are significant applications for which this value is more useful thanNaN (for instance, with). TheJava standard,the.NET Framework method System.Math.Pow ,andPython also treat 0 ⁰ as 1. Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library ; this is the case ofLua andPerl's ** operator(where it is explicitly mentioned that the result of 0**0 is platform-dependent).

Mathematical and scientific software [ edit ]

• SageMath simplifies b ⁰ to 1, even if no constraints are placed on b . It takes 0 ⁰ to be 1, but does not simplify 0 ^x for other x .

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• Maple distinguishes between integers 0, 1, ... and the corresponding floats 0.0, 1.0, ... (usually denoted 0., 1., ...). If x does not evaluate to a number, then x ⁰ and x ^0.0 are respectively evaluated to 1 ( integer ) and 1.0 ( float ); on the other hand, 0 ^x is evaluated to the integer 0, while 0.0 ^x is evaluated as 0. ^x . If both the base and the exponent are zero (or are evaluated to zero), the result is Float(undefined) if the exponent is the float 0.0; with an integer as exponent, the evaluation of 0 ⁰ results in the integer 1, while that of 0. ⁰ results in the float 1.0.

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• Macsyma also simplifies b ⁰ to 1 even if no constraints are placed on b , but issues an error for 0 ⁰ . For x > 0, it simplifies 0 ^x to 0.

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• Mathematica distinguishes between integers as exact, and floating point numbers as approximate. It simplifies b ⁰ to 1, even if no constraints are placed on b , does not simplify 0 ^x , and takes 0 ⁰ to be "Indeterminate".
• PARI/GP also distinguishes between integer and floating-point types: When the exponent 0 is of integer type, the expression, such as 0^0 or 0.^0, is simplified to 1.When the exponent is not of integer type, PARI/GP treats x ^ n as the transcendental function , so that 0^0. yields a domain error.
• APL , Matlab , Magma , GAP ,singular, andGNU Octave evaluate 0 ⁰ as 1.

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• Microsoft Excel treats =0^0 as #NUM!.
• R evaluates as .
• Stata evaluates as but no reference to this behaviour appears in the official documentation.

References [ edit ]