Complex Numbers

The Imaginary Unit

Some equations, like $x^2 = -1$, have no real solution. Mathematicians extend the reals by introducing the imaginary unit $i$, defined by $i^2 = -1$. A complex number takes the form $z = a + bi$, with real part $a$ and imaginary part $b$.

Arithmetic treats $i$ as a symbol alongside real numbers:

Common pitfall: Always remember $i^2 = -1$, not $+1$. A product like $(bi)(di) = -bd$ contributes to the real part, whereas addition strictly keeps real and imaginary parts separate.

Geometry of Complex Numbers

Geometrically, a complex number is a point in the plane with its real part on the horizontal axis and imaginary part on the vertical.

Key geometric terms:

Multiplying complex numbers adds their arguments and multiplies their moduli. Thus, multiplying by a unit complex number is a rotation.

This deep link powers signal processing, computer graphics, and quantum computing. It ensures every polynomial equation is solvable, known as the Fundamental Theorem of Algebra.