User:FactSpewer/Draft of Exponentiation

Zero to the zero power

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

Most authors agree with the statements related to 0⁰ in the two lists below, but come to differing conclusions when it comes to defining 0⁰ or not: see the next subsection.

In most settings not involving continuity, interpreting 0⁰ as 1 simplifies formulas and eliminates the need for special cases in theorems (see the next paragraph for some settings that do involve continuity). For example:

• Regarding 0⁰ as an empty product of zeros suggests a value of 1.
• The combinatorial interpretation of 0⁰ is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
• Equivalently, the set-theoretic interpretation of 0⁰ is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.
• It greatly simplifies the theory of polynomials and power series that a constant term can be written ax⁰ for an arbitrary x. For example:
  • The formula for the coefficients of a product of polynomials would lose much of its simplicity if constant terms had to be treated specially.
  • Identities like $\textstyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}$ and $\textstyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$ are not valid for x = 0 unless 0⁰ = 1.
  • The binomial theorem $\textstyle (1+x)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}$ is not valid for x = 0 unless 0⁰ = 1.
• In differential calculus, the power rule $\textstyle {\frac {d}{dx}}x^{n}=nx^{n-1}$ is not valid for n = 1 at x = 0 unless 0⁰ = 1.

On the other hand, 0⁰ must be handled as an indeterminate form in settings where the exponent varies:

• When f(x) and g(x) are positive-valued functions approaching 0 (as x approaches a real number or ∞), the function f(x)^g(x) need not approach 1. In fact, depending on f and g, the limit of f(x)^g(x) can be any real number between 0 and 1, or it can be undefined. One expresses this ambiguity by saying that 0⁰ is an indeterminate form. The source of this ambiguity is the fact that the two-variable function x^y, continuous on the region defined by x ≥ 0, y ≥ 0, (x,y) ≠ (0,0), cannot be extended to a continuous function on a region including (0,0), no matter how 0⁰ is defined. The rule in calculus that lim_{x →a} f(x)^g(x) = (lim_x→a f(x))^{lim_x→a g(x)} whenever both sides of the equation are defined would fail if 0⁰ were defined.
• The function z^z is defined for nonzero complex numbers z by choosing a branch of log z and setting z^z := e^z log z, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0. There is no holomorphic function defined in a neighborhood of 0 that agrees with z^z for all positive real numbers z.

History of differing points of view

Different authors interpret the situation above in different ways:

• Some argue that the best value for 0⁰ depends on context, and hence that defining it once and for all is problematic. According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness."
• Others argue that 0⁰ is 1. According to p. 408 of Knuth (1992), it "has to be 1".

The debate has been going on at least since the early 1800s. At that time, most mathematicians agreed that 0⁰ = 1, but in 1821 Cauchy listed 0⁰ along with expressions like 0/0 in a table of undefined forms. In the 1830s Libri published an unconvincing argument for 0⁰ = 1, and Möbius sided with him, erroneously claiming that $\lim _{x\to 0^{+}}f(x)^{g(x)}=1$ whenever $\lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}g(x)=0.$ A commentator who signed his name simply as "S" provided a counterexample, and this quieted the debate for some time, with the apparent conclusion of this episode being that 0⁰ should be undefined. More details can be found in Knuth (1992).

Treatment in programming languages and calculators

Computer programming languages that evaluate 0⁰ to 1 include bc, Haskell, J, Java, MATLAB, ML, Perl, Python, R, Ruby, Scheme, and SQL. In the .NET Framework, the method System.Math.Pow treats 0⁰ to be 1.

Microsoft Excel issues an error when it evaluates 0⁰.

Microsoft Windows' Calculator and the calculator in Google search evaluate 0⁰ to 1.

Maple simplifies a⁰ to 1 and 0^a to 0, even if no constraints are placed on a, and evaluates 0⁰ to 1.

Mathematica simplifies a⁰ to 1, even if no constraints are placed on a. It does not simplify 0^a, and it takes 0⁰ to be an indeterminate form.