Complex Numbers (HL)

A complex number extends the real numbers by introducing the imaginary unit $i$, defined by $i^2 = -1$. This single invention lets every quadratic equation have solutions: for example $x^2 + 1 = 0$ gives $x = \pm i$. In Cartesian (rectangular) form a complex number is written $z = a + bi$, where the real number $a$ is the real part $\operatorname{Re}(z)$ and $b$ is the imaginary part $\operatorname{Im}(z)$. Two complex numbers are equal only when their real parts match and their imaginary parts match, a fact called equating real and imaginary parts that turns one complex equation into two real ones.

Arithmetic in Cartesian form follows ordinary algebra with the rule $i^2 = -1$. Addition and subtraction act componentwise. Multiplication expands like a binomial: $(a+bi)(c+di) = ac + adi + bc