Introduction

Why We Need a Larger Number System

In earlier years you solved linear equations in one and two variables, and then quadratic equations of the form $ax^2+bx+c=0$ using the quadratic formula. That formula asks you to compute $\sqrt{b^2-4ac}$ — and it works perfectly as long as the discriminant $b^2-4ac$ is non-negative. But look at the simplest possible case: $x^2+1=0$, i.e. $x^2=-1$. Since the square of any real number is always zero or positive, no real number can ever square to $-1$. Every quadratic with a negative discriminant runs into exactly this wall.

Rather than declaring such equations "unsolvable," mathematicians chose to enlarge the number system itself. The mathematician Euler was the first to give the missing square root of $-1$ a name: the symbol $i$. Later, the Irish mathematician William Rowan Hamilton made the idea fully rigorous — instead of treating $i$ as a mysterious symbol you are simply told to accept, he defined a complex number as an ordered pair of real numbers $(a,b)$, governed by precise rules for addition and multiplication. This sidesteps any hand-waving about what "the square root of $-1$" really is: a complex number is just a pair, in exactly the same spirit as a point in a plane being a pair of coordinates.

What This Chapter Covers

Starting from Hamilton's ordered-pair definition, this chapter builds up the full algebra of complex numbers: the operations of addition, subtraction and multiplication on pairs; the $a+ib$ form and how it connects back to ordered pairs; the modulus and amplitude (argument) of a complex number; and finally the geometric picture — plotting a complex number as a point in the Argand plane and reading it in polar form. Once this machinery is in place, the equation $x^2=-1$ finally has a solution, and so does every quadratic your discriminant test used to reject.