Computing with complex numbers

Computing with complex numbers works exactly as with algebraic expressions, plus a single rule: wherever i² appears, write -1.

Addition and subtraction

Component by component:

(3 + 2i) + (1 - 5i) = (3 + 1) + (2 - 5)i = 4 - 3i

Multiplication

Expand, then replace i²:

(2 + 3i)(1 - i) = 2 - 2i + 3i - 3i²
                = 2 + i - 3×(-1)
                = 2 + i + 3
                = 5 + i

The one reflex to acquire: -3i² is +3, not -3. That sign change is the most frequent mistake of the chapter.

The conjugate

The conjugate of z = a + ib is obtained by flipping the sign of the imaginary part:

z = a + ib      ->      z̄ = a - ib

Its decisive property: the product of a complex number by its conjugate is a positive real number.

z × z̄ = (a + ib)(a - ib) = a² - (ib)² = a² + b²

This is the identity (x + y)(x - y) = x² - y² applied with y = ib. The result, a² + b², contains no i at all.

Other rules, all easy to remember:

conjugate of a sum      = sum of the conjugates
conjugate of a product  = product of the conjugates
z is REAL             <=>  z̄ = z
z is PURELY IMAGINARY <=>  z̄ = -z

The modulus

|z| = √(a² + b²)          hence      z × z̄ = |z|²

This generalises absolute value: |z| measures the "size" of z, and the next chapter shows it is its distance to the origin. The modulus is always positive or zero, and it is multiplicative: |zz'| = |z| × |z'|.

Division: the technique to remember

How do you get rid of an i in a denominator? By multiplying top and bottom by the conjugate of the denominator:

  3 + i         (3 + i)(1 + 2i)         3 + 6i + i + 2i²
--------  =  ----------------------  =  ------------------
  1 - 2i       (1 - 2i)(1 + 2i)              1² + 2²

             3 + 7i - 2       1 + 7i        1     7
          =  -----------  =  --------   =  --- + --- i
                  5              5          5     5

The denominator becomes |1 - 2i|² = 5, a real number: the fraction is back in algebraic form. This is the chapter's most frequent computation.

The formula table

z = a + ib                z̄ = a - ib               |z| = √(a² + b²)

z + z̄ = 2 Re(z)           z - z̄ = 2i Im(z)          z z̄ = |z|²

1/z = z̄ / |z|²            |z z'| = |z| |z'|

Summary

• Compute as usual, replacing i² by -1.
• The conjugate z̄ = a - ib satisfies z z̄ = a² + b² = |z|², a positive real.
• The modulus |z| = √(a² + b²) generalises absolute value and is multiplicative.
• To divide, multiply numerator and denominator by the conjugate of the denominator.
• z real ⟺ z̄ = z; z purely imaginary ⟺ z̄ = -z.