Complex Numbers

A complex number has the form $a+bi$, where $a$ and $b$ are real and$i^2 = -1$. Addition and subtraction work componentwise; multiplication distributes like binomials, using $i^2=-1$ to simplify:

Conjugates and division

The conjugate of $a+bi$ is $a - bi$. Multiplying a complex number by its conjugate always produces a real number:

This is exactly how you divide complex numbers: multiply numerator and denominator by the denominator's conjugate to clear $i$ from the bottom.

The modulus $|a+bi| = \sqrt{a^2+b^2}$ measures distance from the origin in the complex plane — it's the same quantity that appears under the square root above.

Complex roots of quadratics

When the discriminant of $ax^2+bx+c=0$ is negative, the quadratic formula still works — it just produces two complex conjugate roots, since $\sqrt{-k} = i\sqrt{k}$ for $k>0$. Vieta's formulas still hold exactly as before, even though the roots are no longer real.

Practice problems