Practice section 1 of 9

In secondary school, $-1$ and $1$ are the roots of $x^2-1=0$. What are the roots of $x^2+1=0$? Prove that this equation has no real solution, then express its roots using $i^2=-1$.

What is the relation between the set of real numbers and the set of complex numbers?

Complete the real and imaginary parts for the source examples. The first row is already given.

Find the values of x and y such that $x-3+5i=4-i(2+y)$, where $x,y\in\mathbb R$.

Given $z_1=1+3i$ and $z_2=4+3i$, mark and label clearly on Argand diagrams the points representing:

(i) $3z_1$ and $-2z_2$;

(ii) $z_1z_2$;

(iii) $-z_2$;

(iv) $z_1+z_2$;

(v) $z_1-z_2$; and

(vi) $iz_1$, $i^2z_1$ and $-iz_1$.

For each pair of complex numbers $z_1$ and $z_2$,

(a) $z_1=2+3i$, $z_2=3-4i$;

(b) $z_1=3-i$, $z_2=4-i$;

(c) $z_1=-2+5i$, $z_2=1-3i$;

simplify (i) $z_1+z_2$, (ii) $z_1-z_2$ and (iii) $z_1z_2$.