Significant figures, rounding and estimating
What this lesson covers
- The rules for identifying significant figures are: all non-zero digits are significant zeros appearing anywhere between two non-zero digits are significant leading zeros are not significant trailing zeros are significant if there is a decimal point present, and are not significant otherwise exact numbers have an unlimited number of significant figures.
- For numbers written in scientific notation as a × 10^k, the number of significant figures is determined by applying the rules to the value of a.
A number says how well you know it
- Your calculator will give you twelve digits for anything.
- Almost none of them mean anything about the real thing.
- Significant figures are how you say which ones do.
One length, three claims
- Four metres. Four point two metres. Four point two zero zero.
- Those are three different claims about the same wall.
- The zeros are not decoration. They say what you measured with.
| Written | Means measured to | Using |
|---|---|---|
| 4 m | nearest metre | a pace |
| 4.2 m | nearest 10 cm | a tape |
| 4.200 m | nearest mm | a good tape |
| The zeros | are not decoration | they are a claim |
Which digits count
- Every non-zero digit counts, always.
- Zeros are the whole problem, and there are three cases.
- In the middle they count. In front they never do.
| Digit | Does it count? |
|---|---|
| Any non-zero digit | always counts |
| A zero between two of them | counts |
| Zeros in front | never count |
| Zeros at the end | count only after a point |
Why zeros in front are different
- In 0.004, the two zeros do not tell you about the 4.
- They only say where the decimal point is.
- Precision is about the digits you measured, not the size.
Counting them in practice
- Nought point nought nought four two nought has three figures.
- One thousand two hundred and five has four. The inside zero counts.
- Start counting at the first digit that is not a zero.
| Number | What counts | How many |
|---|---|---|
| 0.004 20 | the 4, 2 and 0 | 3 figures |
| 0.000 5 | just the 5 | 1 figure |
| 1 205 | the inside zero counts | 4 figures |
| Front zeros | only place the point | 0 figures |
The case nobody can settle
- A crowd of exactly 4200 people has four figures.
- 4200 rounded from a survey might only have two.
- Written bare, the number cannot tell you which.
| Where it came from | Really has | Rule says |
|---|---|---|
| 4200 people counted | exact | all 4 count |
| 4200 from rounding | could be 2 or 3 | unclear |
| The rule says | no point, no credit | 2 figures |
| Which is | a convention, not a fact |
Scientific notation settles it
- The front number carries the figures.
- The power of ten carries the size. They are separate jobs.
- So 4.2, 4.20 and 4.200 times ten cubed are all 4200.
| Written | Value | Figures |
|---|---|---|
| 4.2 × 10³ | 4200 | 2 figures |
| 4.20 × 10³ | 4200 | 3 figures |
| 4.200 × 10³ | 4200 | 4 figures |
| Same size | different claim | no ambiguity |
Rounding to a number of figures
- Count from the first non-zero digit, then look at the next one.
- Pi to three figures is three point one four.
- Nought point nought nine nine nine to two figures carries to 0.10.
| Round this | Counting | Gives |
|---|---|---|
| 3.14159 to 3 s.f. | count 3, 1, 4 | 3.14 |
| 0.004 672 9 to 2 s.f. | count 4, 6 | 0.0047 |
| 0.0999 to 2 s.f. | the 9s carry | 0.10 |
| Start counting | at the first non-zero |
Figures are not decimal places
- Decimal places count from the point, whatever the number is.
- Figures count from the number, wherever the point is.
- For a small number those give wildly different answers.
| The question | The answer |
|---|---|
| 0.004 67 to 2 d.p. | 0.00 |
| 0.004 67 to 2 s.f. | 0.0047 |
| Places count from the point | figures count from the number |
| Both answers are right | to different questions |
Estimate before you calculate
- Forty-eight times nought point two one.
- Round both: fifty times nought point two is ten.
- Now you know the answer should be near ten before you start.
| Working | Rounded | About |
|---|---|---|
| 48 × 0.21 | round to 50 × 0.2 | about 10 |
| Calculator says | 10.08 | sensible |
| If it said 100.8 | a point is misplaced | check it |
| The estimate | catches size errors | not small ones |
What the estimate actually catches
- Calculators do not make small mistakes. People do.
- A mistyped point or a wrong button is out by a factor of ten.
- An estimate spots that instantly. Nothing else will.
How precise should you be?
- Money goes to the cent. Height goes to the centimetre.
- A bus timetable to the second would be useless.
- The context decides, not the calculator display.
| Situation | Rounded to | Looks like |
|---|---|---|
| Money | to the cent | $14.27 |
| A person’s height | to the centimetre | 171 cm |
| A bus timetable | to the minute | 8:14 |
| 7 people shared by 3 | not 2.333333 | people are whole |
Too many digits is its own mistake
- Seven people shared between three is not 2.333333 people.
- Writing digits you did not measure is claiming what you do not know.
- It reads as careful and it is the opposite.
What to take away
- Seven lines, and the first one is the reason for the rest.
| A number claims its own precision | that is what figures record |
| Non-zero digits | always count |
| Zeros in the middle | count |
| Zeros in front | never count |
| Zeros at the end | count after a decimal point |
| Scientific notation | removes the ambiguity |
| Estimate first | it catches the big mistakes |
Your turn: four questions
- Pause here and answer all four.
- Question 1 has three kinds of zero in it.
- Question 4 cannot be done without scientific notation.
| Question | |
|---|---|
| 1 | How many figures in 0.003 050? |
| 2 | Round 0.028 47 to 2 significant figures. |
| 3 | Estimate 39 × 0.48. Is 18.72 sensible? |
| 4 | Write 4200 with exactly 3 figures. |
Show answer
1. Four. The two zeros in front do not count, the 3 counts, the zero between the 3 and the 5 counts, the 5 counts, and the final zero counts because it comes after a decimal point. 2. 0.028. Start counting at the 2, so the first two figures are 2 and 8, and the next digit is a 4, which rounds down. 3. Round to 40 times 0.5, which is 20. The calculator answer of 18.72 is close to 20, so it is sensible. An answer of 187.2 or 1.872 would not be. 4. 4.20 × 10³. Writing 4200 on its own cannot say three figures, which is exactly the problem scientific notation solves.
The answers
- Four figures: the 3, the middle zero, the 5 and the last zero.
- Nought point nought two eight, because the next digit is a 4.
- Forty times nought point five is twenty, so 18.72 is sensible.
- Four point two zero times ten cubed.
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