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Chapter 4 · Complex Numbers and Quadratic Equations

An equation with no real solution, and the symbol invented to solve it

Why the real numbers are not enough13 min

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13 min.

Square twenty-one ordinary numbers and not one comes out negative. That is evidence, not proof. The proof is three cases — positive, negative, nothing — none below zero.

The idea

x² + 1 = 0 is not hard over the real numbers — it is impossible, and the reason is a one-line argument about signs that rules out every real number at once, so no cleverness will ever find a root. Faced with that, the book does not search harder; it builds a larger number system in which one new object, written i, is declared to satisfy i² = −1. What makes this construction honest rather than wishful is that nothing else is assumed: every number in the new system is written from two ordinary real numbers, and all the arithmetic to come is defined using real arithmetic plus that single relation.

What you should be able to do

  • State why x² = −1 has no solution among the real numbers, giving the sign argument rather than an appeal to a calculator or a graph
  • Distinguish "no solution" from "no solution in this system", and say why the qualifier matters
  • Write down the defining relation for the symbol i and identify it as the one new assumption being made
  • Recognise a number written in the form a + ib, and state that a and b are both ordinary real numbers
  • Read the chapter's three sample complex numbers and say what a and b are in each
  • Explain, using the Historical Note, how Hamilton removed the mystery from a+ib by recasting it as a pair of ordinary reals taken in a fixed order
  • Check that a pair of numbers involving a root of a negative quantity satisfies a stated sum-and-product condition

Words to know

TermDefinition in one lineFirst introduced
complex numbera quantity written from two real numbers a and b in the form a + ibprinted in this chapter, §4.2, p. 76
real numbera number on the ordinary number line, the system this chapter starts fromassumed from earlier classes; used throughout §4.1–4.2, p. 76
non-negativezero or greater — the property of every real square that blocks x² = −1printed in this chapter, §4.1, p. 76
ordered pairtwo numbers written in a fixed order, so that (a, b) and (b, a) differprinted in this chapter, in the Historical Note, p. 88
square roota quantity whose square is the given numberprinted in this chapter, §4.3.6 heading, p. 79
imaginary numbersthe older name for the new quantities, which the Historical Note reports Hamilton set out to avoid needingprinted in this chapter, in the Historical Note, p. 88
the unit ithe new object satisfying i² = −1, from which the whole system is builtthe explanation's shorthand; the chapter introduces the symbol without giving it a name of its own
number system extensionenlarging a system of numbers so that an equation unsolvable in the old one gains a solutionan added phrasing; §4.1 performs the move on p. 76 but supplies no name for it

Where people slip up

  • "There must be some real number that squares to −1; we just haven't found it." The sign argument is not a failed search, it is a proof of absence: a real number is positive, negative or zero, and each of the three cases squares to something that is not negative. Nothing is left to search.
  • "i is imaginary, so the arithmetic built on it is make-believe." The chapter's own Historical Note gives the reply: Hamilton recast the form a + ib as the pair (a, b) taken in order, so the object is two perfectly ordinary reals, and the word imaginary is a historical label rather than a description.
  • "In a + ib, the b is imaginary." Both a and b are ordinary reals. What is new is the symbol they are combined with, not the numbers themselves.
  • "√(−1) is just notation, so I can do anything with it." It carries exactly one property, i² = −1. Every later rule in the chapter has to be derived from that plus ordinary real arithmetic — and §4.3.6 shows a familiar surd rule breaking precisely because someone assumed more than that one property.
  • "Extending a number system means the old rules are gone." The reals sit inside the new system as the numbers with b = 0, and real arithmetic is what the new arithmetic is defined in terms of.

Worked answers: Exercise 4.1 · Miscellaneous Exercise

Transcript1,898 words

Here is an equation: x squared plus one equals nothing. Move the one across and it says something plainer. x squared is minus one. Every method you have ends in the same place — find a number whose square is minus one. So try. Nothing squared is nothing. One squared is one. Minus one squared is also one. A half squared is a quarter. Root two squared is two. Ten squared is a hundred.

Twenty one ordinary numbers were tried here, spread across the line, and every single square came out at nothing or above. Twenty of them above, one exactly nothing, and not one of them below. And that settles precisely nothing. Twenty one failures is a failed search, and a failed search only tells you that you did not find one. It does not tell you there is none to find. Stop there and you have a hunch.

So the first thing to do is turn the hunch into an argument. Stop searching, and start arguing. Take a number on the line — not a particular one, any one at all. It falls into one of exactly three cases: below nothing, exactly nothing, or above nothing. There is no fourth case, and that is the whole trick. Now square each case in turn. Below times below is above. Nothing times nothing is nothing. Above times above is above.

Three cases, and in none of them does the answer come out below nothing. Set that against the search. Twenty one numbers tried, none below: that is evidence. Three cases considered, none below: that is a proof. The second one covers every point on the line at once, including every point nobody will ever write down. And the sign rule it leans on was checked rather than believed. Set the checker against a deliberately broken rule, one that gets unlike signs the wrong way round, and it finds all two hundred of the pairs it should.

Set it against the real one and it finds none. So x squared equals minus one has no solution. Almost. Read what was actually proved, word for word. Not: this equation has no solution. What was proved is: this equation has no solution among the numbers on the line. Those are two different sentences, and the difference between them is the door. The first is a statement about the equation.

The second is a statement about the line — about which numbers you happen to be holding. And you have been through this before, twice, probably without noticing. Take three away from two. Among the counting numbers there is no answer, so the negatives were brought in, and then there was. Divide one by three. Among the whole numbers there is no answer, so the fractions were brought in, and then there was.

Both times the equation did not change at all. The system changed. Nobody calls either of those cheating, so the third time should not feel like cheating either. There are two ways to respond, and both are honest. The first is to call it impossible and walk away. That is respectable, and it is also final — every question of that shape is closed forever. The second is to build a bigger system: one that contains the whole line unchanged, and one thing more.

But the second is only honest under a condition, and the condition is strict. You may add exactly one new object, and you may grant it exactly one new property. Everything after that has to be worked out — from ordinary arithmetic and that single property, and from nothing else. If you help yourself to a second property later, when the first one runs out, you have not built anything.

You have written down what you wanted and called it a result. One object, one property. Watch what that costs, and watch what it buys. Call the new object i. Now: what property? The obvious answer is that its square is minus one — but that is the answer, and answers are worth less than reasons. So ask it as a question instead. Suppose the new symbol squares to some ordinary number, and try nine of them: minus four, minus three, minus two, minus one, nothing, and then one, two, three and four.

Ask each one: does the system it builds contain a number whose square is minus one? Four of the nine do. And they are exactly the four where the square is below nothing. In the other five the search comes back empty-handed and refuses, five times out of nine. So the thing that matters is not the minus one at all. It is that the square is negative in the first place.

Now a sharper question. In how many of the nine is the new symbol itself the missing solution, rather than merely something you could build one out of? Exactly one — the one where the square is minus one. That is the reason it is minus one. Not magic, and not a convention: a choice, with a reason you can count. Now write down what the new system actually holds.

Every number in it looks like a plus i b, where a and b are two ordinary numbers off the line. Both of them ordinary. What is new is the symbol they are combined with, not the numbers themselves. So multiply two of them, the way you would multiply anything. Expand the brackets with ordinary algebra and you get four terms. Only the last of the four has anything new in it, because only the last one has i twice.

Replace that by minus one, and collect. What comes out is a c minus b d, plus i times a d plus b c. That was never a rule you had to be handed. It was worked out, from ordinary algebra and one substitution, and checked across six hundred and twenty five combinations without a single disagreement. And here is the test that it really is a consequence rather than an extra assumption.

That same formula survives at only one of the nine candidate systems. Change what the symbol squares to, and the formula breaks — which is exactly what a consequence is supposed to do. Three of these numbers, read part by part. Two plus i three: its a is two, and its b is three. Minus one, plus i root three: its a is minus one, and its b is root three.

Four, plus i times minus one eleventh: its a is four, and its b is minus one eleventh. That is six ordinary numbers between them. Four of the six are whole numbers. Two of the six are below nothing — the minus one and the minus one eleventh. One of the six is not a ratio of whole numbers at all, and that is root three. Not one of the six is strange.

Every one is a number you have handled since primary school. That matters, because the word attached to these historically is imaginary, and it has done a great deal of damage. It is a name with a history, not a description of anything. One new property, and everything else derived. Both halves of that are worth checking. First half: the old numbers are still in there, and still themselves. Any ordinary number sits in the new system as a plus i nothing.

Multiply two of those together and you get exactly what you always got — across the sample, not one pair stopped behaving. Second half, and this is the one people skip: the new symbol does not license everything. There is a rule you have used for years — root a times root b is root a b. Try it on negatives. Root minus one, times root minus one. The rule says root of one, which is one.

The arithmetic says i times i, which is minus one. Over sixty four pairs of negatives, that rule survives exactly none of them. In all sixty four the answer comes out as the exact opposite of what the rule predicted. One property, not two. The rule you were not given, you do not get. Around eighteen thirty, William Rowan Hamilton answered the sceptic, and the answer is almost annoying in how simple it is.

a plus i b, he said, is the pair a, b. Two ordinary numbers, written in a fixed order. That is the object, and there is nothing else to it. The i is bookkeeping — it marks which of the two is which. And the order is not decoration. Take twenty five pairs built from a short list of numbers, write each one the other way round, and ask how many come back unchanged.

Five of the twenty five. And they are exactly the five whose two parts were equal to begin with. For all the rest, the order is carrying real information. So there is nothing left to be sceptical about. You are not being asked to believe in a strange number, only to keep track of two ordinary ones at once, and to remember which is which. Nearly three centuries before Hamilton, in fifteen forty five, Cardan ran into two conditions at once.

Find two numbers that add to ten and multiply to forty. The pair he produced was five plus root minus fifteen, and five minus root minus fifteen. He wrote them down, and then he called them useless. Do not take that pair on trust. Search the new system for a number whose square is minus fifteen. Exactly two of them exist there. Take either one. Now add the pair. The two roots are opposites, so they cancel, and what is left is ten — exactly ten, with nothing of the new symbol in it at all.

Now multiply the pair. That comes to twenty five, less the square of the root. The square is minus fifteen, so it is twenty five plus fifteen. Forty. Again with nothing of the new symbol left behind. Both conditions, exactly, with nothing rounded. And here is the sharp part. Search the ordinary numbers for one that does the same job — one where twenty five, less its square, is forty.

There are none. Not one, anywhere on the line. The numbers Cardan threw away are the only numbers that work. So where does that leave you. You have one new object, carrying exactly one property. You have a form to write numbers in, a plus i b, with two ordinary numbers doing all of the work. You have multiplication, because it was derived rather than assumed. And you have a pair of numbers three centuries old that finally do the job they were thrown out for failing.

What you do not have yet is a number system. Adding and subtracting have to be defined. Division has to be defined, and it is not obvious. Size has to be given a meaning again, because bigger stopped making sense the moment the line stopped being enough. And there is a picture waiting: those two ordinary numbers are coordinates, and the whole system lies flat on a plane. All of that is still ahead, and none of it needs a second assumption.

The line was not enough. One symbol, one property, and two ordinary numbers is the whole of what was added.

Where this fits

Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.

Comes up again in

Either side of this one

The book

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