1
Every non-zero digit counts. Zeros depend on where they sit.
| Kind of zero | Counts? | Example |
|---|
| between digits | yes | 107 — 3 s.f. |
|---|
| leading, before any digit | never | 0.0045 — 2 s.f. |
|---|
| trailing, after a decimal point | yes | 2.50 — 3 s.f. |
|---|
| trailing, in a whole number | ambiguous | 2500 — usually 2 s.f. |
|---|
2500
2, 5
2 s.f. by convention
2.50
2, 5, 0
3 s.f.
0.0450
4, 5, 0
3 s.f.
Why a written zero can matter
2.5 says “somewhere between 2.45 and 2.55”. 2.50 claims more: between 2.495 and
2.505. The extra zero is a statement about how carefully the number was measured,
which is why it counts.
2
Scientific notation removes the ambiguity.
| Written | Significant figures |
|---|
| 2.5 × 10³ | 2 |
|---|
| 2.50 × 10³ | 3 |
|---|
| 2.500 × 10³ | 4 — all of them deliberate |
|---|
All three equal 2500. Only the way they are written says how much of that is
trustworthy — which is exactly what significant figures record.
3
Rounding to n figures: find the nth, then look right.
0.004073
the leading zeros are not figures
figures: 4, 0, 7, 3
the 4 is the first
2nd figure is 0; the next digit is 7
7 ≥ 5, so round up
= 0.0041
2 significant figures
Keep the place value
The answer is 0.0041, not 41 or 0.41. Rounding changes how precise a number is,
never how big it is — so the zeros holding the decimal point in position all stay.
3847 to 2 s.f.
38|47 → down
3800, not 38
0.2996 to 3 s.f.
299|6 → up
0.300 — keep the zeros
7.0499 to 3 s.f.
7.04|99 → look at one digit only
7.05
4
An answer cannot be more precise than its inputs.
| Operation | Match the | Example |
|---|
| × and ÷ | fewest significant figures | 4.5 × 2.13 = 9.585 → 9.6 |
|---|
| + and − | fewest decimal places | 12.5 + 3.42 = 15.92 → 15.9 |
|---|
Measuring a room as 4.5 m by 2.13 m and reporting 9.585 m² claims millimetre
accuracy you never had. Two figures in, two figures out.
Round at the end, not at every step. Rounding partway and then continuing
lets small errors compound — carry the extra digits and cut them once.
5
Variations: a subtraction that loses figures, an exact count, and a round-up that rolls over.
Close numbers subtracted: 15.38 m − 15.2 m.
15.38 − 15.2 = 0.18
the raw difference
fewest decimal places: 1 (from 15.2)
+ and − follow decimal places
= 0.2 m
only 1 significant figure is left
What is different: both inputs have 3 or 4 figures, yet the answer has 1. Subtracting near-equal measurements wipes out figures.
An exact count: 6 identical boxes each weigh 2.35 kg. Report the total.
6 × 2.35 = 14.1
the 6 is counted, not measured
= 14.1 kg
3 s.f., from 2.35 alone
What is different: a count of whole objects is exact and does not limit the answer. Treating 6 as one figure gives 10 kg and throws away a good measurement.
A round-up that rolls over: 99.96 to 3 significant figures.
99.9|6
third figure is the 9 in the tenths
6 ≥ 5, so 99.9 → 100.0
every 9 rolls over
= 1.00 × 10²
“100” alone would read as 1 s.f.
What is different: the carry adds a digit, so the figures shift one place left. Scientific notation is the clean way to show all three.
6
One-figure estimates catch the big mistakes.
48.7 × 21.3
the real calculation
≈ 50 × 20
round each to 1 s.f.
= 1000
the size of the answer
exact: 1037.31
about 3.7% away — close enough to check with
2 s.f. · 3 s.f. · 3 s.f. · 0.0041 · about 1000
What an estimate is for
Not for choosing between 1037 and 1039 — for spotting 103.7 or 10 373. Decimal
point errors and dropped zeros are the expensive mistakes, and a five-second
estimate finds every one of them.
On the exam
Expect rounding to n significant figures (leading zeros never count, and the place value
must stay), “which has exactly three”, and a measured product or sum reported to the
right precision. Multiply or divide → fewest significant figures; add or subtract → fewest
decimal places. Decide the rule before you calculate.
✓
Check: does the rounded answer still mean the same thing?
0.0041
vs 0.004073
same size, 2 figures ✓
9.6
vs 9.585
2 figures, as the inputs allow ✓
1037.31
estimate said ~1000
right order of magnitude ✓
Remember the flow
Count from the first non-zero digit · Round once, at the end · Estimate to check the size
Leading zeros never count; trailing zeros after a point always do.
