Numbers & Functions

8 Numbers That Shouldn't Exist

CHAPTER 8 Numbers That Shouldn’t Exist We’ve reached the end, and the end is also the oldest beginning. Everything this book planted comes due in this chapter.

Cast your mind all the way back to Chapter 1. There was a pattern, the deepest one in the whole book: every time an operation broke, we invented the numbers that made it work. Subtraction broke, so we built the negatives. Division broke, so we built the fractions. Taking roots broke, so we built the irrationals. And then, in Chapter 4, we hit a wall and did something we’d never done: we stopped. The quadratic formula handed us the square root of a negative number, and we had no number for it. We’re going to open that door now — and when we do, every loose thread in this book pulls tight at once.

8.1 The wall, one more time

Here is the wall, in its simplest form. We want a number whose square is −1: x2 = −1.

Among all the real numbers, there is no such number. A positive squared is positive. A negative squared is also positive ((−1) × (−1) = +1). Zero squared is zero. Nothing on the entire real line squares to give a negative. The square-root machine jams, exactly as it did in Chapter 3.

For two and a half thousand years, this was the end of the conversation. “No solution.” But you know, by now, that this is never really the end. It’s an invitation. Every other wall in this book fell the same way — not by working harder within the old numbers, but by inventing the new number the problem was crying out for. So let’s do it one final time, and do it boldly.

8.2 The bold move

We define a new number — call it i — whose entire job is to be the thing that squares to −1: i2 = −1.

That’s the whole definition. We’re not claiming to have found i hiding among the reals — it isn’t there. We’re declaring it into existence, the same way the ancients declared −2 into existence when subtraction demanded it. The letter i stands for “imaginary” — a slightly unfortunate name, a leftover insult from mathematicians who first distrusted it. There is nothing more imaginary about i than there is about −2. You have never held negative two apples in your hand either. Both are inventions that earn their keep by making the mathematics work. A number built using i — like 3 + 2i — is called a complex number. And here is the first sign we’ve done something right: with i in hand, every quadratic now has roots. Remember x2 + 2x + 5 = 0 from Chapter 4, the one with no real roots? The quadratic formula gives x = −2 ± √4 −20 2 = −2 ± √−16 2 .

That √−16 used to be the wall. No longer: √−16 = √ 16 · √−1 = 4i. So x = −2±4i 2 = −1 ± 2i. Two roots, −1 + 2i and −1 −2i. The quadratic that “had no solutions” had two all along — we just lacked the numbers to write them down. The wall is gone.

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