Negative numbers and the number line

Every number we have counted with so far has been positive — or zero. But zero is not the edge of the number world: a thermometer shows frost, an account can drop below the line, a lift goes down to a level underground. All of those are described by negative numbers: numbers smaller than zero, written with a minus sign — $-3$, $-12$, $-250$.

The number line runs both ways

The number line is a straight line with zero marked on it and evenly spaced steps. To the right of zero lie the positive numbers, to the left the negative ones. The further right, the greater the number; the further left, the smaller.

Zero is neither positive nor negative — it is the reference point. Everything is counted from it, in both directions.

The opposite of a number

Every number has an opposite — the number on the other side of zero, at the same distance. The opposite of $5$ is $-5$, and the opposite of $-5$ is $5$:

A pair of opposites always adds up to zero:

That is why paying $340$ into an account whose balance is $-340$ leaves it at exactly zero.

Comparing negative numbers

There is one rule and it never changes: of two numbers, the greater one is the one further to the right. With positives that matches intuition, but with negatives it often does not:

• $-3 > -7$, even though seven is a bigger digit than three,
• every positive number is greater than every negative one: $1 > -1000$,
• zero is greater than every negative number: $0 > -4$.

The easiest check is a thermometer: $-3$ °C is a milder frost than $-7$ °C, so $-3$ is the greater number.

Adding and subtracting with signs

On the number line, adding is a step to the right and subtracting a step to the left. The only new thing is that we may now walk through zero and keep going:

Any subtraction can be rewritten as adding the opposite:

And out of that comes the rule that causes the most trouble — subtracting a negative:

Two minus signs side by side cancel and leave an addition. That is why the difference between night and noon, $4 - (-6)$, is $10$ degrees.

The distance between two numbers

The distance between two numbers is simply the number of steps between them — always a positive quantity, because a distance has no direction. To find it, subtract the number further left from the number further right.

From $-6$ to $4$ there are $4 - (-6) = 10$ steps. From $-9$ to $-2$ there are $-2 - (-9) = 7$ steps. Subtract them the other way round and you get the same figure with a minus in front — in that case just drop the sign.

Multiplying and dividing — the sign rule

Multiplying by a negative number reverses the direction along the line. The whole rule follows from that:

In other words:

• different signs — a negative result: $(-6) \cdot 7 = -42$,
• matching signs — a positive result: $(-6) \cdot (-7) = 42$.

Division follows exactly the same rule, because division undoes multiplication: $(-42) : 7 = -6$, while $(-42) : (-7) = 6$.

The quickest way in practice: work the result out on the numbers without their signs, then count the minus signs in the calculation. An even number of them gives a positive result, an odd number a negative one.

Common mistakes

• Comparing negatives by their digits — $-7$ is not greater than $-3$; bigger digits put the number further left, which makes it smaller.
• Losing a minus when subtracting a negative — $5 - (-3)$ is $8$, not $2$.
• Mixing the sign rule into addition — minus times minus makes plus applies to multiplication and division; in addition $-6 + (-7) = -13$, because we walk left twice.
• Treating zero as a negative number — zero has no sign and is greater than every negative number.