Why are 0! = 1? Shouldn't 0! = 0
[From: ] [author: ] [Date: 11-11-24] [Hit: ]
0! is the product of the integers from 1 to 0.Since there arent any integers between 1 and 0, this is the empty product (the product of no numbers), which is 1.It may be confusing that the product of no numbers is 1.......
x^m / x^n = x^(m - n)
or x(m - n) = x^m / x^n
Exponents can then be extended to integers (zero and negative included) so that this identity holds. You can get all the other properties:
taking m = n
x^(m - m) = x^m / x^m
x^0 = 1
also if m - n < 0, for example if m = 0, n = -3:
x^(-3) = x^0 / x^3
x^(-3) = 1/x^3
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0! = 1. n! is the product of the integers from 1 to n. So, 0! is the product of the integers from 1 to 0. Since there aren't any integers between 1 and 0, this is the empty product (the product of no numbers), which is 1.
It may be confusing that the product of no numbers is 1. One of the reasons for this definition is to satisfy the following consistency rule:
(the product of the numbers in A) * (the product of the numbers in B) = (the product of the numbers in A ∪ B), (*)
whenever A and B are disjoint sets.
Then, setting B to the empty set in (*) gives
(the product of the numbers in A) * (the empty product) = (the product of the numbers in A)
for all sets A, so the empty product must be the multiplicative identity, which is 1. This is analogous to the sum of no numbers, which is the additive identity, or 0.
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PROOF that 2=1 :)
Given x = 1 and y = 1, then
x = y
Multiplying each side by x,
x^2 = xy
Subtracting y^2 from each side,
x^2 - y^2 = xy - y^2
Factoring each side,
(x + y)(x - y) = y(x - y)
Dividing out the common term, (x - y) results in
x + y = y
Substituting the values of x and y,
1 + 1 = 1
or
2 = 1
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There is no "simple" answer. There is a formal definition that states 0! is an empty product, or nullary product. It is the result of multiplying no factors. It is "by definition" equal to the multiplicative identity 1
I'm afraid it's one of those things you have to learn, like x^0 = 1 no matter what value x is.
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several things using 0 and 1 don't quite follow the rules, so these things are "defined as...", or are "undefined" (dividing by 0). They are what they are, and you just have to learn them.
