AHL 1.12 (HL)—Complex numbers

Represent a complex number and read its geometry.

Write z=a+bi with i²=−1; Re(z)=a, Im(z)=b, modulus is distance from origin and conjugation reflects across the real axis.

Worked example
For z=3−4i, |z|=5 and its conjugate is 3+4i.

Worked example
What does conjugation change? the sign of the imaginary part only.

Common boundary
The modulus is not the imaginary part.

Argand and argument example: $z=3-4i$ is the point $(3,-4)$ on the Argand diagram. Its modulus is $|z|=\sqrt{3^2+(-4)^2}=5$ and its principal argument is $\arg z=\tan^{-1}(-4/3)\approx-53.13^\circ$ because the point lies in quadrant IV. Always use the quadrant, not inverse tangent alone, to select the argument.