Watch: Arithmetic & the conjugate Opens a YouTube search: pick a video that clicks for you Definition i² = −1 ; z = a + bi a = real part Re(z), b = imaginary part Im(z). Conjugate z = a + bi ⟹ z̄ = a − bi z·z̄ = a² + b² (a real number). Modulus |z| = √(a² + b²) Distance from the origin on the Argand plane. Division (multiply by conjugate) \frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{c^2+d^2} Multiply top and bottom by the conjugate of the denominator. Worked example Dividing complex numbers Simplify (3 + 2i) / (1 − i). Multiply top and bottom by the conjugate (1 + i): = (3 + 2i)(1 + i) / [(1 − i)(1 + i)] = (3 + 3i + 2i + 2i²) / (1 + 1). = (3 + 5i − 2) / 2 = (1 + 5i)/2. ✓ ½ + (5/2)i.