1.1 Q-2

Exercise 1.1 — Question 2

This exercise covers basic operations on complex numbers, equality of complex numbers, and evaluation of powers of $i$.

Key Concepts Review

Complex Number: A complex number is written as $z=a+bi$ where $a,b\in \mathbb{R}$ and $i=\sqrt{-1}$.

In ordered pair notation: $z=(a,b)$

• $a=\text{Re}(z)$ is the real part
• $b=\text{Im}(z)$ is the imaginary part

Equality of Complex Numbers: $z_1=a+bi$ and $z_2=c+di$ are equal if and only if: $a=c\text{and}b=d$

That is, both the real parts AND the imaginary parts must be separately equal.

Powers of $i$: Since $i^4=1$, powers of $i$ repeat in a cycle of 4: $i^1=i,i^2=-1,i^3=-i,i^4=1$

To find $i^n$: divide $n$ by $4$ and use the remainder $r$, so $i^n=i^r$.

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Worked Examples

Example 1: Simplify $i^{23}$

$23=4\times 5+3⟹i^{23}=i^3=-i$

Example 2: Simplify $i^{100}$

$100=4\times 25+0⟹i^{100}=i^0=1$

Example 3: Find $x$ and $y$ if $(3x-2)+(2y+1)i=4+5i$

Equating real parts: $3x-2=4⟹x=2$

Equating imaginary parts: $2y+1=5⟹y=2$

Example 4: Simplify $i^{4k+2}$ for any integer $k$

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

Example 5: Simplify $i^{4k+3}$ for any integer $k$

$i^{4k+3}=(i^4{)}^k\cdot i^3=1\cdot (-i)=-i$

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