Empty product

In mathematics, the empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is one, just as the empty sum -- the sum of no numbers -- is zero. This fact is useful in discrete mathematics, algebra, and the study of power series. Two often-seen instances are a⁰ = 1 (any number raised to the power of zero is one) and 0! = 1 (the factorial of zero is one).

One justification

Imagine a calculator that can only multiply. It has an "ENTER" key and a "CLEAR" key. If "21" is displayed, and "4" is entered, then the display will show "84", since 21 × 4 = 84. If one wants to know the value of

3 × 4 × 7,

then one presses "CLEAR"; then enters "3"; then enters "7"; then enters "4". One hopes then to see "84" displayed.

What number must then appear just after "CLEAR" alone has been pressed? It is tempting to say "0", by analogy with conventional calculators. But if "0" is displayed, then after "3" is entered, the display will show the product of 0 and 3, so it will show "0"; and then when "7" is entered, it will again show "0"; and then when "4" is entered it will likewise show "0", rather than "84". Only if "1" is displayed after "CLEAR" has been pressed, will the calculator perform as advertised. Therefore, when no numbers have been multiplied, the product is 1.

A special case: 0⁰

Some accounts say that any non-zero number raised to the power of 0 is 1. This point is somewhat context-dependent. If f(x) and g(x) both approach 0 from above as x approaches some number, then f(x)^g(x) may approach some number other than one, or fail to converge. In that sense, 0⁰ is an indeterminate form. A case in which the limit is not 1 (but 1/2 instead) is f(x) := 2^-1/x and g(x) = x, as x approaches 0 from above. However, if the plane curve along which the ordered pair (f(x), g(x)) moves through the positive quadrant towards (0,0) is bounded away from tangency to either of the two coordinate axes, then the limit is necessarily one. Thus it may be said that in a sense, the limit is almost always 1. Furthermore, if the functions f and g are analytic at the point that the variable approaches, then the value will converge to 1, unless f is constant in some neighborhood of that point.

However, for other purposes, such as those of combinatorics, set theory, the binomial theorem, and power series, one should take 0⁰ = 1. From the combinatorial point of view, the number n^m is the cardinality of the set of functions from a set of size m into a set of size n. If both sets are empty, then there is just one such mapping: the empty function. From the power-series point of view, identities such as

$e^{0}=\sum _{n=0}^{\infty }{\frac {0^{n}}{n!}}={\frac {0^{0}}{0!}}+{\frac {0^{1}}{1!}}+{\frac {0^{2}}{2!}}+{\frac {0^{3}}{3!}}+\cdots$

are not valid unless 0⁰, which appears in the numerator of the first term of such series, is 1. A striking instance is the fact that the Poisson distribution with expectation 0 concentrates probability 1 at 0. That does not agree with the usual formula for the probability mass function of the Poisson distribution unless 0⁰ = 1.

A consistent point of view incorporating all of these aspects is to accept that 0⁰ = 1 in all situations, but the function h(x,y) := x^y is not continuous. Then 0⁰ is still an indeterminate form, because we don't know the value of the limit of f(x)^g(x) (in the example above), but that's a statement about limits, not about the value of 0⁰, which is still 1. (More nuanced approaches are possible, but this view is simple and will always work.)

External link

• sci.math FAQ: What is 0⁰?