Negative numbers: adding, subtracting and multiplying integers

Picture a number line

Negative numbers are the numbers below zero. On a number line they sit to the left of 0: −1, −2, −3 and so on. The further left a number is, the smaller it is, so −8 is less than −3, even though 8 is more than 3. A thermometer is the same line standing up: −8 °C is colder than −3 °C.

Every number has an opposite the same distance from 0 on the other side. The opposite of 6 is −6, and the opposite of −6 is 6. A number's distance from 0 is its absolute value, so |−6| = 6.

Adding and subtracting integers

Adding means moving along the line. Adding a positive number moves right, and adding a negative number moves left, so 4 + (−9) starts at 4 and moves 9 left to −5. Subtraction is the reverse: subtracting a positive moves left, and subtracting a negative moves right.

That is why subtracting a negative gives the same answer as adding. Taking away a debt of $10 leaves you $10 better off. In symbols, 3 − (−10) = 3 + 10 = 13. If you rewrite every subtraction as adding the opposite, you only need two rules.

• Same signs: add the sizes and keep the sign. −15 + (−8) = −23.
• Different signs: subtract the sizes and keep the sign of the larger. −15 + 8 = −7, and 15 + (−8) = 7.
• Subtracting a negative: −2 − (−9) = −2 + 9 = 7.
• Subtracting a positive from a negative: −6 − 4 = −6 + (−4) = −10.

Multiplying negative numbers

Multiplying is repeated adding. 3 × (−4) means three lots of −4, which is −12. So a positive times a negative is negative.

For a negative times a negative, follow a pattern. 3 × (−4) = −12, 2 × (−4) = −8, 1 × (−4) = −4 and 0 × (−4) = 0. Each time the first number goes down by 1, the answer goes up by 4. Keep going and −1 × (−4) = 4, then −2 × (−4) = 8. A negative times a negative is positive.

• Same signs give a positive: 6 × 7 = 42 and −6 × (−7) = 42.
• Different signs give a negative: −6 × 7 = −42 and 6 × (−7) = −42.
• Division follows the same rules: −42 ÷ 6 = −7 and −42 ÷ (−6) = 7.
• With several factors, count the negatives. An even number of them gives a positive answer and an odd number gives a negative one: −2 × (−3) × (−5) = −30.

Watch the brackets with powers. (−3)² means −3 × (−3), which is 9. But −3² means the negative of 3², which is −9. A few questions in the practice set test exactly this.

Negative numbers in real life

Temperatures, bank balances, heights above and below sea level, and golf scores all use negative numbers. To find a change, subtract the start from the end. If it is −6 °C at night and 9 °C at noon, the temperature rose 9 − (−6) = 15 degrees. A diver at −12 m who rises 5 m is then at −12 + 5 = −7 m.

How it is taught in school

In the US, Common Core introduces negative numbers in grade 6: using them for real quantities, placing them on a number line, ordering them and understanding absolute value (6.NS.C.5 to 6.NS.C.7). Adding and subtracting integers comes in grade 7 (7.NS.A.1), along with multiplying and dividing them, where the standard points to products such as (−1)(−1) = 1 (7.NS.A.2).

In England, Year 4 pupils count backwards through zero, Year 5 pupils interpret negative numbers in context, and Year 6 pupils calculate intervals across zero, such as from −4 to 7. Using all four operations with positive and negative numbers is part of Key Stage 3.

How parents can help

• Use a thermometer or a weather app. Ask how many degrees warmer 5 °C is than −3 °C.
• Draw a number line and walk along it with a finger. Moving makes the rules make sense before they are memorized.
• Talk about money owed. A debt of $5 is −5, and paying off part of it moves you toward zero.
• Ask your child to explain why −8 is less than −3. If they can picture the number line, they won't need a trick.
• Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.