Imaginary & Complex Numbers

No real number squares to a negative, so mathematicians defined a new one: i, with i² = −1.

That makes √(−25) = 5i, since (5i)² = 25i² = −25.

A complex number combines a real and an imaginary part: a + bi. In 3 − 4i, the real part is 3 and the imaginary part is −4.

Powers of i cycle every four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then it repeats. To find a high power, divide the exponent by 4 and use the remainder.

Add and subtract by combining like parts. Multiply with FOIL, then replace i² with −1.

The imaginary unit is defined by i² = −1, and ℂ is the set {a + bi : a, b ∈ ℝ}. This extension makes ℂ algebraically closed: every non-constant polynomial has a root, which is the Fundamental Theorem of Algebra.

Powers of i are periodic with period 4, so i^n = i^{n mod 4}.

Arithmetic follows the field axioms with the single extra rule i² = −1. Multiplication uses FOIL and then that substitution.

Division uses the complex conjugate a − bi. Since (a + bi)(a − bi) = a² + b² is real and non-negative, multiplying numerator and denominator by the conjugate rationalises the denominator — structurally identical to rationalising √ denominators with conjugates.

One caution: the identity √a·√b = √(ab) requires non-negative arguments. Writing √(−4)·√(−9) = √36 = 6 is wrong; converting to 2i·3i = 6i² = −6 first is the correct route. Always extract i before multiplying radicals of negatives.