Question

If the square root of - i can be written as $\pm$ (a - ib). Find the value of a $\times$ b.
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Answer 1

Let $\sqrt{-}i=a-ib$
Square on both sides
$-i=a^2-b^2-2abi$
$a^2-b^2=0$ ...(1)
$2ab=+1$ ...(2)
$a^2+b^2=\sqrt{(a^2-b^2{)}^2+4a^2{b}^2}$
$a^2+b^2=\sqrt{0+1}=1$
$a^2+b^2=1$ ....(3)
$a^2-b^2=0$
$2a^2=1$
$a=\pm \frac{1}{\sqrt{2}}$
$b=\pm \frac{1}{\sqrt{2}}$
Square root is $\pm \frac{(1-i)}{\sqrt{2}}$
$a\times b=\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=\frac{1}{2}$