1.1 Q-3

Exercise 1.1 — Question 3

This question practices basic operations on complex numbers and applying the equality condition for complex numbers.

Key Concepts

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

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: $z_1=z_2⟺a=c\text{ and }b=d$

Powers of $i$: Since $i^4=1$, powers of $i$ repeat in a cycle of 4: $i^1=i,i^2=-1,i^3=-i,i^4=1$ For any integer $n$, divide $n$ by 4 and use the remainder $r$: $i^n=i^r$.

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 real numbers $x$ and $y$ such that $(2x-1)+(y+3)i=5+7i$.

Applying the equality condition:

• Real parts: $2x-1=5⟹2x=6⟹x=3$
• Imaginary parts: $y+3=7⟹y=4$

Example 4: Perform the operation $(3+2i)+(1-5i)$.

$=(3+1)+(2-5)i=4-3i$

Example 5: Perform the operation $(2+3i)(1-2i)$.

$=2(1)+2(-2i)+3i(1)+3i(-2i)$ $=2-4i+3i-6i^2$ $=2-i-6(-1)$ $=8-i$