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- Is $0$ a natural number? - Mathematics Stack Exchange
Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century The Peano Axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then you take $0$ to be a natural number
- factorial - Why does 0! = 1? - Mathematics Stack Exchange
$\begingroup$ The theorem that $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ already assumes $0!$ is defined to be $1$ Otherwise this would be restricted to $0 <k < n$ A reason that we do define $0!$ to be $1$ is so that we can cover those edge cases with the same formula, instead of having to treat them separately
- 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?
- algebra precalculus - Zero to the zero power – is $0^0=1 . . .
Whereas exponentiation by a real or complex number is a messier concept, inspired by limits and continuity So $0^0$ with a real 0 in the exponent is indeteriminate, because you get different results by taking the limit in different ways
- Is $0^\infty$ indeterminate? - Mathematics Stack Exchange
Is a constant raised to the power of infinity indeterminate? I am just curious Say, for instance, is $0^\infty$ indeterminate?
- What is the value of $i^0$? - Mathematics Stack Exchange
But: I know what I am writing about I have a PhD mathematics, and have seen all these arguments by people who let $0^0$ undefined, and I have seen even more arguments by people who define $0^0=1$ and these arguments have convinced me And probably they will also convince you once you open yourself to them Think before downvoting! $\endgroup$
- I have learned that 1 0 is infinity, why isnt 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
- What is the meaning of $\\mathbb{N_0}$? - Mathematics Stack Exchange
There is no general consensus as to whether $0$ is a natural number So, some authors adopt different conventions to describe the set of naturals with zero or without zero Without seeing your notes, my guess is that your professor usually does not consider $0$ to be a natural number, and $\mathbb{N}_0$ is shorthand for $\mathbb{N}\cup\{0\}$
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