Empty product
In arithmetic, 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 zeroth power is one) and 0! = 1 (the factorial of zero is one).
More generally, given an operation of multiplication on some collection of objects, the empty product is the result of multiplying no objects together. It is generally defined to be the identity element with respect to the given operation, if such exists. For example, the empty direct product of (isomorphism classes of) groups is (the isomorphism class of) the trivial group, since every group is isomorphic to its direct product with the trivial group.
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2 Another justification 3 0 raised to the 0th power 4 Nullary intersection 5 External link |
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
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.
Mutiplication used to be done by adding common logarithms because of the property
Some accounts say that any non-zero number raised to the 0th power 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 value 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.
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 nm is the size of the set of functionss from a set of size m into a set of size n.
If both sets are empty (size 0), then there is just one such mapping: the empty function.
From the power-series point of view, identities such as
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) := xy 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.)
For similar reasons, the intersection of an empty set of subsets of a set X is conventionally equal to X.One justification
then one presses "CLEAR"; then enters "3"; then enters "4"; then enters "7".
One hopes then to see "84" displayed.Another justification
but if the sums are empty, i.e. zero, then the RHS becomes 100=1.
The same would occur using natural logarithms since e0=1, and similarly with other bases.0 raised to the 0th power
are not valid unless 00, which appears in the numerator of the first term of such a 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.Nullary intersection