Before this page: be able to place negative numbers and decimals in order from ordering positive and negative numbers, and read the value of a digit from place value in integers and decimals.
A number is an arrow: right for positive, left for negative
Every number can be drawn as an arrow on the number line. A positive number points right and a negative number points left. The arrow's length is the number's distance from zero, its magnitude:
−4 + 6 − 5
The first arrow
−4 is an arrow four units long, drawn from zero and pointing left.
A long arrow is not counted unit by unit: −27 + 40 = 13, because 27 of the 40 units reach zero and 13 remain.
Example: work out −7 + 3
Start at −7 and add 3.
- Start at −7. Seven units to the left of zero.
- Adding 3 moves 3 right. The 3 is positive, so its arrow points right.
- −7 + 3 = −4 Three units from −7 reach −4, which is still left of zero because 3 is smaller than 7.
A decimal moves by the same rules. −2.5 + 1.8 = −0.7, because an arrow 1.8 long is not enough to reach zero from −2.5 and leaves 0.7 of the distance still to go.
Subtracting a number is adding it with its sign changed
Changing the sign of 5 gives −5 and changing the sign of −3 gives 3: subtracting 5 adds −5, and subtracting −3 adds 3.
In 3 − (−2), the arrow for −2 points left and is laid with its tip on 3, so its tail lands on 5, the same move adding 2 makes: 3 − (−2) = 3 + 2 = 5.
3 − (−2) = 3 + 2 = 5
5 − (−3) = 5 + 3 = 8
- Subtracting a negative number increases the total: 5 − (−3) = 8.
- Adding a negative number decreases it: 5 + (−3) = 2.
The first sign is the operation and the second belongs to the number. Read together, the pair is one operation: − (−9) is + 9, and + (−9) is − 9.
Example: work out −4 − (−9)
Subtract −9 from −4.
- The signs between the numbers are − and −. A subtraction sign followed by a negative number.
- −4 − (−9) = −4 + 9 Subtracting −9 adds 9, the same number with its sign changed.
- −4 + 9 = 5 Four of the nine units reach zero, and five remain.
The difference between two numbers is the distance between them
Zero falls between −6°C and 4°C, so the distance from one to the other is made of two parts. The difference is that distance, counted as a positive amount whichever reading is the larger.
Example: the rise from −6°C to 4°C
−6°C to 4°C
The colder reading
−6°C stands six degrees below zero.
Subtracting the lower reading from the higher one gives that distance without counting: 4 − (−6) = 4 + 6 = 10, a rise of 10 degrees. Subtracting the other way round gives −10, a fall of the same 10 degrees.
Example: a balance moving from −£85.40 to £32.10
Work out how much was paid in.
- The amount paid in is the rise from −£85.40 to £32.10. −£85.40 is an overdraft of £85.40, and £32.10 is in credit, so zero lies between them.
- 32.10 − (−85.40) = 32.10 + 85.40 The rise is the higher balance minus the lower, and subtracting a negative adds.
- 32.10 + 85.40 = 117.50 £85.40 clears the overdraft and £32.10 is left, so £117.50 was paid in.
Common mistakes
- Adding the two numbers and keeping the minus −8 + 3 is not −11. Adding 3 moves 3 to the right, towards zero, so the answer is closer to zero than −8 is: −8 + 3 = −5.
- Reading two signs as a single minus In 7 − (−4) the second sign belongs to the 4, so the number being subtracted is −4, and subtracting it adds 4. The answer is 7 − (−4) = 7 + 4 = 11, not 3.
- Ignoring the side of zero each reading sits on The difference between −3°C and 5°C is not 2 degrees. The gap runs 3 degrees up to zero and 5 degrees beyond it: 5 − (−3) = 8 degrees.
Summary
- An arrow points right for a positive number and left for a negative one, and its length is the number's distance from zero.
- −7 + 3 = −4: adding less than the distance to zero leaves the answer negative.
- Subtracting a number is adding it with its sign changed, so 5 − (−3) = 5 + 3 = 8; adding a negative runs left, so 3 + (−7) = 3 − 7 = −4.
- The difference between two numbers is the distance between them, and subtracting the lower from the higher gives it: 4 − (−6) = 10.
- Decimals move by the same rules, so −2.5 + 1.8 = −0.7.