1.1 Q-6

Exercise 1.1 — Question 6

Powers of $i$

Recall that $i=\sqrt{-1}$, so the powers of $i$ cycle with period 4:

$i^1=i,i^2=-1,i^3=-i,i^4=1$

For any integer $n$, divide $n$ by 4 and use the remainder $r$:

$i^n=i^r,r=n\mathrm{mod}4$

Equality of Complex Numbers

Two complex numbers $z_1=a+bi$ and $z_2=c+di$ are equal if and only if their real parts are equal and their imaginary parts are equal:

$z_1=z_2⟺a=c\text{ and }b=d$

This condition is used to find unknown real values by equating real and imaginary parts separately.

Worked Examples

Example 1: Evaluate $i^{23}$.

$23=4\times 5+3⟹i^{23}=i^3=-i$

Example 2: Evaluate $i^{100}$.

$100=4\times 25+0⟹i^{100}=i^0=1$

Example 3: Evaluate $i^{4k+2}$ for any non-negative integer $k$.

$i^{4k+2}=(i^4{)}^k\cdot i^2=1^k\cdot (-1)=-1$

Example 4: If $(2x-1)+(y+3)i=5+7i$, find $x$ and $y$.

Equating real parts: $2x-1=5⟹x=3$

Equating imaginary parts: $y+3=7⟹y=4$