AA HL • Lesson rewrite

Complex Numbers

A fuller introduction to \(a+bi\), operations, conjugates, modulus, argument and the Argand diagram.

Quality Repair Patch 1: incorrectly placed videos have been removed from this topic until the video library is fully verified.

1. Why complex numbers?

The real number system has no solution to \(x^2=-1\). Complex numbers extend the number system by defining a new number \(i\) such that:

\[i^2=-1\]

This lets us solve equations such as \(x^2+1=0\), whose solutions are \(x=\pm i\).

2. Cartesian form

A complex number is usually written as:

\[z=a+bi\]

The number \(a\) is the real part and \(b\) is the imaginary part.

\[\operatorname{Re}(z)=a\]
\[\operatorname{Im}(z)=b\]

3. Argand diagram

An Argand diagram represents \(z=a+bi\) as the point \((a,b)\), with the real part on the horizontal axis and the imaginary part on the vertical axis.

ReImz=a+biabθ
The modulus is the distance from the origin; the argument is the angle from the positive real axis.

4. Modulus, argument and conjugate

Modulus\[|z|=\sqrt{a^2+b^2}\]
Argument\[\arg(z)=\theta\]
Conjugate\[\overline z=a-bi\]

The conjugate reflects a complex number in the real axis. It is especially useful because:

\[z\overline z=(a+bi)(a-bi)=a^2+b^2\]

5. Worked example

Let \(z=3+4i\).

  1. \(\operatorname{Re}(z)=3\), \(\operatorname{Im}(z)=4\).
  2. \(|z|=\sqrt{3^2+4^2}=5\).
  3. \(\overline z=3-4i\).
  4. \(\arg(z)=\tan^{-1}(4/3)\), since the point is in the first quadrant.