Which is bigger, −7 or −5?

−5 is bigger. On a number line the further left a number sits, the smaller it is, and −7 is further left than −5, so −7 < −5. It feels backwards because 7 is a bigger digit than 5, but the minus sign reverses the order: with negatives, the larger the digit the smaller the number. A temperature of −7°C is colder (lower) than −5°C.

It is −9. A negative times a positive is negative, so −3 × 3 = −9. Multiplying by 3 makes the size 9, and the single minus sign stays, because a negative and a positive give a negative. Dropping the sign to get 9 loses the direction of the answer.

It is −2. Subtracting 7 from 5 takes you below zero: start at 5 and step 7 to the left and you land on −2. The answer is negative because you took away more than you started with. Writing 2 ignores the sign and treats the subtraction as if it could not go below zero, which it can.

GCSE Maths Foundation · AQA · Order of operations & negatives

Ordering and signs of negatives: −7 < −5, and keep the sign

Q: What is −3 × 3?

It is −9. A negative times a positive is negative, so −3 × 3 = −9. Multiplying by 3 makes the size 9, and the single minus sign stays, because a negative and a positive give a negative. Dropping the sign to get 9 loses the direction of the answer.

Q: What is 5 − 7?

It is −2. Subtracting 7 from 5 takes you below zero: start at 5 and step 7 to the left and you land on −2. The answer is negative because you took away more than you started with. Writing 2 ignores the sign and treats the subtraction as if it could not go below zero, which it can.

Negatives trip students two ways. First, they order them by digit size — judging $-7$ to be larger than $-5$ because $7 > 5$ — when on a number line the further left a number sits, the smaller it is, so $-7 < -5$ . Second, they lose the sign in a calculation, taking $-3 \times 3$ as 9 rather than $-9$ , or $5 - 7$ as 2 rather than $-2$ .

The thirty-second fix: read order off the number line — further left is smaller — and keep the sign through every calculation, because a negative times a positive is negative and subtracting more than you have lands you below zero. So $-7 < -5$ , $-3 \times 3 = -9$ and $5 - 7 = -2$ .

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How to spot it in your own work

You ordered negatives by digit size, putting $-7$ as larger than $-5$ , when further left on the line is smaller, so $-7 < -5$ .
You dropped the sign in a product, writing $-3 \times 3 = 9$ instead of $-9$ — a negative times a positive is negative.
You stopped a subtraction at zero, taking $5 - 7$ as 2 rather than stepping below zero to $-2$ .
You read a temperature or a number line the wrong way round, calling $-7$ “warmer” or “higher” than $-5$ when it is colder and lower.

An exam question that triggers it

Here is a canonical AQA Foundation trigger (non-calculator paper, order the numbers):

The temperatures recorded one night were −5°C, 3°C, −7°C and −1°C.

Write these temperatures in order, coldest first.

The misconception is to order by digit size — putting $-7$ after $-5$ because 7 is bigger than 5 — which gets the two coldest the wrong way round. The minus sign reverses the order.

Read it off the number line: the further left a number sits, the smaller it is, so coldest to warmest is $-7,\ -5,\ -1,\ 3$ .

Why students fall for this

For years “bigger” has meant “more digits, higher count”, and that instinct carries straight into negatives — so $-7$ looks larger than $-5$ because 7 beats 5. But the minus sign reverses the order: a number is larger when it sits further right on the line, and $-7$ is further left than $-5$ , so $-7 < -5$ . Temperature is the everyday anchor — −7°C is colder, and so lower, than −5°C.

The signs go missing in arithmetic for a similar reason: students compute the size and forget the direction. Multiplying by a positive does not remove a minus, so $-3 \times 3 = -9$ — three lots of $-3$ . And a subtraction can take you below zero: $5 - 7$ steps 7 to the left of 5 and lands on $-2$ , because you took away more than you had.

AQA Foundation papers exploit both directly: ordering a list of negatives and positives (often as temperatures), and directed-number arithmetic where the sign of the answer is the whole point.

Worked example — a subtraction below zero. Work out $5 - 7$ .

Start at 5 on the number line and step 7 places to the left:

5 - 7 = -2

The trap is to do $7 - 5 = 2$ and stop at zero, as if a subtraction could not go negative. It can: taking away more than you have lands you below zero, so the answer is $-2$ .

The fix: Order off the number line — further left is smaller — and keep the sign through every calculation

Order by position, not by digit. On the number line the further left a number sits, the smaller it is. So $-7 < -5 < -1 < 3$ , even though 7 is the biggest digit.

Use temperature as a check. Colder means lower means smaller. −7°C is colder than −5°C, which confirms $-7 < -5$ .

Keep the sign in a product. A negative times a positive is negative: $-3 \times 3 = -9$ . Work out the size, then attach the sign — one minus survives.

Let a subtraction go below zero. $5 - 7$ steps below zero to $-2$ . If you take away more than you started with, the answer is negative — never clamp it at zero.

Worked example

Order $-5,\ 3,\ -7,\ -1$ coldest first, then work out $-3 \times 3$ . These are the two traps: ordering by digit size, and dropping the sign.

Place each number on the line. Going left to right the positions are $-7,\ -5,\ -1,\ 3$ — the further left, the smaller.
Read off the order.
$-7 < -5 < -1 < 3$
The trap puts $-5$ before $-7$ because 5 looks smaller than 7, but the sign reverses that.
Work out the size of the product. $3 \times 3 = 9$ .
Attach the sign. A negative times a positive is negative, so
$-3 \times 3 = -9$
not the $9$ you get by dropping the minus.

The same number-line picture settles the subtraction: $5 - 7 = -2$ , because stepping 7 to the left of 5 lands you two places below zero — not the $2$ from doing $7 - 5$ and ignoring the sign.

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Common questions

Which is bigger, $-7$ or $-5$ ?: $-5$ is bigger. On a number line the further left a number sits, the smaller it is, and $-7$ is further left than $-5$ , so $-7 < -5$ . It feels backwards because 7 is a bigger digit than 5, but the minus sign reverses the order: with negatives, the larger the digit the smaller the number. A temperature of −7°C is colder (lower) than −5°C.
What is $-3 \times 3$ ?: It is $-9$ . A negative times a positive is negative, so $-3 \times 3 = -9$ . Multiplying by 3 makes the size 9, and the single minus sign stays, because a negative and a positive give a negative. Dropping the sign to get 9 loses the direction of the answer.
What is $5 - 7$ ?: It is $-2$ . Subtracting 7 from 5 takes you below zero: start at 5 and step 7 to the left and you land on $-2$ . The answer is negative because you took away more than you started with. Writing 2 ignores the sign and treats the subtraction as if it could not go below zero, which it can.

Related misconceptions

Order of operations: × and ÷ before + and −The neighbouring number skill: do multiply and divide before add and subtract, and finish brackets first, so 60 ÷ 2 + 4 = 34.
Squaring a negative numberWhere the sign rules meet powers: a negative times a negative is positive, so (−8)² = 64, and the bracket decides between (−4)² = 16 and −4² = −16.

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