Algebra of Complex Numbers

i is defined as the square root of −1. This is not a real number — no real number squares to give a negative. The symbol i (iota) extends the real number system to make equations like x² + 1 = 0 solvable.

Every complex number is written as a + ib where a is the real part and b is the imaginary part. Re(z) = a, Im(z) = b. When b = 0, z is purely real. When a = 0 and b ≠ 0, z is purely imaginary.

Re(z) and Im(z) are both real numbers — b is the imaginary part, not ib. A common error is writing Im(z) = ib. Note: Im(z) = 0 iff z is real; Re(z) = 0 and Im(z) ≠ 0 iff z is purely imaginary.

Two complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal. This splits one complex equation into two real equations — a powerful technique used throughout the chapter.

Real parts add with real parts, imaginary parts add with imaginary parts. Addition of complex numbers is commutative and associative. The additive identity is 0 + 0i = 0. The additive inverse of a + ib is −a − ib.

Subtract real parts and imaginary parts separately. Subtraction is equivalent to adding the additive inverse: (a + ib) − (c + id) = (a + ib) + (−c − id).

Expand like ordinary binomials, then replace i² with −1. The −bd term comes entirely from i² = −1. Multiplication is commutative, associative, and distributes over addition. The multiplicative identity is 1 + 0i = 1.

Multiply numerator and denominator by the conjugate of the denominator (c − id). This makes the denominator real: (c + id)(c − id) = c² + d². Valid when c + id ≠ 0, i.e., c and d are not both zero.

The powers of i cycle with period 4. This cycle repeats forever: i⁵ = i, i⁶ = −1, and so on. To find iⁿ for any positive integer n, divide n by 4 and use the remainder.

Reduce the exponent modulo 4. If n mod 4 = 0 → 1; if 1 → i; if 2 → −1; if 3 → −i. For negative exponents: i⁻¹ = −i, i⁻² = −1, i⁻³ = i, i⁻⁴ = 1 — the same cycle in reverse.