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Is $0$ a natural number?  Mathematics Stack Exchange
generally speaking 0 is not natural at all. It is special in so many respects; people naturally start counting from 1; the harmonic sequence 1 / n is defined for any natural number n; the 1 st number is 1; in making limits, 0 plays a role which is symmetric to ∞, and the latter is not a natural number.
factorial  Why does 0! = 1?  Mathematics Stack Exchange
The theorem that (n k) = n! k!(n−k)! (n k) = n! k! (n − k)! already assumes 0! 0! is defined to be 1 1. Otherwise this would be restricted to 0
complex analysis  What is $0^{i}$?  Mathematics Stack Exchange
$$\lim_{n\to 0} n^{i} = \lim_{n\to 0} e^{i\log(n)} $$ I know that $0^{0}$ is generally undefined, but can equal one in the context of the empty set mapping to itself only one time. I realize that in terms of the equation above, the limit does not exist, but can $0^{i}$ be interpreted in a way to assign it a value?
Is $0^\infty$ indeterminate?  Mathematics Stack Exchange
If we type those expressions into Mathematica, however, it tells us that 0^infinity is 0 and 1^infinity is indeterminate. – user3680. Commented Oct 9, 2013 at 23:42. All the answers here assume 0∞ 0 ∞ is 0 + ∞ 0 + ∞. But Wikipedia doesn't assume that. It claims that 0∞ 0 ∞ is an indeterminate form because 0 + ∞ 0 + ∞ has the ...
elementary set theory  Is this statement ∅∈{0} true or false ...
3. ∅ ∈ {0} ∅ ∈ {0} if and only if ∅ = 0 ∅ = 0. Whether this is true depends on which 0 0 you're talking about. Assuming you're referring to the normal definitions of N N, Z Z, Q Q, etc., I think your first bulleted point only adds confusion for the OP. N N is not universally taught to exclude the number 0 0.
I have learned that 1/0 is infinity, why isn't it minus infinity?
1 x 0 = 0. Applying the above logic, 0 / 0 = 1. However, 2 x 0 = 0, so 0 / 0 must also be 2. In fact, it looks as though 0 / 0 could be any number! This obviously makes no sense  we say that 0 / 0 is "undefined" because there isn't really an answer. Likewise, 1 / 0 is not really infinity. Infinity isn't actually a number, it's more of a concept.
definition  Why is $x^0 = 1$ except when $x = 0$?  Mathematics Stack ...
This gives you: xa ⋅ 1 = xa ⋅ x0 = xa + 0 = xa. When the base is also zero, it's not possible to define a value for 00 because there is no value that is consistent with all the necessary constraints. For example, 0x = 0 and x0 = 1 for all positive x, and 00 can't be consistent with both of these.
algebra precalculus  Zero to the zero power – is $0^0=1 ...
For the same reason, the sum of any empty list is zero, and the product is one. This is when a product or sum of an empty list is applied to a number, it leaves it unchanged. Thus if the product = 1, then we immediately see why . Without this property, one could prove that , by the ruse that there are zero zeros in the product on the left (zero ...
limit when zero divided by infinity  Mathematics Stack Exchange
33. 0∞ 0 ∞ is not an indeterminate form. On the contrary, those limits tell you that the limit of the entire quotient is 0 0. This may be easier to see if you rewrite to. limx→∞ f(x) 1 h(x) lim x → ∞ f (x) 1 h (x) where limx→∞ f(x) = 0 lim x → ∞ f (x) = 0 and limx→∞ 1 h(x) = 0 lim x → ∞ 1 h (x) = 0, and the product ...
What does it mean to have a determinant equal to zero?
Apr 9, 2013 at 6:21. 12. "When the determinant of a matrix is zero, the system of equations associated with it is linearly dependent; that is, if the determinant of a matrix is zero, at least one row of such a matrix is a scalar multiple of another." If the determinant is zero, one of the rows doesn't need to be a scalar multiple of the others.
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