Introduction

When we first learn about square roots, a natural question comes up: what is $\sqrt{-1}$? If your teacher told you it doesn't exist, they were completely wrong both right and wrong at the same time. It doesn't exist as a real number. But mathematicians decided that was a perfectly good reason to invent a new kind of number.

We define the imaginary unit $i$ by the equation

Numbers formed by multiplying $i$ by a real number — like $2i$, $-5i$, or $\pi i$ — are called pure imaginary numbers. They cannot be simplified any further; $2i$ is already in its simplest form.

Warning: The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ breaks down for negative numbers. For example,

$$\sqrt{-1}\sqrt{-1} \neq \sqrt{(-1)^2} = 1$$

The correct answer is

$$i \cdot i = i^2 = -1$$

Never combine square roots of negative numbers under one radical.

A complex number is any number of the form

The set of all complex numbers is denoted $\mathbb{C}$. This is a strict superset: every real number is a complex number (take $b=0$), and every pure imaginary number is a complex number (take $a=0$).

For a complex number $z = a + bi$:

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

Note that $b$ itself is real — it is the coefficient of $i$.

So,

not $-7i$.

Powers of $i$

If we keep multiplying $i$ by itself, we get a cycle that repeats every $4$ steps:

The key rule is:

where the remainder is taken in $\{0,1,2,3\}$ and $i^0=1$.

2019 AMC 10A Problem 4

What is $i^{2019}$?

Since

$$2019 = 4 \cdot 504 + 3$$

we get

$$i^{2019}=i^3=\boxed{-i}$$

The mod-$4$ trick is mechanical once you see it — and it appears on AMC/AIME more often than you'd expect.

Algebra of Complex Numbers

Addition and Subtraction

Add or subtract real and imaginary parts separately:

This is exactly like vector addition componentwise.

Multiplication

Expand using FOIL and replace $i^2$ with $-1$:

You do not need to memorize this formula.

Example: Compute $(2+3i)(1-4i)$.

Equating Real and Imaginary Parts

Two complex numbers are equal iff their real parts and imaginary parts are equal:

This means one complex equation is secretly two real equations.

Example: Find real $x,y$ such that

Expanding:

Equating parts:

Multiply the first equation by $2$ and the second by $3$:

Adding:

Substituting back:

Conjugates

The conjugate of

is

Geometrically, this reflects the point across the real axis.

The key identity is:

The product is always a nonnegative real number.

Two useful identities:

Properties of conjugation:

Division

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

More generally,

Example: Compute

Multiply top and bottom by $1+4i$:

Expand the numerator:

Denominator:

Therefore,

Worked Examples

Example 1: Compute $i^{2023}$.

Since

$$2023=4\cdot505+3$$

we get

$$i^{2023}=i^3=-i$$

Example 2: If $z+6i=iz$, find $z$.

Rearranging:

$$z-iz=-6i$$

$$z(1-i)=-6i$$

Thus,

$$z = \frac{-6*i}{1-i} \cdot \frac{1+i}{1+i} = \frac{-6*i(1+i)}{2} = \frac{-6*i+6}{2} = 3-3i$$

Example 3: Let

$$z+\overline{z}=6$$

and

$$z\overline{z}=13$$

Find $z$.

Write

$$z=a+bi$$

Then

$$2a=6 \implies a=3$$

Also,

$$a^2+b^2=13$$

so

$$9+b^2=13 \implies b^2=4 \implies b=\pm2$$

Therefore,

$$z=3+2i \quad\text{or}\quad z=3-2i$$

Example 4 (AIME flavor): Find all complex $z$ such that

$$z^2=2i$$

Write

$$z=a+bi$$

Then

$$z^2=(a^2-b^2)+2abi$$

Equating parts:

$$a^2-b^2=0$$

$$2ab=2 \implies ab=1$$

Since $ab>0$, we must have $a=b$.

Then

$$a^2=1$$

giving

$$z=1+i \quad\text{or}\quad z=-1-i$$

Strategy Checklist

Task	Technique
Simplify $i^n$	Compute $n \bmod 4$
Solve for complex variables	Expand and equate real/imaginary parts
Eliminate $i$ from denominator	Multiply by conjugate
Extract $\operatorname{Re}(z)$ or $\operatorname{Im}(z)$	Use conjugate identities
Relate $z$ and $\overline{z}$	Use $z\overline{z}=a^2+b^2$
Solve equations like $z^2=w$	Write $z=a+bi$ and equate parts

Common Pitfalls

• Treating $\operatorname{Im}(z)$ as $bi$ instead of $b$
• Forgetting that $i^2=-1$
• Incorrectly combining square roots of negative numbers
• Confusing $z+\overline{z}$ with $2|z|$
• Forgetting to multiply numerator and denominator by the conjugate
• Changing the sign of the real part when conjugating

Remarks

Everything in this module has been purely algebraic. But there is a beautiful geometric picture underneath all of it. In the next module, we interpret

as the point $(a,b)$ in the plane and discover that conjugation is reflection, addition is vector addition, and multiplication becomes rotation and scaling.

Seeing the geometry makes many of the identities here feel obvious almost inevitable.

Practice Problems