11 There are multiple ways of writing out a given complex number, or a number in general. Usually we reduce things to the "simplest" terms for display -- saying $0$ is a lot cleaner than saying $1-1$ for example. The complex numbers are a field. This means that every non-$0$ element has a multiplicative inverse, and that inverse is unique.
Possible Duplicate: 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
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.
A -1 A means that first we apply A transformation then we apply A -1 transformation. When we apply A transformation we reach some plane having some different basis vectors but after apply A -1 we again reach to the plane have basis i ^ (0,1) and j ^ (1,0).
Is there a formal proof for $(-1) \\times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
The reason why $1^\infty$ is indeterminate, is because what it really means intuitively is an approximation of the type $ (\sim 1)^ {\rm large \, number}$. And while $1$ to a large power is 1, a number very close to 1 to a large power can be anything.....
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.
