1.3 Negative numbers — Learn

Prerequisites: ordering on a number line; the four operations (section 1.2).

By the end of this section you can: order negative numbers; add, subtract, multiply and divide with negative numbers; and interpret a negative result as a loss, a debt or a fall.

1. The number line

Negative numbers are to the left of zero. The further left, the smaller.

Example 1. The temperature falls from $3\,^\circ\text{C}$ to $-4\,^\circ\text{C}$. How many degrees did it fall?

The gap from $3$ down to $0$ is $3$ degrees, and from $0$ down to $-4$ is another $4$ degrees, so the fall is $3 + 4 = 7\,^\circ\text{C}$.

2. Adding and subtracting

Think of a number line, or of money and debt.

Subtracting a negative is the same as adding:

Watch out. "Two negatives make a positive" applies to multiplying and dividing, not to adding. $(-3) + (-5) = -8$, not $+8$. And $-3 - 5 = -8$: you move 5 further left, not right.

3. Multiplying and dividing

For multiplication and division, decide the sign first, then work with the sizes:

Example 2. Work out $(-6) \times 4$ and $(-6) \times (-4)$.

$(-6) \times 4 = -24$ (different signs, so negative). $(-6) \times (-4) = 24$ (same signs, so positive).

The same rules apply to division: $(-20) \div 5 = -4$ and $(-20) \div (-5) = 4$.

4. Negative numbers in context

A profit is positive; a loss is negative. A balance below zero is a debt. If takings are less than costs, the profit is negative.

Try it: profit and loss

Predict first: in the chapter problem the stall spent £92.80. If the stall only sold 60 cans (at £1.20) and 6 boxes (at £3.00), will the result be a profit or a loss? Decide before you move anything.

Then use the tool: set "Bought" to 120, "Cost each" to 0.65, "Price each" to 1.20 and "Sold" to 60 to see this case, and try your own values.

Explain: when the readout says the profit is negative, what does that mean in words? Why is a negative profit called a loss?

5. Progress check

With 60 cans sold:

The stall would have made a loss of £2.80. A negative profit is a loss.

You should now be able to

• order positive and negative numbers;
• add and subtract with negatives by moving along a number line;
• multiply and divide using the sign rules;
• explain what a negative answer means in a money context.