The Complex Number System and Its Properties

Complex numbers exist because x² = -1 has no answer among the reals: no number times itself gives a negative. Define i so that i² = -1, and suddenly every polynomial has the solutions it should have. A complex number is written a + bi, with a and b both real. The real numbers are the complex numbers where b = 0 - they are not a separate system, just the b = 0 line sitting inside the bigger plane. That is also why 7 counts as complex: it is 7 + 0i. A number like 5i is complex too, with a = 0; it is called pure imaginary, but pure imaginary is a type of complex number, not its opposite.

The conjugate of z = x + bi is x - bi, flip the sign of the imaginary part only. Multiply them out: (x + bi)(x - bi) = x² - (bi)² = x² + b^2. The cross terms xbi and -xbi cancel, and -(bi)² becomes +b² since i² = -1. The result is always real, which is the whole reason conjugates matter: they are the tool for clearing i out of a denominator or confirming a root pattern.

This is also why polynomial degree and root count always match once complex numbers are allowed. A degree-n polynomial with real coefficients has exactly n zeros counting multiplicity, and any nonreal zeros come in conjugate pairs - so m pairs use up 2m of those n slots, leaving n - 2m real zeros. The same algebra extends geometrically: treating a + bi as the point (a, b) turns complex multiplication by i into a 90 degree rotation, which is exactly what a particular 2x2 rotation matrix does to a vector.