Significant Figures
Rounding: decimal places and significant figures
In this note
- 1.9Round to the nearest 10, 100, 1000, million or whole number.
- 1.9Round to a given number of decimal places (d.p.).
- 1.9Round to a given number of significant figures (s.f.), including numbers with zeros.
Next: estimation and rounding in context. Bounds: the Limits of Accuracy chapter.
Formula bank
The one rounding rule
General methodHow to round any number
- Find the last digit you keep (the rounding digit).
- Look at the NEXT digit only (the decider).
- 5 or more → round the kept digit up (carry into the next place if it was 9). 4 or less → keep it.
- Whole-number places you drop become zeros; decimal places you drop are removed.
Example Write 2.6963 correct to 2 d.p.
Keep 2.69, decider 6 → round up, which carries.
- 2.69 | 63
- 9 + 1 = 10: carry
Answer 2.70
Try it yourself
Show answer
Answer 2700
Show answer
Answer 12.345
Show answer
Answer 39 000 000
Significant figures
Significant figures count from the first non-zero digit, so 3 s.f. works equally well for a huge or a tiny number.
General methodHow to round to any number of significant figures
- Find the first non-zero digit: that is the 1st s.f. (zeros before it don't count).
- Count along (zeros in the middle DO count) to the last s.f. you keep.
- Use the rounding rule on the next digit.
- Fill whole-number places with zeros to keep the size; keep a final decimal zero if it is one of your s.f. (19.97 → 20.0).
| Number | To | Answer | Why |
|---|---|---|---|
| 0.040 705 | 3 s.f. | 0.0407 | 4, 0, 7 | 0 |
| 83 651 | 1 s.f. | 80 000 | keep the size |
| 3050 | 2 s.f. | 3100 | 3, 0 | 5 → up |
| 19.97 | 3 s.f. | 20.0 | the .0 shows 3 s.f. |
Example Write 0.004 708 3 correct to 3 s.f.
Start at the 4; keep 4, 7, 0; decider 8 → up.
- 0.00470 | 83
Answer 0.00471
Try it yourself
Show answer
Answer 6.08
Show answer
Answer 810
Show answer
Answer 0.0007
Ways they test it
Type 1 Round one number several ways
"Write 7 465 123 (a) to the nearest thousand (b) to 3 s.f." (1 mark each)
Spot it the same number, different instructions.
- Find the kept digit for each.
- Decider, round, keep the size.
For example 7 465 123
- nearest 1000: 7 465 000
- 3 s.f.: 7 470 000
Answer 7 465 000; 7 470 000
Type 2 Tiny numbers to s.f.
"Write 0.085 49 correct to 3 s.f."
Spot it leading zeros.
- Start counting at the first non-zero digit.
For example 0.085 49
- 8, 5, 4 | 9 → up
Answer 0.0855
Type 3 Carries and zeros
"Write 199.6 to 3 s.f." / "0.002 995 1 to 2 s.f."
Spot it 9s before the decider.
- Carry.
- Show the zeros that are significant.
For example 0.002 995 1 to 2 s.f.
- 2, 9 | 9 → up → 30
Answer 0.0030
Type 4 Calculate, then round
"Calculate … Give your answer correct to 2 s.f."
Spot it a calculation with an accuracy.
- Write the full display (mark).
- Then round (mark).
Mistakes to avoid
- ✗ 0.004 708 to 3 s.f. = 0.005✓ 0.00471Leading zeros don't count.
- ✗ 83 651 to 1 s.f. = 8✓ 80 000Keep the size.
- ✗ 2.6963 to 2 d.p. = 2.7✓ 2.70Show both decimal places.
- ✗ 2.4501 → 2.451 → 2.45 → 2.5 (rounding in steps)✓ round once, from the originalNever round twice.
Summary
- Look at the NEXT digit only: 5 or more rounds up.
- d.p. counts from the point; s.f. from the first non-zero digit.
- Keep the size with zeros; keep accuracy zeros (2.70, 20.0).
- Round once, from the original number.
If you see …
- leading zerosskip them when counting s.f.
- a 9 being rounded upcarry.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 50
Show answer
Answer 150
Show answer
Answer 6000
Medium
Guide me
What is it asking? 2 s.f.
What do we know? start at the 3
Plan 3, 0 | 9 → up
Show working
- 0.030 | 96 → 0.031
Answer 0.031
Guide me
What is it asking? 2 d.p.
What do we know? 8.99 | 7
Plan round up and carry twice
Show working
- 8.99 + 0.01 = 9.00
Answer 9.00
Guide me
What is it asking? 3 s.f.
What do we know? 6, 0, 2 | 5
Plan 5 → up
Show working
- 6.02 | 5 → 6.03
Answer 6.03
Hard
Guide me
What is it asking? 3 s.f.
What do we know? 7, 4, 6 | 5
Plan round up, keep the size
Show working
- 746 | 5123 → 747, then zeros
Answer 7 470 000
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 7 470 000 | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? 2 s.f.
What do we know? 2, 9 | 9
Plan round up, carry, show the zero
Show working
- 0.0029 | 951 → 0.0030
Answer 0.0030
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 0.0030 | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? 2 d.p.
What do we know? 15.99 | 5
Plan round up, carry all the way
Show working
- 15.99 + 0.01 = 16.00
Answer 16.00
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 16.00 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can round to the nearest 10, 100, 1000 or million and to decimal places.
- I can round to significant figures, including numbers with zeros.
- I can handle carries through 9s and keep the right zeros.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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