complex no

juantheron · 2011-11-23 07:30:04

$gif.latex?\hspace{-16}$If%20$\mathbf{a\;,b\in%20C}\;,$%20Then%20Prove%20that%20$\mathbf{\frac{a\bar{b}+\bar{a}b}{2\mid%20a%20\mid%20\mid%20b%20\mid%20}\in%20\left[-1\;,1\right]}$$

gnitsuk · 2011-11-24 02:42:34

let

then

and

so

Hence:

We need to show that this is always between -1 and 1 inclusive, that is to say we need to show that the denominator always has greater or equal absolute value than that of the numerator, i.e. that:

i.e. that

Which is clearly true as the LHS is always > 0.

One thing, we have proven the inequality for ALL complex numbers except one, the number 0 + 0i, for this number the inequality is not true as the denominator would be zero.

Last edited by gnitsuk (2011-11-24 23:55:53)

Math Is Fun Forum

#1 2011-11-23 07:30:04

complex no

#2 2011-11-24 02:42:34

Re: complex no

Board footer