Significant figures
Why every digit matters
Measure a pencil with your school ruler. The tip lands between 7.3 and 7.4 centimetres, a little closer to 7.3.
You are sure about the 7 and the 3. The next digit is a guess: maybe 7.33, maybe 7.34.
The reliable digits plus the first uncertain digit are called significant figures. So 7.34 cm has three significant figures.
Here’s another one. A stopwatch gives the period of a pendulum as 1.62 seconds. The 1 and the 6 are reliable; the 2 is uncertain. Three significant figures again.
Writing extra digits does not make a measurement better. If you write 7.3412 cm, you are claiming a precision your ruler never had.
Changing units does not change the count. Watch the decimal point slide: 2.308 cm, 23.08 mm and 0.02308 m. The same four digits stay lit every time.
The clearest way to show significant figures is scientific notation: a number between 1 and 10 times a power of ten. 0.02308 m becomes 2.308 × 10⁻² m.
The power of ten never counts as a significant figure. Next, we’ll look at the exact rules for which zeros count.
Counting, rounding and calculating
Rule one: all non-zero digits are significant. Rule two: zeros between non-zero digits are significant too. So 2.308 has four.
Rule three: zeros before the first non-zero digit are not significant. In 0.002308, watch the leading zeros fade: they only place the decimal point. Four significant figures.
Rule four: trailing zeros in a number with a decimal point are significant. 3.500 has four, and 0.06900 has four.
Rule five: trailing zeros with no decimal point are not significant. 4700 has only two. To show three, write 4.70 × 10³.
Exact numbers, like the 2 in 2πr or 5 students in a room, have unlimited significant figures.
Rounding off: if the digit you drop is more than 5, the last kept digit goes up. 2.746 becomes 2.75. If it is less than 5, nothing changes. 2.743 becomes 2.74.
If the dropped digit is exactly 5, check the digit before it. Even? Leave it: 2.745 becomes 2.74. Odd? Raise it: 2.735 becomes 2.74.
When you multiply or divide, keep as many significant figures as the least precise number. When you add or subtract, keep as many decimal places as the least precise number.
For example, 4.237 g divided by 2.51 cm³. The calculator shows 1.688047…, but you report 1.69 g cm⁻³, just three figures.
Try it: which digits count?
- Type any measurement, such as 0.0450 or 6.02 × 10²³.
- Amber boxes are significant; dashed boxes are not. The numbers under the boxes match the rules listed.
- Use the slider to round, and watch what happens when the dropped digit is exactly 5.
Try it: calculating with measurements
- Pick an example, then change the numbers.
- Multiply or divide: count significant figures. Add or subtract: count decimal places.
- Compare what the calculator shows with what you should report.
Quick check
How many significant figures are there in 0.007030 m?
- 2
- 3
- 4
- 7
Answer
C. The zeros before the 7 only place the decimal point, so they do not count. The zero between 7 and 3 counts, and the trailing zero after the decimal point counts. That gives 7, 0, 3, 0: four significant figures. In scientific notation: 7.030 × 10⁻³ m.
Adding masses
Three masses are 436.32 g, 227.2 g and 0.301 g. How should their total be reported?
- 663.821 g
- 663.82 g
- 663.8 g
- 664 g
Answer
C. The sum is 663.821 g. When adding, keep as many decimal places as the least precise number. 227.2 g has only one decimal place, so the answer is 663.8 g.
Summary
- Scientific notation:
a × 10ᵇ, 1 ≤ a < 10 - × or ÷:
keep the fewest significant figures - + or −:
keep the fewest decimal places
- Significant figures are the reliable digits plus the first uncertain digit.
- Zeros before the first non-zero digit never count. Trailing zeros count only if there is a decimal point.
- Changing units or the power of ten does not change the number of significant figures.
- When rounding off a 5, an even digit before it stays the same and an odd one goes up: 2.745 → 2.74, 2.735 → 2.74.