Chapter 12, Problem 39E

Exercises 35 through 40 are a review of complex number arithmetic. Recall that $\left(1\right)$ to add two complex numbers you simply add the real parts and the imaginary parts: e.g., $\left(2+3i\right)+\left(5+2i\right)=7+5i$; $\left(2\right)$ to multiply two complex numbers you multiply them as if they were polynomials and use the fact that $i^2=-1$: e.g., $\left(2+3i\right)\left(5+2i\right)=10+4i+15i+6i^2=4+19i$. Finally, if you know how to multiply two complex numbers then you also know how to square them, since ${\left(a+bi\right)}^2=\left(a+bi\right)\left(a+bi\right)$.

a. Plot the points corresponding to the complex numbers $\left(1+i\right)$, $i\left(1+i\right)$, $i^2\left(1+i\right)$, and $i^3\left(1+i\right)$.

b. Plot the points corresponding to the complex numbers $\left(3-2i\right)$, $i\left(3-2i\right)$, $i^2\left(3-2i\right)$, and $i^3\left(3-2i\right)$.

c. What geometric effect does multiplication by $i$ have on a complex number?