Need for Complex Numbers

Need for Complex Numbers

Consider the simple-looking equation $x^2+1=0$, i.e. $x^2=-1$. Within the real numbers this equation has no solution, because the square of every real number is never negative: a positive real squares to a positive real, a negative real squares to a positive real (since a negative times a negative is positive), and $0^2=0$. So there is no real $x$ with $x^2=-1$.

The same gap shows up for an entire family of quadratic equations. For $ax^2+bx+c=0$ ($a,b,c$ real, $a\neq0$), the quadratic formula gives

Whenever the discriminant $D=b^2-4ac$ is negative, the formula asks us to take the square root of a negative number — something the real number system simply cannot do. For example, $x^2-2x+5=0$ has $D=4-20=-16<0$, so it has no real root, even though it is a perfectly ordinary-looking quadratic.

$x^2+4=0 \Rightarrow x^2=-4$. No real number squares to $-4$: this equation, too, has no real solution.

To close this gap, mathematicians introduced a new number, denoted $i$ (read as iota), with the single defining property

This is not a real number — it is a genuinely new symbol, adjoined to the real numbers to build a larger system. Once $i$ exists, $\sqrt{-1}=i$, $\sqrt{-4}=\sqrt{4}\cdot\sqrt{-1}=2i$, and more generally $\sqrt{-a}=\sqrt{a}\,i$ for any positive real $a$.

A complex number is any expression of the form $z=a+ib$, where $a$ and $b$ are real numbers. Every real number $a$ is also a complex number (take $b=0$: $a=a+0i$), so the real numbers sit inside this larger system, written $\mathbb{R}\subset\mathbb{C}$. With $i$ available, $x^2+1=0$ now has two solutions, $x=i$ and $x=-i$, and — as this chapter will show — every quadratic equation with real coefficients has a solution once we work inside $\mathbb{C}$, the set of complex numbers. The rest of this chapter develops the arithmetic of these new numbers, their geometric picture in the Argand plane, and how they solve every quadratic equation, whatever the sign of its discriminant.