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Is $0$ a natural number? - Mathematics Stack Exchange

Mar 15, 2013 · 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 …

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 …

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 …

What is the integral of 0? - Mathematics Stack Exchange

Feb 4, 2018 · How about drawing sum upper and lower sums! You won't get very far because you'll be married to the horizontal axis and then, of course, all of the sums are zero and since …

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 …

definition - Why is $x^0 = 1$ except when $x = 0$? - Mathematics …

Jan 22, 2017 · 1) x^a × x^b = x^a+b; for x = 0 and a = 0, you would get 0^0 × 0^b = 0^b = 0, so we can't tell anything -- except confirm that 0^0 = 1 still works here! 2) x^{-a}=1/{x^a} -- so when a …

What does it mean to have a determinant equal to zero?

Nov 27, 2019 · The volume of the parallelepiped determined by the row vectors of the matrix is $0$. The system of homogenous linear equations represented by the matrix has a non-trivial …

What is $\\gcd(0,a)$, where $a$ is a positive integer?

Feb 24, 2019 · Thus it makes no sense to define $\rm\ gcd(0,8)\ $ to be $\,0\,$ or $\,1\,$ since $\,0\,$ is not a common divisor of $\,0,8\,$ and $\,1\,$ is not the greatest common divisor. The …

Limit when denominator = 0 - Mathematics Stack Exchange

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their …

What do zero eigenvalues mean? - Mathematics Stack Exchange

Dec 3, 2014 · If 0 is an eigenvalue, then the nullspace is non-trivial and the matrix is not invertible. Therefore all the equivalent statements given by the invertible matrix theorem that apply to …