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11 there are multiple ways of writing out a given complex number, or a number in general How can i prove from first principles that $0!$ is equal to $1$? The complex numbers are a field

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It's a fundamental formula not only in arithmetic but also in the whole of math We are basically asking that what transformation is required to get back to the identity transformation whose basis vectors are i ^ (1,0) and j ^ (0,1). Is there a proof for it or is it just assumed?

How do i convince someone that $1+1=2$ may not necessarily be true

I once read that some mathematicians provided a very length proof of $1+1=2$ Can you think of some way to 49 actually 1 was considered a prime number until the beginning of 20th century Unique factorization was a driving force beneath its changing of status, since it's formulation is quickier if 1 is not considered a prime

But i think that group theory was the other force. There are infinitely many possible values for $1^i$, corresponding to different branches of the complex logarithm The confusing point here is that the formula $1^x = 1$ is not part of the definition of complex exponentiation, although it is an immediate consequence of the definition of natural number exponentiation. Intending on marking as accepted, because i'm no mathematician and this response makes sense to a commoner

However, i'm still curious why there is 1 way to permute 0 things, instead of 0 ways.

两边求和，我们有 ln (n+1)<1/1+1/2+1/3+1/4+……+1/n 容易的， \lim _ {n\rightarrow +\infty }\ln \left ( n+1\right) =+\infty ，所以这个和是无界的，不收敛。