Complex numbers in Cartesian form

Higher Level

Start at z = 3 + 4i and multiply by i again and again. Press play and watch where it goes, and what stays the same.

3 + 4ithe number

5modulus, |z|

0.927argument, rad

The distance from the origin never changes. Only the direction does.

The vocabulary

For z = a + bi: a is the real part, b the imaginary part (a real number, not bi), z* = a − bi is the conjugate, |z| = √(a² + b²) is the modulus, and the argument is the angle from the positive real axis.

z = 3 + 4i→ |z| = √(9 + 16) = 5 arg z= arctan(4/3) = 0.927 rad z z*= (3+4i)(3−4i) = 25 = |z|²

Arithmetic

add→ parts separately, like vectors multiply→ expand, then use i² = −1 (3+4i)(1+2i)= 3 + 6i + 4i + 8i² = −5 + 10i divide→ multiply top and bottom by the conjugate of the bottom

Dividing works because z z* is always real. Multiplying the bottom by its conjugate clears the i out of the denominator, exactly as multiplying by √2 clears a surd. It is the same trick for the same reason.

“Imaginary” is a bad historical name. These numbers describe alternating current, signal processing and every rotation in two dimensions. The figure above is the reason: complex multiplication is rotation and scaling, which is why engineers use them daily.

Worked example

z = 5 + i and w = 2 − 3i.

Find z/w in the form a + bi.

Multiply top and bottom by the conjugate of w, which is 2 + 3iThe conjugate is chosen so the bottom becomes real. Exactly the surd trick, for the same reason.
Bottom: (2 − 3i)(2 + 3i) = 4 + 9 = 13The i terms cancel and i² makes the last term positive. The result is the modulus squared.
Top: (5 + i)(2 + 3i) = 10 + 15i + 2i + 3i²Expand carefully. Four terms, and the last one is about to change sign.
= 10 + 17i − 3 = 7 + 17iBecause i² = −1, so 3i² is −3.
z/w = (7 + 17i)/13 = ⁷⁄₁₃ + ¹⁷⁄₁₃iSplit the real and imaginary parts over the denominator. Leaving it as a single fraction with i on top is not finished.

Your turn

1. z = 3 + 4i. Find |z|.

2. Find the real part of (3 + 4i)(1 + 2i).

3. Find the imaginary part of i(3 + 4i).

Where the marks go

Using i² = −1 and collecting real and imaginary parts separately.

The imaginary part is a real number. For 3 + 4i it is 4, not 4i.

Multiplying by the conjugate when dividing, and simplifying to a + bi rather than leaving a fraction with i underneath.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

Find the 10th term of 3, 6, 12, 24.

Geometric: 3 × 2⁹ = 1536. Divide consecutive terms to find r; subtracting would send you to the wrong formula.
Which method?

What is the imaginary part of −4 + 3i?

A real number: 3, not 3i. The imaginary part is the coefficient.
Which method?

A test is 99% accurate for a disease affecting 1 in 1000. Someone tests positive. Roughly how likely is it that they have it?

Bayes, and the base rate. About 9%, because there are far more healthy people to get wrong.

The same idea elsewhere

1.13, polar form: Rotation is what the next page makes explicit, and it starts here.
5.18, second order equations: Complex eigenvalues are what make a system spiral rather than settle.

Want a verdict on your own draft?

These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.

Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.

Get a free verdict Full written review, $99

I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.