H2 Maths Notes (JC 1-2): 4) Complex Numbers

Before you begin\ Review your Additional Maths introduction to complex arithmetic so operations with ( a + bi ) feel fluent. Switch the GC to Complex or Polar mode as needed and practice toggling between rectangular and polar displays.

4.1 | Core Concepts and Forms

Representations

• Cartesian form: \( z = a + bi \) with real part ( \Re(z) = a ) and imaginary part ( \Im(z) = b ).
• Polar form: ( z = r (\cos\theta + i\sin\theta) ) where ( r = |z| = \sqrt{a^2 + b^2} ) and ( \theta = \arg(z) ) measured in radians unless stated.
• Euler link: ( z = r (\cos\theta + i\sin\theta) ) corresponds to the shorthand r*exp(i*theta); switch forms to suit the question.

Modulus and argument

• Compute ( |z| = \sqrt{a^2 + b^2} ). State exact surd form until a decimal is required.
• The principal argument satisfies ( -\pi < \theta \leq \pi ). Adjust by adding/subtracting ( 2\pi ) for other branches.
• When plotting, move ( a ) units along the real axis then ( b ) units along the imaginary axis.

Conjugates

• Conjugate ( \overline{z} = a - bi ).
• Useful identity: multiplying a complex number by its conjugate produces the squared modulus (( |z|^2 )).
• Reciprocal form: for non-zero z, multiply numerator and denominator by the conjugate so the denominator becomes the real scalar ( |z|^2 ).

4.2 | Operations and Powers

• Addition/subtraction: combine real and imaginary parts directly.
• Multiplication/division: switch to polar form when arguments are involved; add/subtract arguments and multiply/divide moduli.
• De Moivre's theorem: ( \left( r(\cos\theta + i\sin\theta) \right)^n = r^n(\cos(n\theta) + i\sin(n\theta)) ) for integer ( n ).
• Roots: \( z^{n} = w \) has ( n ) distinct roots [ z_k = \lvert w \rvert^{1/n} \left[ \cos\left( \frac{\arg(w) + 2k\pi}{n} \right) + i\sin\left( \frac{\arg(w) + 2k\pi}{n} \right) \right] ] for ( k = 0, 1, \dots, n-1 ).

Example 1 -- Convert to polar form

Let ( z = -1 + i\sqrt{3} ).

1. Modulus: ( |z| = \sqrt{(-1)^2 + 3} = 2 ).
2. ( \tan\theta = \frac{\sqrt{3}}{-1} = -\sqrt{3} ); quadrant II gives ( \theta = \frac{2\pi}{3} ).
3. Hence ( z = 2 \left( \cos\left( \tfrac{2\pi}{3} \right) + i\sin\left( \tfrac{2\pi}{3} \right) \right) ).

Example 2 -- Cube roots of a complex number

Solve ( z^3 = 8 \left( \cos\left( \tfrac{3\pi}{4} \right) + i\sin\left( \tfrac{3\pi}{4} \right) \right) ).

1. Set \( r = 8 \), ( \theta = \tfrac{3\pi}{4} ).
2. Roots have modulus \( r^{1/3} = 2 \).
3. Arguments: ( \theta_k = \frac{3\pi}{4} \frac{1}{3} + \frac{2k\pi}{3} = \frac{\pi}{4} + \frac{2k\pi}{3} ) for \( k = 0, 1, 2 \).
4. Therefore [ z_k = 2 \left[ \cos\left( \frac{\pi}{4} + \frac{2k\pi}{3} \right) + i\sin\left( \frac{\pi}{4} + \frac{2k\pi}{3} \right) \right]. ]

4.3 | Loci on the Argand Plane

• Circle: ( |z - (a + bi)| = r ) represents a circle centre ( (a, b) ) with radius ( r ).
• Half-plane: ( \Re(z) > c ) or ( \Im(z) \geq d ) describes vertical or horizontal regions.
• Line: ( \arg(z - a) = \theta ) is a ray from ( a ) making angle ( \theta ) with the positive real axis.
• General locus problems often combine modulus and argument; square moduli to remove the square root before expanding.

Example 3 -- Region defined by inequalities

Sketch the region satisfying ( |z - 1 + 2i| \leq 3 ) and ( \arg(z - 1) \in \left[ 0, \tfrac{\pi}{2} \right] ).

1. First condition: circle centre ( (1, -2) ) radius ( 3 ).
2. Second condition: points relative to ( 1 ) lying in the first quadrant (between the positive real and positive imaginary axes).
3. The locus is the sector of the circle that falls within that quadrant.

4.4 | Calculator Workflows

• Use rectangular-polars conversion (r\theta or Abs/Arg) to verify modulus and argument steps; always record the keystrokes in your working.
• Store complex numbers in variables (e.g. Z1) to reuse them for repeated operations like powers or conjugates.
• For equation solving, switch to polynomial mode in complex form, but still justify analytically in the written solution.

4.5 | Exam Watch Points

• State the branch (degrees vs radians) for arguments and stay consistent throughout a question.
• When giving roots, list all distinct values with arguments in ascending order and include the general form if required.
• On locus questions, include a short sentence describing the geometry after the algebra (e.g. "circle centre... radius...").
• For division, multiply numerator and denominator by the conjugate to show the simplification explicitly.

4.6 | Quick Revision Checklist

• [ ] Toggle confidently between Cartesian and polar/euler forms.
• [ ] Apply De Moivre to powers and roots without skipping the argument step.
• [ ] Interpret loci conditions on the Argand plane with a labelled sketch.
• [ ] Explain the effect of conjugation and modulus in plain language after calculations.

Next step: drill sub-topic 4.1 with targeted practice on conversions, loci proofs, and De Moivre applications.