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2 A special case: 00 3 Nullary intersection 4 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.
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 2 instead) is f(x) := 21/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 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
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.A special case: 00
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.Nullary intersection