Appearance
1.2 — Negative Numbers and the Number Line
You have ₹300 in your account and you spend ₹500. The bank does not refuse to answer; it says your balance is -200. Nobody finds this confusing in daily life. Yet the same quantity, presented as 300 - 500 in a maths class, took European mathematicians about four hundred years to accept, and they called such answers "absurd" and "fictitious" the entire time.
The reason for the resistance is worth understanding, because it tells you exactly what a negative number is. If a number means "how many things are in this pile", then a negative number is nonsense — there is no pile with less than nothing in it. The Greeks, whose numbers were lengths of line segments, could not make sense of a negative length and did not try.
The escape is to stop thinking of a number as a quantity and start thinking of it as a position or a change.
1. The number line, and what direction buys you
Draw a straight line. Mark a point on it and call it 0. Choose a spacing and mark 1, 2, 3, \ldots to the right. Now mark the same spacing to the left, and call those points -1, -2, -3, \ldots.
That picture does three jobs at once, and each one dissolves a piece of the confusion.
It gives numbers an order. Further right is bigger. So -2 \lt -1, because -2 sits further left. This is the part people get wrong when they first meet negatives: a debt of ₹500 is worse than a debt of ₹200, so -500 is less than -200, even though five hundred is a bigger number than two hundred in the everyday sense. The everyday sense is talking about the size, which we will name in a moment; the mathematical "less than" is talking about position on the line.
It separates size from direction. The number -7 has a size of seven and a direction of "left". The size is called the absolute value, written |-7| = 7, and read "the absolute value of minus seven is seven". It is simply the distance from zero, with the sign thrown away. Distance is never negative, so |x| is never negative.
It turns addition into movement. Adding a positive number means walking right. Adding a negative number means walking left. That is the whole rule, and it makes every case obvious:
- 5 + 3: start at 5, walk 3 right, land on 8.
- 5 + (-3): start at 5, walk 3 left, land on 2.
- -5 + 3: start at -5, walk 3 right, land on -2.
- -5 + (-3): start at -5, walk 3 left, land on -8.
No rules to memorise. One picture, four answers.
Adding the negatives to the counting numbers gives the integers:
\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}
and the hole we found at the end of Chapter 1.1 is now filled: the integers are closed under subtraction. Any integer minus any integer is an integer, always.
2. Subtraction is adding the opposite
Here is a simplification that pays for itself forever. Define the additive inverse of a number a, written -a, as the number that brings you back to zero:
a + (-a) = 0
The opposite of 5 is -5, because 5 + (-5) = 0. The opposite of -5 is 5, for the same reason. So -(-5) = 5: the opposite of the opposite is the thing itself. On the line, you turned round twice and are facing the original way.
Now define subtraction out of existence:
a - b \;\; \text{means} \;\; a + (-b)
Subtracting is adding the opposite. This is not a trick; it is a genuine reduction of two operations to one, and it means every property of addition automatically applies. 7 - 3 is 7 + (-3). And 7 - (-3) is 7 + (-(-3)) = 7 + 3 = 10, which is why subtracting a negative adds. If someone cancels a ₹3,000 debt of yours, you are ₹3,000 better off. Removing a negative is a gain.
3. Why a negative times a negative is positive
This is the question, and it deserves a proper answer rather than "because that is the rule". There are three answers, at three different depths. Read all three; each one convinces a different part of your brain.
The everyday answer
Think of a video of a man walking. Multiplication has two signs to play with: the direction he walks, and the direction the film is played.
- Walking forward, film played forward: he moves forward. Positive times positive is positive.
- Walking backward, film played forward: he moves backward. Negative times positive is negative.
- Walking forward, film played in reverse: he appears to move backward. Positive times negative is negative.
- Walking backward, film played in reverse: he appears to move forward. Negative times negative is positive.
Two reversals cancel. That is the intuition, and it is genuinely the reason.
The pattern answer
Write out a multiplication table and let the pattern finish itself.
3 \times 3 = 9,\quad 3 \times 2 = 6,\quad 3 \times 1 = 3,\quad 3 \times 0 = 0
Each time the second number drops by one, the answer drops by three. Keep going and there is only one consistent continuation:
3 \times (-1) = -3,\quad 3 \times (-2) = -6
Now do the same on the other side, dropping the first number:
2 \times (-2) = -4,\quad 1 \times (-2) = -2,\quad 0 \times (-2) = 0
Each time the first number drops by one, the answer rises by two. Continue:
(-1) \times (-2) = 2,\quad (-2) \times (-2) = 4
Any other answer would break the pattern that held for every case we already agreed on.
The real answer
The pattern argument is suggestive. Here is the proof, and it rests entirely on the distributive law from Chapter 1.1: a(b + c) = ab + ac.
Start with something nobody disputes:
(-1) \times \big(1 + (-1)\big) = (-1) \times 0 = 0
because anything times zero is zero. Now expand the left-hand side using the distributive law:
(-1) \times 1 \;+\; (-1) \times (-1) = 0
The first term is -1. So:
-1 \;+\; (-1) \times (-1) = 0
Which says that (-1) \times (-1) is the number you add to -1 to get 0. Only one number does that, and it is +1. Therefore:
(-1) \times (-1) = 1
And this is the important part: it could not have been otherwise. If we had insisted that negative times negative is negative, the distributive law would break, and with it every technique of algebra. The sign rule is not a decree. It is the only choice that keeps arithmetic consistent.
The general rules follow immediately, and their shape is easy to remember: an even number of minus signs gives a plus, an odd number gives a minus.
(-3) \times 4 = -12, \qquad (-3) \times (-4) = 12, \qquad (-2)^3 = -8, \qquad (-2)^4 = 16
4. Division, and the one thing you genuinely cannot do
Recall from Chapter 1.1 that division is an inverse question. 12 \div 3 asks "what times 3 gives 12?" — and the answer is 4.
Sign rules for division are inherited from multiplication for free, since it is the same question backwards. -12 \div 3 asks what times 3 gives -12; the answer is -4.
Now ask about zero, carefully, because there are two different questions and only one of them is interesting.
What is 0 \div 5? What times 5 gives 0? Zero does. The answer is 0, and there is nothing strange here.
What is 5 \div 0? What times 0 gives 5? Nothing does — every number times zero is zero, so no number can produce five. There is no answer, not a hidden or infinite one. The question has no solution.
What is 0 \div 0? What times 0 gives 0? Everything does. 1 \times 0 = 0, and 7 \times 0 = 0, and -3.5 \times 0 = 0. There are infinitely many answers, so there is no single one, and defining it would break the rule that a division has one result.
So division by zero is not banned out of superstition. In the first case there are no answers; in the second there are too many. Either way there is nothing for the expression to mean. This is why a calculator shows an error and why a program crashes — Volume I, 1.4 covers what floating-point hardware does with it, which is to produce a special value called Infinity or NaN rather than a number.
The one place "divide by zero" almost makes sense
In calculus (Part 5) you will meet expressions that approach zero divided by something that also approaches zero, and those can have perfectly good answers. That is not division by zero — it is a limit, a genuinely different operation that asks where a quantity is heading rather than what it equals. Chapter 5.1 makes the distinction precise.
5. Reading the notation without stumbling
The minus sign does three different jobs and the same symbol is used for all three. Once you see this, a lot of small confusions clear up.
As an operation between two numbers: 7 - 3. "Seven minus three."
As the sign of a single number: -3. "Negative three." This is one symbol standing for a number, not an instruction.
As "the opposite of": -x. Read as "the opposite of x", not "negative x", because if x is itself negative then -x is positive. If x = -4 then -x = 4. This is the single most common misreading in early algebra: seeing -x and assuming it must be a negative number. It means "flipped", not "negative".
6. Working with the number line: a few worked cases
A temperature drop. It is -3^\circ\text{C} in Shimla and the forecast says it will fall another 5 degrees. Falling means walking left: -3 + (-5) = -8^\circ\text{C}.
A temperature difference. Delhi is at 22^\circ, Shimla at -8^\circ. How much warmer is Delhi? Difference is subtraction: 22 - (-8) = 22 + 8 = 30 degrees. This is the case where subtracting a negative is not a curiosity but the everyday answer — the gap between the two points on the line really is thirty units.
A bank statement. Opening balance ₹4,200. Three transactions: -1,500, +2,000, -3,100.
4200 - 1500 + 2000 - 3100 = 4200 + (-1500) + 2000 + (-3100) = 1600
Rewriting every subtraction as an addition lets you add in any order you like, because addition is commutative and associative. Grouping the credits and the debits separately: (4200 + 2000) - (1500 + 3100) = 6200 - 4600 = 1600. Same answer, less chance of a slip.
Elevation. The Dead Sea shore sits about 430 metres below sea level, so its elevation is -430\ \text{m}. Sea level is not "no height"; it is the agreed origin. That is exactly what zero is on the number line — a chosen origin, not an absence.
7. Absolute value, said properly
We defined |x| as the distance from zero. Written as a rule with cases:
|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \lt 0 \end{cases}
Read the second line carefully in light of the previous section: if x is negative, |x| is the opposite of x, which is positive. If x = -6, then |x| = -(-6) = 6.
The genuinely useful thing about absolute value is that it measures distance between two points:
|a - b| = \text{the distance from } a \text{ to } b \text{ on the line}
and the order does not matter, since |a-b| = |b-a|. The distance from 3 to 10 is |3 - 10| = |-7| = 7, the same as the distance from 10 to 3.
This is the seed of a very large idea. In Chapter 3.4 the distance between two points in a plane will be a formula built from Pythagoras; in Chapter 4.1 the distance between two vectors in any number of dimensions will be a formula built the same way; and in Chapter 12.6 of Volume I, the "distance" between two pieces of text is what a search engine ranks by. All of them are generalisations of |a - b|.
8. Where this shows up in your life
Every bank account, every credit card, every ledger. Double-entry bookkeeping is nothing but the insistence that every transaction appears twice with opposite signs, so the total is always zero. Accountants got there before mathematicians did.
Every thermometer and altimeter. Both use a chosen origin with values on both sides.
Lifts, basements and floor numbering. In much of Europe the ground floor is 0 and basements are -1, -2. In India and the US the ground floor is often 1, which is an ordinal convention rather than a positional one, and is why the two systems disagree about which floor you are on.
Every computer, in a subtler way. Volume I, 1.3 explains two's complement, the representation that lets a processor add and subtract signed numbers with one circuit. It works by choosing the negative numbers so that a + (-a) overflows to exactly zero — the additive-inverse property of this chapter, built into hardware.
Time zones and dates. UTC$+5{:}30$ for India, UTC$-5$ for New York. Adding across midnight wraps around, which is not the number line any more but the clock, and the arithmetic of clocks is Chapter 1.6.
Subtraction is now closed. Division is not — 7 \div 2 still has no answer among the integers. The next chapter invents the numbers it needs.