Introduction to Integers (Negative Numbers)

Numbers below zero

So far in maths you have probably worked only with numbers from zero upwards: 0, 1, 2, 3, and so on. But numbers do not stop at zero. Integers are the whole numbers stretched in both directions — the positive numbers, zero, and a mirror set of negative numbers.

You already meet negatives in real life:

• A temperature of −5 °C (five degrees below freezing).
• A bank balance of −£20 (you owe twenty pounds).
• A submarine −30 metres below sea level.
• A lift going to floor −2 in a car park.

An integer is any of these whole numbers: ... −3, −2, −1, 0, 1, 2, 3 ... Notice there are no fractions or decimals here — those come from other number families like the ones in Decimals Explained.

The number line

The best tool for understanding integers is the number line. Picture a ruler that runs left and right with zero in the middle:

... -4  -3  -2  -1   0   1   2   3   4 ...
       <-- smaller        larger -->

Two rules unlock everything:

• Move right → numbers get bigger.
• Move left → numbers get smaller.

This means −5 is smaller than −1, even though 5 is bigger than 1. The minus flips your intuition: a bigger "size" with a negative sign is actually lower. That is why −20 °C is colder than −2 °C.

Comparing integers

Use the number line to compare. Whichever number is further right is greater.

Comparison	True statement	Why
3 and −5	3 > −5	3 is to the right of −5
−1 and −4	−1 > −4	−1 is to the right of −4
0 and −2	0 > −2	zero is right of any negative

Absolute value

The absolute value of a number is its distance from zero, ignoring the sign. We write it with two bars: |−7| = 7 and |7| = 7. Because distance is never negative, absolute value is always zero or positive.

Think of it as "how far from zero, regardless of direction." Both −7 and +7 sit 7 steps away from zero, so they share the same absolute value.

Adding integers

To add, walk along the number line. Start at the first number, then move right for a positive and left for a negative.

Example 1: −8 + 3. Start at −8, move 3 steps right → −5.

Example 2: −2 + (−6). Start at −2, move 6 steps left → −8.

A quick shortcut once you trust the line:

• Same signs → add the sizes, keep the sign. (−4) + (−5) = −9.
• Different signs → subtract the smaller size from the larger, and take the sign of the larger. (−9) + 4 = −5.

Subtracting integers

Here is the rule that surprises everyone: subtracting a negative is the same as adding.

$$ a - (-b) = a + b $$

Why? Subtracting means removing. Removing a debt makes you better off. If you owe £2 and that debt is taken away, you gain £2. So:

$$ 5 - (-2) = 5 + 2 = 7 $$

Example: −3 − (−7) = −3 + 7 = 4. Example: 2 − 9 = −7 (start at 2, move 9 left).

A handy trick: turn every subtraction into an addition of the opposite, then use the addition rules. So 6 − 10 becomes 6 + (−10) = −4.

Multiplying and dividing integers

For multiplication and division, the size is just the normal product or quotient. The clever part is the sign, and the rule is short:

First sign	Second sign	Result sign
+	+	+
−	−	+
+	−	−
−	+	−

In words: same signs give a positive; different signs give a negative.

• 6 × 4 = 24 → +24
• (−6) × (−4) = +24 (two negatives → positive)
• (−6) × 4 = −24 (different signs → negative)
• (−20) ÷ 5 = −4, and (−20) ÷ (−5) = +4

Why do two negatives make a positive? Multiplying by a negative reverses direction on the number line. Reverse once and you face the negative way; reverse a second time and you are pointing positive again. Two flips bring you back.

A worked mix

Evaluate −4 + 6 × (−2).

1. Multiplication first (order of operations): 6 × (−2) = −12.
2. Now add: −4 + (−12) = −16.

The order-of-operations habit you may have met in Algebra Basics matters just as much with negatives.

Practice activity

Try these on paper. Sketch a number line whenever you feel unsure.

1. Put in order, smallest first: 2, −7, 0, −1, 5.
2. Find |−12| and |9|.
3. Calculate −5 + 8.
4. Calculate −3 − (−10).
5. Calculate (−7) × 3 and (−24) ÷ (−6).

Answers: 1) −7, −1, 0, 2, 5 2) 12 and 9 3) 3 4) 7 5) −21 and 4.

Why this matters

Negative numbers are not a strange invention — they describe debts, temperatures, depths, and any change that can go down as well as up. Integers also open the door to the rest of algebra, where solutions are often negative. Keep the number line in your head, remember that subtracting a negative adds, and lock in the sign rule for multiplication. With those three ideas, the whole world of negative numbers becomes simple and predictable.