Cant understand complex number question

mathperson

I have a question complex number question and I dont even know where to begin with it. I particularly dont understand the line 'By showing the m values of ....... and ..... are the same'. What are the 'm values'? And why does it say 'let n be a negative integer'..what is the significance of it being negative? The entire question seems hard to decipher.

MathsManiac

The point is that there is an assumption that you have previously defined what z^(1/n) means if n is a positive integer. You now want to extend the definition to include negative integers. If you assume that n is a negative integer, then -n is a positive integer. So you can then define z^(1/n) in terms of [something]^(1/(-n)).

There's also an assumption that you've previously defined what z^(-1) means.

Regarding your question about "the m values of...":
In complex analysis, z^(1/n) (for positive integer n) is often not considered to be a single-valued function in the same way as it usually is in real analysis. The power of (1/n) gives the "nth root". There are n different complex numbers that could be considered to be an "nth root" of a given complex number. For example, 1-2i and -1+2i are both square roots of -3-4i. There is a way to pick one of these to call "the" square root, but it's not as clean as with a real number. So, it's quite common to say that a complex number has two square roots, three cube roots and so on.

mathperson

Very nice answer, I can actually understand the question now Pity you aren't doing our tutorials.