Extend the number line

The number i satisfies $i^{2}$ = −1. A complex number a + bi is a point (a, b) on the Argand plane. Its conjugate a − bi reflects that point across the real axis.

Multiply complex numbers by expanding and replacing $i^{2}$ by −1. To divide, multiply numerator and denominator by the denominator’s conjugate. The resulting denominator is real.

\displaystyle z\overline z=a^2+b^2,\quad |z|=\sqrt{a^2+b^2}

EXPLORE THE ARGAND PLANEReal part: 3Imaginary part: 2Modulus 3.61 · Argument 33.7°

WORKED EXAMPLE

Simplify $\frac{3+i}{1-i}$ .

Multiply by $\frac{1+i}{1+i}$ .
The numerator becomes 2 + 4i and the denominator becomes 2.
The result is 1 + 2i.

Assumed knowledge

Expand and factorise algebraic expressions, simplify fractions, and use index laws. Solve real quadratic equations and recognise when the discriminant is negative.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Extend the number line

Simplify 3+i1−i\frac{3+i}{1-i}1−i3+i​.

Put the idea to work.

Simplify $\frac{3+i}{1-i}$ .