Exercise 1.1 — Question 1

Key Concepts Reviewed

Definition of a Complex Number (M-11-A-01)

A complex number $z$ is any number of the form: $z=a+ib$ where $a,b\in \mathbb{R}$ and $i=\sqrt{-1}$.

Alternatively, $z$ can be written as an ordered pair $(a,b)$.

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Real and Imaginary Parts (M-11-A-02)

For $z=a+ib$:

• $\text{Re}(z)=a$ is the real part
• $\text{Im}(z)=b$ is the imaginary part (a real number, not $ib$)

Example: For $z=5-3i$, $\text{Re}(z)=5$ and $\text{Im}(z)=-3$.

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Equality of Complex Numbers (M-11-A-03)

Two complex numbers $z_1=a+ib$ and $z_2=c+id$ are equal if and only if: $a=c\text{and}b=d$

That is, their real parts are equal and their imaginary parts are equal.

Example: If $x+iy=3-5i$, then $x=3$ and $y=-5$.

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Basic Operations on Complex Numbers (M-11-A-04)

Let $z_1=a+ib$ and $z_2=c+id$.

Addition: $z_1+z_2=(a+c)+i(b+d)$

Subtraction: $z_1-z_2=(a-c)+i(b-d)$

Multiplication: $z_1\cdot z_2=(ac-bd)+i(ad+bc)$

Division: $\frac{z_1}{z_2}=\frac{(ac+bd)}{c^2+d^2}+i\frac{(bc-ad)}{c^2+d^2},z_2\ne 0$

Powers of $i$

To evaluate $i^n$ for large $n$, divide $n$ by 4 and use the remainder $r$:

Example: $i^{23}$: $23=4\times 5+3$, so $i^{23}=i^3=-i$.