Precision: your answer inherits its worst measurement
What this lesson covers
- A solution to a calculation cannot be more precise than the least precise number used in that calculation.
A calculator cannot measure anything
- It will happily give you eleven point five five square metres.
- But it never saw the room. You did, with a tape.
- An answer cannot be more precise than what went into it.
Precision comes from the equipment
- Pace out a room and you know it to the nearest metre.
- Use a tape and you know it to the nearest ten centimetres.
- How many digits you may write is decided before you write any.
| Method | Good to | You may write |
|---|---|---|
| Pacing it out | nearest metre | 4 m |
| A tape measure | nearest 10 cm | 4.2 m |
| A good steel tape | nearest cm | 4.20 m |
| The tool decides | how many figures | you may write |
One room, measured and multiplied
- Four point two metres by two point seven five metres.
- The calculator says eleven point five five square metres.
- One of those measurements has only two figures in it.
| Step | Working | Result |
|---|---|---|
| Measured | 4.2 m by 2.75 m | |
| Calculator says | 4.2 × 2.75 | 11.55 |
| Figures in | 2 and 3 | |
| So the answer gets | 2 figures | 12 m² |
The rule, in one line
- The answer takes the figures of the least precise input.
- Two figures went in, so two figures come out.
- Twelve square metres. Not eleven point five five.
What four point two actually claims
- It says the wall is nearer to 4.2 than to 4.1 or 4.3.
- So the real length is somewhere from 4.15 to 4.25.
- A measurement is a range. It was never a single value.
| Written | Meaning | Range |
|---|---|---|
| Written 4.2 | rounded to 1 decimal place | |
| So it could be | anything from 4.15 | up to 4.25 |
| Same for 2.75 | 2.745 to 2.755 | |
| A measurement | is a range | not a point |
So how big could the room be?
- Smallest: 4.15 times 2.745, which is 11.39.
- Largest: 4.25 times 2.755, which is 11.71.
- The second decimal place of 11.55 is not shaky. It is unknown.
| Case | Working | Area |
|---|---|---|
| Smallest possible | 4.15 × 2.745 | 11.39 |
| Largest possible | 4.25 × 2.755 | 11.71 |
| So the area is | between those | |
| 11.55 claims | a hundredth | we never measured |
Adding follows a different rule
- Multiplying and dividing carry significant figures through.
- Adding and subtracting carry decimal places instead.
- 12.5 plus 3.25 is 15.75, and you write 15.8.
| Operation | Match the fewest |
|---|---|
| Multiplying or dividing | match the fewest significant figures |
| Adding or subtracting | match the fewest decimal places |
| 12.5 + 3.25 = 15.75 | but 12.5 knew no hundredths |
| So write | 15.8 |
Do not round in the middle
- 24.7 divided by 3 is 8.233 and running.
- Round it to 8.2 now, times by 3, and you get 24.6.
- Keep the extra figures and you get your 24.7 back.
| Working | Doing | Ends at |
|---|---|---|
| 24.7 ÷ 3 | is 8.233… | |
| Round now to 8.2 | then × 3 | 24.6 |
| Keep 8.233 | then × 3 | 24.7 |
| Carry one spare figure | round at the end |
The same number, two answers
- 2.346 to one decimal place is 2.3.
- But to two places first it is 2.35, and then it becomes 2.4.
- One number, two answers, because it was rounded twice.
| Route | Step | Answer |
|---|---|---|
| 2.346 to 1 d.p. | straight there | 2.3 |
| 2.346 to 2 d.p. | first | 2.35 |
| Then to 1 d.p. | rounds up again | 2.4 |
| Two different answers | so round once | at the end |
Four point two zero is not four point two
- The extra zero says you measured to the nearest centimetre.
- It is a claim about your equipment, not a tidier way to write it.
- So never add a zero you did not earn.
| Written | Claims | Figures |
|---|---|---|
| 4.2 m | nearest 10 cm | 2 figures |
| 4.20 m | nearest cm | 3 figures |
| Same length | different equipment | |
| Do not add zeros | you did not measure |
Estimate the measurement too
- A door is about two metres. A step is about thirty centimetres.
- Those let you check a measurement before you trust it.
- A room that comes out at forty metres long is a misread tape.
How precise does the job need?
- A shelf in an alcove needs millimetres or it will not go in.
- Paint gets bought in whole tins, so litres are enough.
- Ask what the number is for before deciding how to round it.
| Job | Precision | Because |
|---|---|---|
| Concrete for a slab | nearest 0.1 m³ | order extra |
| A shelf in an alcove | nearest mm | or it jams |
| Paint for a wall | nearest litre | tins are whole |
| Ask what it is for | then choose |
Writing fewer digits is not being lazy
- Twelve square metres looks less impressive than 11.55.
- It is the more careful answer, not the less careful one.
- Extra digits are a claim about knowledge you do not have.
What to take away
- Seven lines, and the last one is the habit.
| A measurement is a range | not an exact value |
| The instrument decides | how many figures you may write |
| Multiplying | match the fewest significant figures |
| Adding | match the fewest decimal places |
| Round once | at the very end |
| 4.20 is not 4.2 | the zero is a claim |
| The calculator display | is not the answer |
Your turn: four questions
- Pause here and work all four.
- Question 2 is an addition, so it follows the other rule.
- Question 4 is asking you to round only once.
| Question | |
|---|---|
| 1 | Area of 3.5 m by 1.24 m, properly rounded. |
| 2 | Add 15.2 g and 3.45 g, properly rounded. |
| 3 | Why is 4.20 m not the same as 4.2 m? |
| 4 | Round 3.748 to 2 significant figures. |
Show answer
1. 4.3 square metres. The calculator gives 4.34, but 3.5 carries only two significant figures, so the answer gets two. 2. 18.7 grams. This is an addition, so the rule is decimal places, not significant figures: 15.2 is given to one decimal place, so the answer is too. 3. Because 4.20 says the length was measured to the nearest centimetre and 4.2 says only to the nearest ten centimetres. Same length, different claim about the equipment. 4. 3.7. If you round to two decimal places first you get 3.75, and rounding that again gives 3.8, which is wrong. Round once, from the original number.
The answers
- 4.34 rounds to 4.3 square metres, because 3.5 has two figures.
- 18.65 rounds to 18.7 grams, on decimal places.
- The zero claims the nearest centimetre.
- 3.7. Going through 3.75 would give you 3.8, and that is wrong.
Is your child on track?
Take the free placement test