## Are complex numbers complex ?

You  might perhaps think that complex numbers are complex to handle. Quite contrary. They are easily applied to various kinds of engineering problems and are easily handled in pure math concepts compared to real numbers. Which brings me to another point. Mathematicians are perhaps short of rich vocabulary so they name some object as a “ring”, which is not a wedding or engagement ring at all; there is a “field”, which is not a field of maize at all; then there is a “group”, which is just an abstract object and certainly not a group of people!!

Problem:

Prove the identity:

$({n \choose 0}-{n \choose 2}+{n \choose 4}- \ldots)^{2}+({n \choose 1}-{n \choose 3}+{n \choose 5}-\ldots)^{2}=2^{n}$

Solution:

Denote

$x_{n}={n \choose 0}-{n \choose 2}+{n \choose 4}- \ldots$ and

$y_{n}={n \choose 1}-{n \choose 3}+{n \choose 5}-\ldots$

and observe that $(1+i)^{n}=x_{n}+i y_{n}$

Passing to the absolute value it follows that

$|x_{n}+y_{n}i|=|(1+i)^{n}|=|1+i|^{n}=2^{n/2}$.

This is equivalent to $x_{n}^{2}+y_{n}^{2}=2^{n}$.

More later,

Nalin Pithwa

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