Emiliano Guzman

2022-12-05

Writing complex numbers in form $a+bi$
Can $\sqrt{i+\sqrt{2}}$ be expressed as $a+bi$ with $a,b\in \mathbb{R}$? In general, what kinds of expressions can be rewritten in that form?

Laylah Henry

Expert

You can express such an expression in the form $a+ib$
Let
$x+iy=\sqrt{i+\sqrt{2}}\phantom{\rule{0ex}{0ex}}\left(x+iy{\right)}^{2}=\left(\sqrt{i+\sqrt{2}}{\right)}^{2}\phantom{\rule{0ex}{0ex}}{x}^{2}-{y}^{2}+2ixy=i+\sqrt{2}\phantom{\rule{0ex}{0ex}}$
Now you only need to solve the equations ${x}^{2}-{y}^{2}=\sqrt{2}$, and $xy=\frac{1}{2}$ to get the values of $x$ and $y$

Do you have a similar question?