Complex Numbers

Extending the Number Line

Some equations, like $x^2 = -1$, have no real solution — no real number squares to a negative. Mathematicians extend the reals by introducing the imaginary unit $i$, defined by $i^2 = -1$. A complex number then has the form $z = a + bi$, with a real part $a$ and an imaginary part $b$. This modest extension makes every polynomial equation solvable — the Fundamental Theorem of Algebra.

Arithmetic works by treating $i$ as a symbol with $i^2 = -1$:

• Addition — add real and imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
• Multiplication — expand and use $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
• The conjugate of $a + bi$ is $a - bi$; multiplying a number by its conjugate gives the real value $a^2 + b^2$.

Geometrically, a complex number is a point in the plane — real part on the horizontal axis, imaginary part on the vertical. Its modulus $|z| = \sqrt{a^2 + b^2}$ is its distance from the origin, and its argument is the angle it makes. In this view, multiplying complex numbers adds their angles and multiplies their moduli — so multiplication by a unit complex number is a rotation. This deep link between complex arithmetic and rotation powers signal processing (the Fourier transform), computer graphics, and quantum computing.

Common pitfall: mishandling $i^2$, or adding complex numbers as if the parts mix. Remember $i^2 = -1$ (not $+1$), so a product like $(bi)(di) = bd\,i^2 = -bd$ contributes to the real part. And addition keeps the parts separate — real adds to real, imaginary to imaginary; you never add a real part to an imaginary part.