Significant Figures, Made Simple
How many digits you write is a promise about how precisely you measured. Here's how to keep that promise.
⏱️ About 15 min
A cheap kitchen scale reads 2 g; a lab balance reads 2.0000 g. Both might be weighing the same thing, but the second one is making a far bigger promise about how carefully it measured. Significant figures are how scientists write that promise honestly — no inventing precision that the instrument never had.
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The big idea: The digits you keep in a measurement — its significant figures — signal how precisely it was measured. When you calculate with measurements, the result can never be more precise than the least precise value you started with, so simple rules tell you where to round.
Which digits actually count?
Significant figures (sig figs) are the digits in a measurement that carry real information — every digit you actually measured, plus the last one you estimated. A few simple rules cover every case:
- Every non-zero digit is significant. (123 → 3 sig figs.)
- Zeros between non-zero digits count. (1002 → 4 sig figs.)
- Leading zeros never count — they only park the decimal point. (0.0025 → 2 sig figs.)
- Trailing zeros count only if there's a decimal point. (2.50 → 3 sig figs; but 250 is ambiguous — write 2.5 × 10² or 2.50 × 10² to be clear.)
🔑 The trailing-zero rule is the one people miss
A zero on the end after a decimal point is there on purpose — it says 'I measured this place and it really was zero.' So 5.0 (2 sig figs) is a stronger claim than 5 (1 sig fig), even though they're the 'same' number. Writing that extra zero is a promise; don't make it unless you measured it.
✨ Accuracy vs precision — not the same thing
Accuracy is how close you are to the true value. Precision is how close repeated measurements are to each other (and how many sig figs you can report). A scale stuck 10 g high is very precise (same reading every time) but not accurate. Sig figs describe precision, not accuracy — a very precise wrong answer is still wrong.
Rounding to a number of significant figures
To round to a given number of sig figs, count that many significant digits from the left, then look at the next digit: 5 or more rounds up, less than 5 rounds down. Example: 0.0854 to two sig figs → the third digit is 4, so round down → 0.085. To three sig figs, 12.346 → 12.3.
⚠️ Sig figs are not arbitrary rounding
Rounding to a 'nice' number and rounding to significant figures are different things. Sig figs follow from how precisely you measured — you keep exactly as many digits as your data justifies, no more and no fewer. And always round once, at the very end; rounding partway through a multi-step calculation drags errors into your final answer.
Two rules for calculations
When you combine measurements, the answer inherits the precision of your weakest input:
- Multiplication & division: the result keeps the same number of significant figures as the input with the fewest.
- Addition & subtraction: the result keeps the same number of decimal places as the input with the fewest.
\[ 100.0 \div 3.0 = 33.33\ldots \;\longrightarrow\; 33 \quad(\text{2 sig figs, set by } 3.0) \]
Division keeps the fewest significant figures: 3.0 has only 2, so the answer does too.
📝 Worked example: Add these three measured lengths and give the answer to the correct precision: 24.5 cm + 1.25 cm + 3.752 cm.
- This is addition, so use the decimal-places rule (not sig figs).
- The raw sum is 24.5 + 1.25 + 3.752 = 29.502 cm.
- Find the fewest decimal places among the inputs: 24.5 has just one decimal place.
- Round the sum to one decimal place: 29.502 → 29.5.
✓ 29.5 cm — limited to one decimal place by the least precise value, 24.5 cm.
✏️ Practice: How many significant figures are in the measurement 0.004560?
sig figs
Solution
- Leading zeros (the 0.00) never count — they only locate the decimal point.
- The significant digits are 4, 5, 6, and the trailing 0.
- That trailing zero counts because it comes after the decimal point → 4 significant figures.
✏️ Practice: A rectangle measures 4.5 cm by 2.11 cm. Give its area, rounded to the correct number of significant figures.
cm²
Solution
- Area = 4.5 × 2.11 = 9.495 cm² (raw value).
- This is multiplication, so keep the fewest significant figures: 4.5 has 2, 2.11 has 3 → answer gets 2.
- Round 9.495 to 2 sig figs → 9.5 cm².
Check your understanding
1. How many significant figures does 0.03080 have?
Leading zeros (0.0) don't count. The 3, the middle 0, the 8, and the trailing 0 all count (the last zero is after the decimal point), giving 4 significant figures.
2. A balance always reads exactly 5.00 g too high. Its readings are best described as:
It gives the same offset reading every time, so it's precise (reproducible) but not accurate (not near the true value). Precision and accuracy are different ideas — sig figs describe precision only.
3. You compute 6.0 × 2.00 = 12.00 on your calculator. How should you report it?
In multiplication the answer keeps the fewest significant figures of the inputs. 6.0 has just 2 sig figs, so the answer must have 2: report 12. Keeping extra digits would claim precision your measurement never had — sig figs aren't arbitrary rounding.
✅ Key takeaways
- Significant figures are the meaningful digits in a measurement — they signal its precision.
- Non-zeros and 'sandwiched' zeros always count; leading zeros never do; trailing zeros count only with a decimal point.
- Multiplication/division: keep the fewest significant figures. Addition/subtraction: keep the fewest decimal places.
- Accuracy (closeness to truth) and precision (reproducibility) are different — sig figs describe precision.
- Round once, at the end, to exactly the precision your data supports — not to an arbitrary 'nice' number.
➡️ Now you can measure, convert, and report a number honestly. Time to put those skills to work on a property that decides whether things float or sink: density.
Want to test yourself on this?
Try the Chemistry practice test →