9.4 Me, myself, and the square root of i

We have already seen how to take a number such as $\frac{3}{2-i}$ and rewrite it in $a+bi$ format. There are many other numbers—such as $2^i$ and $\text{log}\left(i\right)$ —that do not look like $a+bi$ , but all of them can be turned into $a+bi$ form. In this assignment, we are going to find $\sqrt{i}$ —that is, we are going to rewrite $\sqrt{i}$ so that we can clearly see its real part and its imaginary part.

How do we do that? Well, we want to find some number $z$ such that $z^2=i$ . And we want to express $z$ in terms of its real and imaginary parts—that is, in the form $a+bi$ . So what we want to solve is the following equation:

$(a+bi{)}^2=i$

You are going to solve that equation now. When you find $a$ and $b$ , you will have found the answers.

Stop now and make sure you understand how I have set up this problem, before you go on to solve it.

Now, we are trying to solve the equation $(a+bi{)}^2=i$ . So take the formula you just generated in number 2, and set it equal to $i$ . This will give you two equations: one where you set the real part on the left equal to the real part on the right, and one where you set the imaginary part on the left equal to the imaginary part on the right.