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.
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.
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
This is same as AA -1. It means that we first apply the A -1 transformation which will take as to some plane having different basis vectors. If we think what is the inverse of A -1 ? 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).
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.
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.....
The other interesting thing here is that 1,2,3, etc. appear in order in the list. And you have 2,3,4, etc. terms on the left, 1,2,3, etc. terms on the right. This should let you determine a formula like the one you want. Then prove it by induction.
How do I calculate this sum in terms of 'n'? I know this is a harmonic progression, but I can't find how to calculate the summation of it. Also, is it an expansion of any mathematical function? 1 ...
