How should I calculate the square root of a complex number using a Casio 991EX?

Question

With Casio 991EX how should I calculate the square root of a complex number? It says Math error even though is set in COMPLEX mode

Answer 1

If the calculator does not support complex square roots natively, you can

• convert from Cartesian to polar,

• divide the argument by two,

• square-root the modulus,

• convert back to Cartesian.

---

Alternatively, if there is no conversion,

$$(a+ib)^2=x+iy\iff\begin{cases}a^2-b^2=x,\\2ab=y\end{cases}.$$

Then

$$a^4-a^2b^2=a^4-\frac{y^2}4=xa^2$$ is a biquadratic equation in $a$, giving

$$a=\sqrt{\frac{x+\sqrt{x^2+y^2}}2},\\ b=\frac y{2a}.$$

Answer 2

For $a\neq 0$,

$$\sqrt{a+ib}=\pm\sqrt[4]{a^2+b^2}\left[\cos\frac{\left(\arctan\frac b{a}\right)}2+i\sin\frac{\left(\arctan\frac b{a}\right)}2\right]$$

If $a=0$, $$\sqrt{ib}=\pm\frac{\sqrt{2|b|}}2(1\pm i)$$ where the sign of $i$ is the same as of $b$.