IB Maths AA HLTopic 1 — Number & AlgebraPaper 1 & 2~9 min read
Introduction to Complex Numbers
For years, you’ve been told you can’t square root a negative number. Try to solve x2 = −1 and your teacher would say “no real solutions”. Mathematicians eventually got tired of that limitation, so they invented a new kind of number — called imaginary — that does exactly what you were told was impossible. The result is a much bigger number system called the complex numbers, written ℂ. Once you accept this single new symbol — the letter i, where i2 = −1 — every quadratic in the world has a solution, and a whole new branch of mathematics opens up.
📘 What you need to know
The imaginary unit is defined by i = √(−1), so i2 = −1.
A complex number has the form z = a + bi, where a is the real part and b is the imaginary part.
This is called Cartesian form (sometimes called rectangular form). It’s in the formula booklet.
Notation: Re(z) = a and Im(z) = b. The “i” itself is not part of the imaginary part.
The set of all complex numbers is denoted ℂ. Every real number is also a complex number (with imaginary part zero).
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
To square root a negative number, factor out the −1 and pull it out as i. For example, √(−25) = √(25) · √(−1) = 5i.
The imaginary unit i
Here’s the one definition that unlocks the whole topic. Mathematicians said: “What if there were a number whose square is −1?” They couldn’t find one on the real number line, so they invented a new symbol for it.
The imaginary unit
i = √(−1) ⟺ i2 = −1
That’s it. One symbol, two ways of writing the same fact. Whenever you see i2 in your work, just replace it with −1 and keep going. Whenever you see √(−something), pull out the −1 as i and you’ve got a multiple of i.
Square roots of negatives — the standard trick
√(−k) = √k · √(−1) = √k · i (for k > 0)
So √(−4) = 2i, √(−9) = 3i, √(−16) = 4i, and √(−7) = √7 · i. Easy.
Whenever you’re squaring or unsquaring something negative, the only new rule is “replace √(−1) with i”. That single substitution turns every “no real solutions” into something solvable.
What is a complex number?
A complex number is just a real number plus a multiple of i. The general form is:
Cartesian form of a complex numberz = a + bi where a, b ∈ ℝ✓ in the formula booklet
The two parts have specific names:
Real part
Re(z) = a
The “ordinary” number part — the bit without an i.
Imaginary part
Im(z) = b
The number multiplying i. Note: the i itself is NOT part of Im(z).
For example, take z = 3 + 4i. Here Re(z) = 3 and Im(z) = 4. Notice the imaginary part is just 4 — not 4i. That’s a small detail, but it costs marks if you forget it.
Complex numbers are usually written using the letter z, the way we use x for real numbers. So if you see “let z = 2 − 5i”, that just means “we’re calling this complex number z for short.”
Special cases — purely real and purely imaginary
A complex number doesn’t always need both parts. If the imaginary part is zero, it’s just a regular real number. If the real part is zero, it’s called purely imaginary.
How complex numbers extend everything you’ve seen before
Look at the diagram. Real numbers are nested inside the complex numbers — every real number x is just x + 0i, a complex number with imaginary part zero. The complex numbers add a whole new layer that lives outside the real number line.
Two special types:Purely real: imaginary part is 0 (e.g., 7 = 7 + 0i). Purely imaginary: real part is 0 (e.g., 5i = 0 + 5i).
When are two complex numbers equal?
Two complex numbers a + bi and c + di are equal only when both their real parts match and both their imaginary parts match. There’s no halfway — the parts have to agree separately.
Equality of complex numbersa + bi = c + di ⟺ a = c AND b = d
🤔 Why do real and imaginary parts have to agree separately?
Because i is not a real number — you can’t trade real for imaginary. Real and imaginary live on different “axes” (you’ll meet this idea properly in the Argand-diagram note). Saying 3 + 2i = 3 − 2i would be like saying the points (3, 2) and (3, −2) are the same point. They aren’t.
This rule is incredibly useful in equation-solving. If a question gives you 2x + 3yi = 8 − 9i, you can split it into two separate real equations: 2x = 8 and 3y = −9. Solve each on its own.
Solving equations that “had no real solutions”
Now for the payoff. With i in the toolkit, every quadratic equation now has a solution. The recipe is exactly the same as before — just use the √(−k) = i√k rule whenever a negative number ends up under a square root.
🧭 Recipe — solving x2 = (negative)
Take the square root of both sides, including ±.
Pull the −1 out of the square root as i.
Simplify the remaining real square root.
Write the answers as ±ki (or in Cartesian form if there’s a real part too).
If the equation has the form (x + a)2 = (negative), do the square root step first, then rearrange to isolate x. The final answer will be a complex number in Cartesian form: real part from rearranging, imaginary part from the square root.
Worked examples
WE 1
Simplify √(−25) and √(−18)
Express each square root in terms of i, simplifying as far as possible.
Part (a): √(−25)√(−25) = √(25 · −1) = √25 · √(−1)= 5 · i = 5i√(−25) = 5iPart (b): √(−18)√(−18) = √(18 · −1) = √18 · i√18 = √(9 · 2) = 3√2√(−18) = 3√2 · i (or 3i√2)always pull out the −1 first, then simplify the remaining surd as usual
WE 2
Solve x2 = −36
Find the values of x that satisfy x2 = −36.
Step 1: Square root both sides (don’t forget ±)x = ± √(−36)Step 2: Pull out the −1 as i√(−36) = √36 · √(−1) = 6iStep 3: Write the solutionsx = ±6iin real numbers there were no solutions — in complex numbers, there are two
WE 3
Solve (x − 5)2 = −49 in Cartesian form
Solve the equation (x − 5)2 = −49, giving your answers in Cartesian form.
Step 1: Square root both sidesx − 5 = ± √(−49)x − 5 = ± 7iStep 2: Rearrange to isolate xx = 5 ± 7ix = 5 + 7i or x = 5 − 7ithese are complex numbers in Cartesian form: real part 5, imaginary part ±7
WE 4
Identify Re(z) and Im(z)
For each complex number, write down the real part and the imaginary part:
(a) z = 6 − 11i (b) w = 4i (c) u = −7
Part (a): z = 6 − 11iRe(z) = 6, Im(z) = −11 (NOT −11i)Part (b): w = 4i = 0 + 4iRe(w) = 0, Im(w) = 4 (purely imaginary)Part (c): u = −7 = −7 + 0iRe(u) = −7, Im(u) = 0 (purely real)all three identified ✓remember Im(z) is just the coefficient of i — never include the i itself
WE 5
Find unknowns using equality
Find real numbers x and y such that
(3x − 1) + (2y + 5)i = 8 − 3i
Step 1: Equate the real parts3x − 1 = 83x = 9, x = 3Step 2: Equate the imaginary parts2y + 5 = −32y = −8, y = −4x = 3, y = −4two complex equations split into two real equations — solve each on its own
💡 Top tips
Memorise i2 = −1. This is the only new fact you need. Whenever you see i2 in your working, replace it with −1 and continue.
Always pull out the −1 first when simplifying square roots of negatives. So √(−12) = √(12) · i = 2i√3, not the other way around.
Don’t forget the ± when square-rooting. Every quadratic-style equation gives two complex solutions.
Re(z) and Im(z) are both real numbers. Im(z) is the coefficient of i, not the imaginary term itself. So Im(3 + 4i) = 4, not 4i.
Two complex numbers are equal only when both parts agree. Use this to split a complex equation into two real ones — a really common technique.
Use your GDC’s complex mode for arithmetic checks. Most calculators have a “rectangular” or “Cartesian” setting that handles i directly.
Real numbers are a subset of complex numbers. Every real x is also x + 0i. Any rule that holds for complex numbers also holds for reals.
⚠ Common mistakes
Including i in Im(z). The imaginary part is the number 4, not the term 4i. Im(3 + 4i) = 4, full stop.
Forgetting the ± when square-rooting. x2 = −9 has two solutions, not one.
Treating i like a variable. i is a fixed number with i2 = −1. Don’t try to “factor out” or “cancel” it like a letter.
Trying to compare real and imaginary parts as if they’re the same kind of thing. 3 + 2i ≠ 3 − 2i, even though the real parts agree — the imaginary parts must also match.
Getting i2 wrong. i2 = −1, not 1, not −i, not i. Memorise this one fact above all others.
Mixing up √(−k) with −√k. The first is i√k; the second is just a negative real number.
Forgetting that purely real and purely imaginary numbers are still complex. Every real number x is the complex number x + 0i.
Complex numbers are the gateway to a huge area of mathematics — they show up everywhere from electrical engineering to quantum mechanics to computer graphics. The single new symbol i carries an enormous amount of weight, but at this stage your job is just to get comfortable with the notation: real part, imaginary part, equality, and solving equations that previously seemed impossible. The next note builds on this with arithmetic — adding, subtracting, multiplying, and dividing complex numbers.
Need help with Complex Numbers?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
