1-2 The complex number: equality, addition and multiplication

Definition

• A complex number is z = a + bi where a, b ∈ ℝ and i² = −1.
• a is the real part, written Re(z); b is the imaginary part, written Im(z). Both parts are real numbers.
• If b = 0 the number is real; if a = 0 and b ≠ 0 it is purely imaginary.

Equality of two complex numbers

• a + bi = c + di ⟺ a = c and b = d: equate real parts to real parts and imaginary to imaginary, then solve the two equations.

Addition and subtraction

• (a + bi) + (c + di) = (a + c) + (b + d)i — add the real parts together and the imaginary parts together.

Multiplication

• Multiply as two algebraic expressions, then replace i² by −1.
• (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
• Example: (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i.