Part I · Numbers Chapter 2 of 60
Zero and minus
What is 3 − 5? A seventeenth-century sceptic is sure it's nothing, and he calls “minus times minus” nonsense. Let's argue with him by the rules: every sign rule will have to be defended.
Builds on: 1 · Counting: from notches to bits
You will learn
- add and subtract signed numbers by picturing them as arrows on a line
- prove from the distributive law that minus times minus is plus
- explain why you can't divide by zero, and find the absolute value of a number
1Why does minus times minus make plus?
The shepherd from the last chapter promised his neighbour five sheep, but he had three. How many sheep does he have now? A schoolchild answers without a second thought: minus two, he owes two sheep. But between "obvious to a schoolchild" and "accepted by mathematicians" lie almost two thousand years of argument. Negative numbers were called absurd, fictitious, false, and even today many people remember "minus times minus is plus" as a spell you simply have to learn.
A spell won't do for us. Let's hold a debate. We'll call our opponent the Sceptic. He is a composite of an educated seventeenth-century European, but his objections are not made up: similar arguments were put forward at various times by Diophantus, Michael Stifel, Gerolamo Cardano, René Descartes and Antoine Arnauld. The rules are simple. The Sceptic accepts the natural numbers and the familiar laws of arithmetic. Everything else — zero, minus, the sign rules — has to be defended using only what he has already accepted. Let's write these laws out explicitly: from here on every proof will rely only on them, on the familiar properties of natural numbers and on the definitions of new notions.
For any numbers $a$, $b$, $c$:
- the commutative laws: $a + b = b + a$ and $ab = ba$;
- the associative laws: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$;
- the distributive law: $a(b + c) = ab + ac$;
- zero and one: $a + 0 = a$ and $a \cdot 1 = a$.
For natural numbers the Sceptic has checked these laws a thousand times: heaps of pebbles can be pushed around and poured together, a rectangle of tiles can be turned or cut in two. The distributive law, for example, is a rectangle with sides $a$ and $b + c$, cut into rectangles $a \times b$ and $a \times c$; it is what expanding brackets and long multiplication rest on: $7 \cdot 23 = 7 \cdot 20 + 7 \cdot 3$. Our condition: new numbers are admitted only if these laws hold for them too. Otherwise we would lose expanding brackets, and all of algebra with it.
Round one: is zero a number?
Zero is nothing. A number answers the question "how many?", and "none" is not an answer but the absence of one. You told us yourself that zero was invented for an empty place. A symbol, fine. But a number?
What makes a symbol a number at all? That you can calculate with it by rules and get correct answers. The "empty place" of the last chapter had no rules: it only held a position. The first to give zero rules of arithmetic was the Indian astronomer Brahmagupta. In his treatise Brahmasphutasiddhanta, written in 628, he lists them: a number plus zero is the same number, a number minus zero is the same number, zero minus zero is zero, and the product of zero with any number is zero. From that moment zero is added, subtracted and multiplied on an equal footing with the other numbers.
$$a + 0 = a, \qquad a - 0 = a, \qquad a - a = 0, \qquad a \cdot 0 = 0.$$And zero answers "how many?" perfectly well. How many apples are in an empty basket? How many goals did the team score in a 0–3 defeat? Without zero the subtraction $5 - 5$ would have no result, and arithmetic would trip over nothing. The rule $a \cdot 0 = 0$ also makes sense: taking five coins zero times means taking nothing. And in round four we'll see that it follows from the laws of arithmetic for negative numbers too.
The word itself reached us by a roundabout route. In the last chapter the Arabic sifr, "empty", turned into "cipher" and "zero". The older English word "nought" means simply "not anything", and the Latin nullus, "none", gave us "null". Brahmagupta went beyond addition and multiplication and tried to divide by zero, but that's for round five: there he went wrong.
Very well. Zero is a number, since it's so convenient to calculate with. But I won't go below zero.
Round two: less than nothing
You can't have less than nothing. If I have three apples, I can't give away five, and no arithmetic will fix that. The equation $4x + 20 = 4$ is preposterous: Diophantus himself called it absurd.
Diophantus did write that, in the third century AD. But on the other side of the world people had been counting differently for a long time. In the Chinese treatise The Nine Chapters on the Mathematical Art, compiled about two thousand years ago, the coefficients of equations were laid out with counting rods: red for positive, black for negative. Accountants today write debts in red; the Chinese calculators did the opposite. The rules for adding and subtracting such numbers were set out in detail in the third century by the commentator Liu Hui.
In the same treatise of 628 Brahmagupta calls positive numbers fortunes and negative numbers debts, and writes out the rules: the sum of two debts is a debt, a debt subtracted from zero becomes a fortune, the product of two debts is a fortune. The rule "minus times minus is plus" was stated almost fourteen centuries ago, only without proof.
Europe held out longer. Leonardo of Pisa in his Book of Calculation (1202) interpreted a negative answer to a problem as a debt, but that was an exception. Michael Stifel in 1544 called negative numbers absurd, Cardano in his Great Art (1545) fictitious, and Descartes in his Geometry (1637) false roots of equations. They were fully accepted only in the eighteenth and nineteenth centuries.
Meanwhile "below zero" turns up at every step in everyday life. In winter the thermometer reads $-15^\circ$. The $-1$ button in a lift takes you to the basement. The shore of the Dead Sea lies more than 400 metres below sea level. Your bank balance goes negative when you spend more than you have. In every case there is a reference point with two directions away from it, and a negative number says "this much the other way".
The integers are the natural numbers, zero and the negative numbers $-1, -2, -3, \dots$ Their set is written $\mathbb Z$, from the German Zahlen, "numbers":
$$\mathbb Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}.$$
The numbers $a$ and $-a$ are called opposites: their sum is zero, $a + (-a) = 0$. The opposite of zero is zero itself.
Note that writing $-a$ does not mean the number is negative. If $a = -4$, then $-a = 4$. The minus in front of a letter is read "the opposite of $a$", and which sign it ends up with depends on $a$ itself.
Round three: where minus three lives
I can put three apples on the table. Show me minus three.
We will, only not on a table but on a line. Mark the point 0 on it, choose a direction, "right", and a unit length. The natural numbers take their places to the right at equal steps, and to the left of zero, as in a mirror, their opposites.
A number line is a line with a marked point 0, a chosen positive direction and a unit length. A positive number $a$ is shown as the point to the right of zero at distance $a$, and its opposite $-a$ as the mirror-image point on the left.
Minus three is the point three steps to the left of zero. But it's more useful to think of a number as a shift: an arrow of length 3 pointing left. Adding two numbers means walking their arrows one after the other: from zero along the first, from its tip along the second. Where you stop is the sum. A debt of five coins plus an income of three is an arrow five steps to the left followed by three steps to the right: the result is $-2$, a debt of two coins.
And subtraction? Subtracting $b$ means walking the arrow $b$ backwards. An arrow turned around is the arrow of the opposite number. So every subtraction turns into an addition:
For any integers $a$ and $b$ the difference $a - b$ equals $a + (-b)$.
The idea: check that the number $a + (-b)$ does what a difference is supposed to do. The difference $a - b$ is the number that together with $b$ gives $a$: that's how subtraction is defined, as the inverse of addition. Add $b$ to the number $a + (-b)$: $\bigl(a + (-b)\bigr) + b = a + \bigl((-b) + b\bigr)$ by the associative law, $(-b) + b = 0$ because it's the sum of opposites, and $a + 0 = a$. So $a + (-b)$ qualifies as the difference. No other number does: if $x + b = a$ and $y + b = a$, add $-b$ to both equations and the same three laws give $x = a + (-b) = y$. So $a - b = a + (-b)$.
The picture also shows the rules that school makes you memorise separately. Numbers of the same sign add up to a long arrow: add the lengths and keep the common sign, $-4 + (-3) = -7$. Numbers of different signs cancel each other out: subtract the shorter length from the longer and take the sign of the longer arrow, $-9 + 4 = -5$.
The line also brings order to comparison: of two numbers, the one further right is larger. So $-2 > -5$, even though 5 is larger than 2. It's warmer at $-2^\circ$ than at $-5^\circ$, and a debt of two coins is better than a debt of five.
Work out $-7 + 12$.
An arrow 7 steps to the left, then 12 steps to the right. The signs differ: $12 - 7 = 5$, and the longer arrow's sign is plus. Answer: $5$.
Work out $-8 - (-3)$.
Subtracting $-3$ means adding $3$: $-8 - (-3) = -8 + 3 = -5$.
The arrows are fairly convincing, I admit. A debt plus a debt is a bigger debt; any merchant understands that. I'll grant you addition.
Round four: minus times minus
Multiplication is another matter. Three times a debt of five coins is a debt of fifteen, agreed. But what does it mean to take a debt minus three times? And why do two debts give you a fortune? That's a conjuring trick, not arithmetic.
We won't argue with the first half of the objection. Multiplying by a natural number is repeated addition: $3 \cdot (-5) = (-5) + (-5) + (-5) = -15$. If we want the product not to change when the factors swap places, then also $(-5) \cdot 3 = -15$. Plus times minus is minus, and that needs no tricks.
The hard case is $(-3) \cdot (-5)$. "Taking something minus three times" really does mean nothing, and the Sceptic is right: the answer doesn't follow from multiplication as repeated addition. So we need other grounds. We have three, from the most visual to the most rigorous.
First argument: the table continues itself
Look at the row of the times table for three: $3 \cdot 3 = 9$, $3 \cdot 2 = 6$, $3 \cdot 1 = 3$, $3 \cdot 0 = 0$. Each step to the left takes away three. Continue with the same step: $3 \cdot (-1) = -3$, $3 \cdot (-2) = -6$, exactly what repeated addition gave. Now take the column for $-2$ and walk down it: $3 \cdot (-2) = -6$, $2 \cdot (-2) = -4$, $1 \cdot (-2) = -2$, $0 \cdot (-2) = 0$. Each step down adds two. The next step: $(-1) \cdot (-2) = 2$.
The pattern is persuasive, but remember Moser's circle and Euler's formula, which produced primes forty times in a row and then broke on the forty-first. Coincidences, even many of them, prove nothing. The table shows which rule would be convenient. Why it's the only possible one, the second argument explains.
Second argument: a proof
Remember the terms of the debate. The Sceptic accepted the laws of arithmetic, and we agreed that they must hold for the new numbers too. That requirement is enough to work out $(-1) \cdot (-1)$ without asking anybody's opinion. We'll need two auxiliary facts, lemmas. Both are short, but without them the proof would rest on a promise.
If $x + y = 0$, then $y = -x$.
The idea: add $-x$ to both sides and simplify. From $x + y = 0$ it follows that $(-x) + (x + y) = (-x) + 0$. On the right, by the law of zero, $-x$ remains. On the left, by the associative law, $(-x) + (x + y) = \bigl((-x) + x\bigr) + y$, the sum of opposites $(-x) + x$ is zero, and $0 + y = y$ remains. So $y = -x$: a number that gives zero when added to $x$ can only be the opposite of $x$.
For any number $a$ we have $a \cdot 0 = 0$.
The idea: write zero as $0 + 0$ and expand the brackets. Since $0 + 0 = 0$, the distributive law gives $a \cdot 0 = a \cdot (0 + 0) = a \cdot 0 + a \cdot 0$. Write $x = a \cdot 0$; we have found $x = x + x$. Add $-x$ to both sides: on the left $x + (-x) = 0$, on the right $(x + x) + (-x) = x + \bigl(x + (-x)\bigr) = x + 0 = x$. So $0 = x$, that is, $a \cdot 0 = 0$. Brahmagupta's rule turns out to be not a convention but a consequence of the laws of arithmetic.
$(-1) \cdot (-1) = 1$.
The idea: take a product that is certainly zero, write the zero inside it as $1 + (-1)$ and expand the brackets. The mysterious product will appear by itself, with nowhere to hide.
Let's check other numbers the same way. $(-2) \cdot \bigl(3 + (-3)\bigr) = (-2) \cdot 0 = 0$, and by the distributive law this is $(-2) \cdot 3 + (-2) \cdot (-3) = -6 + (-2)(-3)$. Hence $(-2)(-3) = 6$. Assemble the proof for $(-1) \cdot (-1)$ yourself: the building blocks are the laws of arithmetic, and at every step exactly one fits.
The same argument works for any numbers, not just ones.
For any numbers $a$ and $b$ we have $(-a) \cdot b = -(ab)$ and $(-a) \cdot (-b) = ab$.
The idea is the same as for $(-1) \cdot (-1)$: add the unknown product to a known one so that the distributive law gives zero, and then appeal to the lemma on the uniqueness of the opposite.
Two facts follow at once from the sign rule, and they'll be useful for the rest of the course.
A product of several non-zero numbers is positive if an even number of the factors are negative, and negative if an odd number are.
The idea: strip the minuses off the factors one at a time. By the sign rule $(-a) \cdot b = -(ab)$, that is, the minus on any factor can be moved to the front of the whole product; the commutative and associative laws let us reorder and regroup the factors. Move the minuses of all the negative factors to the front this way. What remains is a product of positive numbers, which is positive, preceded by as many minuses as there were negative factors. Every two minuses cancel, because $-(-x) = x$: the opposite of the opposite is $x$ itself. So with an even number of minuses the result is positive, with an odd number negative. For example, $(-2) \cdot (-3) \cdot (-5) = -30$, while $(-1)^{100} = 1$.
For any number $a$ we have $a^2 \ge 0$, and $a^2 = 0$ only when $a = 0$.
Consider three cases. If $a > 0$, then $a^2 = a \cdot a$ is a product of positive numbers, so it is positive. If $a < 0$, then $a = -c$ with $c > 0$, and by the sign rule $a^2 = (-c)(-c) = c \cdot c > 0$. If $a = 0$, then $a^2 = 0$ by the lemma on multiplying by zero. For example, $3^2 = 9$ and $(-3)^2 = 9$. So no number whose square is $-1$ exists among the integers, or indeed among any numbers on the line.
The sign rule was not invented "for convenience". If negative numbers are to obey the distributive law, there's no other option: $0 = (-1) \cdot \bigl(1 + (-1)\bigr) = -1 + (-1)(-1)$ forces $(-1)(-1) = 1$. Declaring $(-1)(-1) = -1$ would give $0 = -2$.
Third argument: turning around
Back to the arrows. Multiplying by 2 stretches an arrow to twice its length. What does multiplying by $-1$ do? It turns $3$ into $-3$ and $-3$ into $3$: the arrow keeps its length but points the other way. You can picture it as a rotation about zero through $180^\circ$. Then multiplying by $-1$ twice means rotating by $180^\circ$ and another $180^\circ$, a full turn. The arrow is back where it started: $(-1) \cdot (-1) = 1$.
The rotation picture looks like a toy, but it hides the next step for all of mathematics. If multiplying by $-1$ is a rotation through $180^\circ$, is there a number that rotates by $90^\circ$ when you multiply by it? Two such rotations would make a half turn, so the square of this number would be $-1$. There's no such number on the line, but in the plane there is. It's the imaginary unit $i$, and we'll meet it in chapter 15.
One last objection, and it's not mine but Monsieur Arnauld's. You say that $-1$ is less than $1$. Then in the proportion $-1 : 1 = 1 : (-1)$ the smaller is to the larger as the larger is to the smaller. How can that be?
Antoine Arnauld raised this argument in the seventeenth century, and Leibniz took it seriously. Let's calculate: $(-1) : 1 = -1$ and $1 : (-1) = -1$. Both sides of the proportion are equal, so the proportion holds. What breaks is not arithmetic but habit: "the smaller divided by the larger is less than one" is a rule for positive numbers. Multiplying and dividing by a negative number reverses inequalities. From $2 < 5$, multiplying by $-1$ gives $-2 > -5$: a half turn swaps "further left" and "further right". Let's prove this carefully. Recall that $a < b$ means "$b$ is to the right of $a$", that is, the difference $b - a$ is positive.
If $a < b$, then $-a > -b$.
The idea: compare the direction of the arrow from $a$ to $b$ with that of the arrow between their reflections.
We know that $a < b$. What can we say about the numbers $-a$ and $-b$?
Opposite numbers are reflections in zero, and reflection reverses order. If $a$ is to the left of $b$, then $-a$ is to the right of $-b$, whatever the signs: $2 < 5$ and $-2 > -5$; $-3 < 1$ and $3 > -1$.
Round five: dividing by zero
Since you are so bold, be consistent. Allow division by zero as well: $1 : 0$ is infinity, and that's that. John Wallis argued exactly this way in 1656, and even concluded that the ratio of a positive number to a negative one is greater than infinity.
Now it's our turn to be sceptics. Division is the inverse of multiplication: $a : b = c$ means that $c \cdot b = a$. Check any example: $12 : 3 = 4$ because $4 \cdot 3 = 12$. Let's try to divide by zero the same way.
If $a \ne 0$, there is no number $c$ with $c \cdot 0 = a$. If $a = 0$, every $c$ satisfies $c \cdot 0 = 0$. So the quotient $a : 0$ is undefined for every $a$.
The idea: the lemma on multiplying by zero settles everything. The quotient $a : 0$ would have to be a number $c$ with $c \cdot 0 = a$. But by the lemma $c \cdot 0 = 0$ for every $c$. If $a \ne 0$, we get $0 = a$, a contradiction, so no suitable $c$ exists: for example, $6 : 0$ would mean $c \cdot 0 = 6$, but any number times zero gives zero, not six. If $a = 0$, the equation $c \cdot 0 = 0$ holds for every $c$ — for 5, for $-17$ and for 0 — and there are no grounds to pick one of them as the answer. In the first case there is no quotient, in the second there are too many. That's why division by zero is undefined: not because someone forbade it, but because no number can play the part of the answer.
What about infinity? People suggest it because dividing by small numbers makes the quotient grow: $1 : 0.1 = 10$, $1 : 0.01 = 100$, $1 : 0.001 = 1000$. But approach zero from the other side: $1 : (-0.1) = -10$, $1 : (-0.001) = -1000$. The quotient heads off into the negatives without any bound. If $1 : 0$ had a value, it would have to be huge and positive and huge and negative at the same time. How to talk about such behaviour carefully, without dividing by zero, is explained by the theory of limits (chapter 25).
Brahmagupta thought that $0 : 0 = 0$, and that rule did not stand the test of time. Wallis, having accepted $1 : 0 = \infty$, ended up with negative numbers larger than infinity: a fine illustration of where division by zero leads.
Sometimes division by zero is hidden so cleverly that you can't see it. Here is a "proof" that $2 = 1$.
What is $0 : 7$?
Here zero is being divided, and dividing by seven is fine. $0 : 7 = 0$ because $0 \cdot 7 = 0$. What's forbidden is dividing by zero; zero divided by any other number is zero.
Round six: absolute value
Your line says that $-5$ is less than $3$. My common sense says a debt of five coins is bigger than a debt of three, and certainly "bigger" than three coins. Who is right?
You both are; you're talking about different things. A number has a position on the line and it has a size: how far it is from zero, in whichever direction. Position is compared with "less than". Size is measured by the absolute value.
The absolute value of a number $a$ (also called its modulus) is the distance from the point $a$ to zero on the number line. The notation $|a|$ was introduced by Karl Weierstrass in 1841.
The absolute value of a difference is the distance between points: $|a - b|$ shows how far $a$ is from $b$. Between $-5$ and $3$ there are exactly $|-5 - 3| = |3 - (-5)| = 8$ steps. The temperature difference between $-15^\circ$ and $+4^\circ$ is $|4 - (-15)| = 19$ degrees.
For any numbers $a$ and $b$: $|a| \ge 0$, and $|a| = 0$ only for $a = 0$; $|-a| = |a|$; $|ab| = |a| \cdot |b|$; $|a - b|$ is the distance between the points $a$ and $b$ on the line.
The first two properties follow directly from the definition. If $a \ge 0$, then $|a| = a \ge 0$; if $a < 0$, then $|a| = -a$, and the opposite of a negative number is positive. Only zero has absolute value zero: a positive number is its own absolute value, and a negative one has the positive $-a$. The numbers $a$ and $-a$ are mirror images in zero, so they are the same distance from it: $|-a| = |a|$.
For a product, go through the signs. If $a$ and $b$ are both non-negative, then $ab \ge 0$ and $|ab| = ab = |a| \cdot |b|$. If exactly one of them is negative, say $a < 0 \le b$, then by the sign rule $ab = -\bigl((-a)b\bigr)$ with $(-a)b \ge 0$; so $|ab| = (-a)b = |a| \cdot |b|$. If both are negative, then $ab = (-a)(-b) > 0$ and $|ab| = (-a)(-b) = |a| \cdot |b|$.
Finally, the arrow from the point $b$ to the point $a$ is the number $a - b$: walking it from $b$, we arrive at $b + (a - b) = a$. The length of the arrow is the absolute value of this number, that is, $|a - b|$.
The most useful property of absolute value concerns sums. Walk the arrow $a$ first, then the arrow $b$, and you end up at the point $a + b$. If the arrows point in different directions, the second part of the walk eats into the first, and you end up closer to zero than the distance you walked. This is called the triangle inequality: in the plane, where arrows can go at an angle, it says that a side of a triangle is shorter than the sum of the other two (more on this in the chapter on vectors). On the line the triangle is "squashed flat", but the inequality is the same.
For any numbers $a$ and $b$ we have $|a + b| \le |a| + |b|$. Equality holds if and only if $a$ and $b$ don't have opposite signs, that is, $ab \ge 0$.
The idea: the distance from the start of a walk to its end is no more than the length of the walk itself.
Work out $|-8| - |3 - 10|$.
$|-8| = 8$. In the second absolute value, work out the difference first: $3 - 10 = -7$, and $|-7| = 7$. In total $8 - 7 = 1$.
All the rules of the debate are now gathered in one place. Check how well you've got the hang of them: the trainer starts with addition, and at level three it offers expressions with absolute values, brackets and powers.
I give up. Your numbers behave decently: they add like arrows, they multiply by a law I held to myself, and you won't let even yourselves divide by zero. So be it.
Where next
The integers have closed the gap this chapter opened with: you can always add, subtract and multiply them, and the result is again an integer. Division doesn't work out like that, even without zero. $12 : 3 = 4$, but $13 : 3$ is not a whole number, and $13$ is divisible by nothing except one and itself. Which numbers divide which? Are there "atoms" among them, numbers from which all the others are built by multiplication? That is chapter 3.