GCSE Maths: Negative numbers (Higher)
Part of the GCSE Maths number basics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Negative numbers
A negative number is a number less than zero. We use them for temperatures below freezing, for money you owe (an overdraft), for floors below ground level in a lift, and for golf scores below par. On a number line, zero is in the middle, positive numbers go to the right, and negative numbers go to the left. A number is bigger the further right it is. So −8 < −3 < 0 < 2. Notice that −8 looks bigger than −3 but it is actually smaller: it is further below zero.
Adding and subtracting. Start at the first number on the number line. Adding moves you to the right and subtracting moves you to the left. So −3 + 5 = 2 (move 5 right) and 2 − 9 = −7 (move 9 left, crossing zero). When two signs are next to each other, replace them with one: + (−) becomes −, and − (−) becomes +. Think of a debt: adding a debt of £5 leaves you £5 worse off, and taking away a debt of £5 leaves you £5 better off. To find the difference between two values (for example two temperatures) subtract the lower from the higher: 6 − (−3) = 6 + 3 = 9.
Multiplying and dividing. First ignore the signs and work out the answer. Then decide the sign: if the two numbers have the same sign the answer is positive; if they have different signs the answer is negative. So 6 × (−4) = −24, (−6) × (−4) = +24, −24 ÷ 4 = −6 and −24 ÷ (−4) = 6. Why is a negative times a negative positive? Look at the pattern (−4) × 3 = −12, (−4) × 2 = −8, (−4) × 1 = −4, (−4) × 0 = 0. Each time the second number goes down by 1 the answer goes up by 4, so the next answers must be 4, 8, 12 for (−4) × (−1), (−4) × (−2), (−4) × (−3).
Key ideas
- Negative numbers are below zero. The further right on the number line, the bigger the number.
- Adding moves right, subtracting moves left. + (−) is −, and − (−) is +.
- Difference between two values = higher value − lower value.
- Multiply or divide: same signs give a positive answer; different signs give a negative answer.
Method. Make the numbers easy to compare, then order them smallest to largest.
- Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.Negative: −3, −8, −1.5
Not negative: 5, 0, 2 - Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.−3 = −3.0
5 = 5.0
−8 = −8.0
0 = 0.0
−1.5 = −1.5
2 = 2.0 - Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 1 decimal place, so we can compare them as if they were whole numbers. (The minus signs are dealt with next.)3: 30, 5: 50, 8: 80, 0: 0, 1.5: 15, 2: 20
- Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).−8, −3, −1.5, 0, 2, 5
Check. Check each neighbouring pair: −8 < −3; −3 < −1.5; −1.5 < 0; 0 < 2; 2 < 5. ✓
Watch out. For negative numbers, the number with the bigger digits is the smaller number.
Method. The difference between two temperatures is the higher temperature minus the lower one.
- Decide which temperature is higher and subtract. The difference tells us how many degrees apart they are. We always subtract the lower value from the higher value.21 − (−18)
- Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 18, you are 18 better off. So "− (−18)" is the same as "+ 18".21 − (−18) = 21 + 18
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at 21 on the number line and move 18 steps to the right (adding moves up).
21 + 18 = 39
Check. Count on the number line: from −18 up to 0 is 18, and from 0 up to 21 is 21. 18 + 21 = 39. ✓
Watch out. Do not just subtract the numbers and ignore the minus sign: 21 − 18 = 3 is wrong here because −18 is below zero.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and multiply the numbers. First find the size of the answer, using the numbers as if they were positive.6 × 7 = 42
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.One number is negative and the other is positive: the signs are different, so the answer is negative.
- Write the answer. (−6) × 7 = −42
Check. Check with the inverse: −42 ÷ 7 = −6. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and divide the numbers. First find the size of the answer, using the numbers as if they were positive.24 ÷ 4 = 6
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.Both numbers are negative: the signs are the same, so the answer is positive.
- Write the answer. (−24) ÷ (−4) = 6
Check. Check with the inverse: 6 × (−4) = −24. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Method. Use a pattern: reduce the second number by 1 each time and see how the answers change.
- Start with a multiplication you know and make a pattern. Keep the first number as −4. Multiply it by 3, 2, 1, 0 and then by the negative numbers, going down by 1 each time.−4 × 3 = −12
−4 × 2 = −8
−4 × 1 = −4
−4 × 0 = 0
−4 × (−1) = 4
−4 × (−2) = 8
−4 × (−3) = 12 - Look at how the answers change. Each time the second number goes down by 1, the answer goes up by 4. This pattern carries on past zero.−12, −8, −4, ... each step is +4
- Read off the answer. The pattern shows that multiplying two negative numbers gives a positive number.−4 × (−3) = 12
Check. The sign rule agrees: two negative numbers multiplied give a positive. 4 × 3 = 12. ✓
Watch out. The pattern only works if you keep the first number the same and change the second by 1 each time.
Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).
Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.
- Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.(−5 × 4) + (−12 ÷ (−3))
- Brackets first. Multiply: −5 × 4. This is the only calculation left in this part.(−5 × 4) + (−12 ÷ (−3))
−5 × 4 = −20
= −20 + (−12 ÷ (−3)) - Brackets first. Divide: −12 ÷ (−3). This is the only calculation left in this part.−20 + (−12 ÷ (−3))
−12 ÷ (−3) = 4
= −20 + 4 - Add: −20 + 4. Only addition or subtraction is left, so do it now.−20 + 4
−20 + 4 = −16
= −16
Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives −16. ✓
Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.
Watch out
−3 − 4 is −7, not 1. Both numbers take you left. And the sign rules for × and ÷ do not work for + and −: −3 + (−4) = −7, not +7.Exercise 1E Negative numbers
Non-calculatorMethod. Make the numbers easy to compare, then order them smallest to largest.
- Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.Negative: −5, −8, −1
Not negative: 3, 0 - Count the digits in each number. Compare the sizes first, ignoring the minus signs. More digits means a bigger size.5: 1 digit, 3: 1 digit, 8: 1 digit, 0: 1 digit, 1: 1 digit
- Where the numbers have the same number of digits, compare column by column from the left. The first column where two numbers differ decides which is bigger.5, 3, 8, 0, 1
- Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).−8, −5, −1, 0, 3
Check. Check each neighbouring pair: −8 < −5; −5 < −1; −1 < 0; 0 < 3. ✓
Watch out. For negative numbers, the number with the bigger digits is the smaller number.
Method. Make the numbers easy to compare, then order them smallest to largest.
- Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.Negative: −0.5, −0.45, −0.55
Not negative: 0.4 - Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.−0.5 = −0.50
−0.45 = −0.45
−0.55 = −0.55
0.4 = 0.40 - Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 2 decimal places, so we can compare them as if they were whole numbers. (The minus signs are dealt with next.)0.5: 50, 0.45: 45, 0.55: 55, 0.4: 40
- Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).−0.55, −0.5, −0.45, 0.4
Check. Check each neighbouring pair: −0.55 < −0.5; −0.5 < −0.45; −0.45 < 0.4. ✓
Watch out. For negative numbers, the number with the bigger digits is the smaller number.
Method. Make the numbers easy to compare, then order them largest to smallest.
- Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.Negative: −12, −2, −21
Not negative: 7 - Count the digits in each number. Compare the sizes first, ignoring the minus signs. More digits means a bigger size.12: 2 digits, 2: 1 digit, 7: 1 digit, 21: 2 digits
- Where the numbers have the same number of digits, compare column by column from the left. The first column where two numbers differ decides which is bigger.12, 2, 7, 21
- Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).7, −2, −12, −21
Check. Check each neighbouring pair: 7 > −2; −2 > −12; −12 > −21. ✓
Watch out. For negative numbers, the number with the bigger digits is the smaller number.
Method. For two negative numbers, the one closer to zero is bigger. Compare the sizes, then reverse the symbol.
- Both numbers are negative: ignore the minus signs for a moment. We compare how far each is from zero.3 and 5
- Compare the sizes. The number further from zero is the smaller number when it is negative.3 < 5 so the order reverses for negatives
- Write the final comparison. Reverse the symbol.−3 > −5
Check. On a number line the bigger number is to the right. ✓
Watch out. −7 is smaller than −2, even though 7 is bigger than 2.
Method. Negative numbers are below zero, so they are smaller than positive numbers.
- Look at the signs. A negative number is always smaller than a positive number or zero, because it is below zero on the number line.−7 is negative, 2 is not.
- Decide. The negative one is smaller.−7 < 2
Check. On a number line −7 is to the left of 2. ✓
Method. For two negative numbers, the one closer to zero is bigger. Compare the sizes, then reverse the symbol.
- Both numbers are negative: ignore the minus signs for a moment. We compare how far each is from zero.0.4 and 0.04
- Compare the sizes. The number further from zero is the smaller number when it is negative.0.4 > 0.04 so the order reverses for negatives
- Write the final comparison. Reverse the symbol.−0.4 < −0.04
Check. On a number line the bigger number is to the right. ✓
Watch out. −7 is smaller than −2, even though 7 is bigger than 2.
Method. For two negative numbers, the one closer to zero is bigger. Compare the sizes, then reverse the symbol.
- Both numbers are negative: ignore the minus signs for a moment. We compare how far each is from zero.12 and 21
- Compare the sizes. The number further from zero is the smaller number when it is negative.12 < 21 so the order reverses for negatives
- Write the final comparison. Reverse the symbol.−12 > −21
Check. On a number line the bigger number is to the right. ✓
Watch out. −7 is smaller than −2, even though 7 is bigger than 2.
Method. Find 0 on the scale, then count the marks below it.
- Find 0 on the scale. Marks above 0 are positive and marks below 0 are negative.0°C
- Count down to A. A is 7 small marks below 0, so it is a negative temperature.A = −7°C
Check. 7 below zero is −7. ✓
Method. Find 0 on the scale, then count the marks above it.
- Find 0 on the scale. 0°C
- Count up to B. B is 4 small marks above 0, so it is positive.B = 4°C
Check. 4 above zero is 4. ✓
Method. The colder place has the lower temperature, which is the lower point on the thermometer.
- Compare the two temperatures. A negative number is always less than a positive number.−7 < 4
- Decide. The lower temperature is colder.A is colder.
Check. A is below zero but B is above zero. ✓
Method. The difference is the higher temperature minus the lower temperature.
- Write the subtraction. Higher minus lower.4 − (−7)
- Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 7, you are 7 better off. So "− (−7)" is the same as "+ 7".4 − (−7) = 4 + 7
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at 4 on the number line and move 7 steps to the right (adding moves up).
4 + 7 = 11
Check. From −7 up to 0 is 7 degrees, and from 0 to 4 is 4 degrees: 7 + 4 = 11. ✓
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.−3 + 5
- Look at the signs. The number being added is positive, so there are no double signs to fix.−3 + 5
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at −3 on the number line and move 5 steps to the right (adding moves up).
3 steps take you to 0, then 2 more steps take you to 2.
−3 + 5 = 2
Check. Reverse the calculation: 2 − 5 = −3. ✓
Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.4 − 9
- Look at the signs. The number being subtracted is positive, so there are no double signs to fix.4 − 9
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at 4 on the number line and move 9 steps to the left (subtracting moves down).
4 steps take you down to 0, then 5 more steps take you to −5.
4 − 9 = −5
Check. Reverse the calculation: −5 + 9 = 4. ✓
Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.−7 − 2
- Look at the signs. The number being subtracted is positive, so there are no double signs to fix.−7 − 2
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at −7 on the number line and move 2 steps to the left (subtracting moves down).
You start at or below zero and go further down, so the answer is more negative: 7 + 2 = 9, and with a minus sign that is −9.
−7 − 2 = −9
Check. Reverse the calculation: −9 + 2 = −7. ✓
Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.6 − (−4)
- Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 4, you are 4 better off. So "− (−4)" is the same as "+ 4".6 − (−4) = 6 + 4
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at 6 on the number line and move 4 steps to the right (adding moves up).
6 + 4 = 10
Check. Reverse the calculation: 10 + (−4) = 6. ✓
Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.−5 + (−3)
- Deal with the two signs next to each other: adding a negative. Adding a negative number takes away: if you add a debt of 3 you are 3 worse off. So "+ (−3)" is the same as "− 3".−5 + (−3) = −5 − 3
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at −5 on the number line and move 3 steps to the left (subtracting moves down).
You start at or below zero and go further down, so the answer is more negative: 5 + 3 = 8, and with a minus sign that is −8.
−5 − 3 = −8
Check. Reverse the calculation: −8 − (−3) = −5. ✓
Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.−2 − (−8)
- Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 8, you are 8 better off. So "− (−8)" is the same as "+ 8".−2 − (−8) = −2 + 8
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at −2 on the number line and move 8 steps to the right (adding moves up).
2 steps take you to 0, then 6 more steps take you to 6.
−2 + 8 = 6
Check. Reverse the calculation: 6 + (−8) = −2. ✓
Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and multiply the numbers. First find the size of the answer, using the numbers as if they were positive.6 × 7 = 42
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.One number is negative and the other is positive: the signs are different, so the answer is negative.
- Write the answer. (−6) × 7 = −42
Check. Check with the inverse: −42 ÷ 7 = −6. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and multiply the numbers. First find the size of the answer, using the numbers as if they were positive.8 × 5 = 40
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.One number is negative and the other is positive: the signs are different, so the answer is negative.
- Write the answer. 8 × (−5) = −40
Check. Check with the inverse: −40 ÷ (−5) = 8. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and multiply the numbers. First find the size of the answer, using the numbers as if they were positive.9 × 4 = 36
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.Both numbers are negative: the signs are the same, so the answer is positive.
- Write the answer. (−9) × (−4) = 36
Check. Check with the inverse: 36 ÷ (−4) = −9. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and divide the numbers. First find the size of the answer, using the numbers as if they were positive.24 ÷ 4 = 6
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.One number is negative and the other is positive: the signs are different, so the answer is negative.
- Write the answer. (−24) ÷ 4 = −6
Check. Check with the inverse: −6 × 4 = −24. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and divide the numbers. First find the size of the answer, using the numbers as if they were positive.36 ÷ 9 = 4
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.One number is negative and the other is positive: the signs are different, so the answer is negative.
- Write the answer. 36 ÷ (−9) = −4
Check. Check with the inverse: −4 × (−9) = 36. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Multiply or divide the numbers as if they were positive, then fix the sign: same signs give positive, different signs give negative.
- Ignore the signs and divide the numbers. First find the size of the answer, using the numbers as if they were positive.45 ÷ 5 = 9
- Decide the sign of the answer. Same signs (both positive or both negative) give a positive answer. Different signs give a negative answer.Both numbers are negative: the signs are the same, so the answer is positive.
- Write the answer. (−45) ÷ (−5) = 9
Check. Check with the inverse: 9 × (−5) = −45. ✓
Watch out. The sign rule is for multiplying and dividing only. In an addition, −3 + (−4) = −7, not +7.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Start with the first value. Add for a rise and subtract for a fall, using a number line to cross zero carefully.
- Write the start and the changes as a calculation. A rise means add and a fall means subtract. start = −8°C
- Rise of 15°C: add 15. Use a number line: add moves right (up), subtract moves left (down).Start at −8 on the number line and move 15 steps to the right (adding moves up).
8 steps take you to 0, then 7 more steps take you to 7.
−8 + 15 = 7 - Write the answer with its unit. 7°C
Check. Work backwards: 7, −8. ✓
Watch out. Going below zero is not the same as going back up. Count the steps to zero, then the remaining steps below it.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Start with the first value. Add for a rise and subtract for a fall, using a number line to cross zero carefully.
- Write the start and the changes as a calculation. A rise means add and a fall means subtract. start = −8°C
- Rise of 15°C: add 15. Use a number line: add moves right (up), subtract moves left (down).Start at −8 on the number line and move 15 steps to the right (adding moves up).
8 steps take you to 0, then 7 more steps take you to 7.
−8 + 15 = 7 - Fall of 9°C: subtract 9. Use a number line: add moves right (up), subtract moves left (down).Start at 7 on the number line and move 9 steps to the left (subtracting moves down).
7 steps take you down to 0, then 2 more steps take you to −2.
7 − 9 = −2 - Write the answer with its unit. −2°C
Check. Work backwards: −2, 7, −8. ✓
Watch out. Going below zero is not the same as going back up. Count the steps to zero, then the remaining steps below it.
Method. Subtract the bill from the balance. A negative balance means Jo owes the bank money.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.120 − 175
- Look at the signs. The number being subtracted is positive, so there are no double signs to fix.120 − 175
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at 120 on the number line and move 175 steps to the left (subtracting moves down).
120 steps take you down to 0, then 55 more steps take you to −55.
120 − 175 = −55
Check. −55 + 175 = 120. ✓
Watch out. A negative balance (overdrawn) means money is owed.
Method. Paying in adds to the balance, so add 60 to her negative balance.
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.−55 + 60
- Look at the signs. The number being added is positive, so there are no double signs to fix.−55 + 60
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at −55 on the number line and move 60 steps to the right (adding moves up).
55 steps take you to 0, then 5 more steps take you to 5.
−55 + 60 = 5
Check. 5 − 60 = −55. ✓
Watch out. −55 + 60 is positive: the payment is bigger than the debt, so she has some money left.
Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).
Method. Start with the first value. Add for a rise and subtract for a fall, using a number line to cross zero carefully.
- Write the start and the changes as a calculation. A rise means add and a fall means subtract. Floors below ground are negative.start = −2
- Rise of 7: add 7. Use a number line: add moves right (up), subtract moves left (down).Start at −2 on the number line and move 7 steps to the right (adding moves up).
2 steps take you to 0, then 5 more steps take you to 5.
−2 + 7 = 5 - Fall of 4: subtract 4. Use a number line: add moves right (up), subtract moves left (down).Start at 5 on the number line and move 4 steps to the left (subtracting moves down).
5 − 4 = 1 - Write the answer with its unit. 1
Check. Work backwards: 1, 5, −2. ✓
Watch out. Going below zero is not the same as going back up. Count the steps to zero, then the remaining steps below it.
Method. Find the distance between the numbers, halve it, and add it to the smaller number.
- Find the distance between the two numbers. On the number line, count from the lower to the higher number.3 − (−7) = 10
- Halve the distance. Halfway means half of the distance from either end.10 ÷ 2 = 5
- Add the half-distance to the lower number. Start at the lower number and move to the right.−7 + 5 = −2
Check. −2 to 3 is 5 and −7 to −2 is 5. They are equal. ✓
Method. Use a number line: subtracting moves left.
- Start at −3. −3 is 3 below zero.start = −3
- Subtract 4: move 4 to the left. Moving left takes you further below zero.−3 − 4 = −7
- Say what Tom did wrong. He ignored the minus sign on the 3 and worked out 4 − 3 = 1.Tom is not correct.
Check. −7 + 4 = −3. ✓
Watch out. Both numbers take you to the left, so the answer is more negative.
Method. Replace the double sign and compare.
- Replace the double sign. Adding a negative number is the same as subtracting the positive number.−5 + (−3) = −5 − 3
- Work out the answer. Both numbers take you to the left of −5 by 3.−5 − 3 = −8
Check. −8 + 3 = −5. ✓
Method. Follow BIDMAS: the multiplications are done before the subtraction.
- Do the multiplications first. Multiplication has priority over subtraction. Use the sign rules.(−3) × 4 = −12 (different signs)
(−2) × (−5) = +10 (same signs) - Rewrite the calculation. Replace each product by its value.−12 − 10
- Subtract. Both numbers take you left on the number line.−12 − 10 = −22
Check. −22 + 10 = −12. ✓
Method. Brackets first, then multiply.
- Work out the first bracket. Adding a negative number moves left.−2 + (−7) = −9
- Work out the second bracket. 4 − 9 moves 9 to the left from 4, crossing zero.4 − 9 = −5
- Multiply the two results. Two negative numbers multiplied give a positive number.(−9) × (−5) = 45
Check. 45 ÷ (−5) = −9. ✓
Method. Keep going with the pattern past zero and see what happens to the answers.
- Continue the pattern. Each time the second number goes down by 1, the answer goes up by 5.(−5) × 1 = −5
(−5) × 0 = 0
(−5) × (−1) = 5
(−5) × (−2) = 10
(−5) × (−3) = 15
(−5) × (−4) = 20 - Explain. The pattern increases by 5 each time and it carries on, so a negative times a negative must be positive.(−5) × (−4) = 20
Check. 5 × 4 = 20, and the two negatives make a positive. ✓
Method. Find the total fall over the 6 hours, then subtract it from the starting temperature.
- Work out the total fall. A steady rate means the same amount each hour. Multiply the rate by the number of hours (6 pm to midnight is 6 hours).2.5 × 6 = 15
- Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.3 − 15
- Look at the signs. The number being subtracted is positive, so there are no double signs to fix.3 − 15
- Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.Start at 3 on the number line and move 15 steps to the left (subtracting moves down).
3 steps take you down to 0, then 12 more steps take you to −12.
3 − 15 = −12
Check. −12 + 15 = 3, the starting temperature. ✓
Method. List the pairs of negative numbers that multiply to 12, then pick the pair with the right sum.
- List pairs with a product of 12. Two negative numbers multiply to give a positive number, so pairs of negative numbers can work.−1 × −12 = 12
−2 × −6 = 12
−3 × −4 = 12 - Work out the sum of each pair. Add the two negative numbers: the answer is more negative.−1 + (−12) = −13
−2 + (−6) = −8
−3 + (−4) = −7 - Choose the pair whose sum is −7. −3 and −4
Check. (−3) × (−4) = 12 and −3 + (−4) = −7. ✓
Method. Think about moving left twice on the number line.
- Start with a negative number. It is to the left of zero.e.g. −2
- Add another negative number. Adding a negative number moves further left, so we stay below zero.−2 + (−5) = −7
- Decide. This will happen every time.Always true.
Check. −1 + (−1) = −2. ✓
Method. Use the sign rule for multiplying.
- Test an example. Same signs give a positive answer.(−3) × (−4) = 12
- Decide. This is positive for every pair of negative numbers.Never true.
Check. (−1) × (−1) = 1. ✓
Method. Try a positive number and a negative number.
- Try a positive number. 5 − 2 = 3, which is less than 5.3 < 5 (true)
- Try a negative number. 5 − (−2) = 7, which is more than 5.7 > 5 (false)
- Decide. It depends on the number subtracted.Sometimes true.
Check. 5 − (−2) = 5 + 2 = 7. ✓