2006 SMT/Advanced Topics Problems/Problem 1

Problem

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of $i$. What two answers should he give?

Solution

Remember from Euler's Formula that $\cos\theta+i\sin\theta=e^{i\theta}$.

We have $i=\cos\left(\frac{\pi}{2}\right)+i\sin\left(\frac{\pi}{2}\right)=e^{\pi i/2}$.

Therefore, $\sqrt{i}=i^{1/2}=(e^{\pi i/2})^{1/2}=e^{\pi i/4}=\cos\left(\frac{\pi}{4}\right)+i\sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$.

Of course, we also have to consider the negative of this, so the two answers are $\boxed{\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}}$.

See Also

2006 SMT/Advanced Topics Problems

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