Pure · Complex numbers
Chapter 1 · 3
The idea
The Argand diagram, modulus and argument
A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.
In this lesson — start anywhere
Pure · Complex numbers
The Argand diagram, modulus and argument
Picturing a complex number z = a + bi as a point (or vector) on the Argand diagram, its modulus |z| = √(a²+b²) as the distance from the origin, and its argument arg z as the angle from the positive real axis — taken as the principal value in (−π, π], which means getting the quadrant right, not just arctan(b/a). Geometric effects: the conjugate reflects in the real axis, addition is vector addition, and |z₁ − z₂| is the distance between the two points.
Why it works
A number that needs a plane
A real number lives on a number line. A complex number needs two coordinates — the real part and the imaginary part — so it lives on a plane. The Argand diagram is exactly that plane: the horizontal axis is the real axis and the vertical axis is the imaginary axis. The number is plotted at the point , and you can equally think of it as the vector arrow from the origin to that point. So sits across and up; sits in the bottom-left.Keep reading — free
The rest of the explanation, plus 5 worked examples you step through move by move.
Start freeTakes a minute — no card.