Too many digits
A balance gives a metal sample’s mass as 15.42 g, and a graduated cylinder gives its volume as 6.3 mL. Your calculator says the density is 2.447619048 g/mL. But the cylinder only reads to 0.1 mL, and the last digit of 6.3 is already an estimate, so the answer can’t possibly be known to nine decimal places. The measurements that go into a calculation decide how many significant figures the answer can have.
Quick review: counting significant figures
Lesson 1.1.5 defined significant figures as all the digits you know for certain plus the one estimated digit.
🔧 Math tool card · Significant FiguresSkip it if you already know this
The counting rules: nonzero digits count; zeros between nonzero digits count; leading zeros don’t count; trailing zeros after a decimal point count. Trailing zeros in a whole number are ambiguous. American textbooks often add a decimal point to show they count (“100.” has 3 significant figures), but scientific notation is clearer everywhere.
How many significant figures are in 2.050 × 10⁻³?
In scientific notation, look only at 2.050: the 2, the 0, the 5, and the trailing 0 all count, for 4.
Multiplying and dividing: count significant figures
When you multiply or divide, the answer has as many significant figures as the measurement with the fewest significant figures:
15.42 has 4 significant figures and 6.3 has only 2, so the answer keeps 2. Think of it this way: an answer can’t be more precise, relatively, than the roughest measurement that went into it.
Adding and subtracting: count decimal places
When you add or subtract, round the answer to the same decimal place as the measurement with the fewest decimal places (the least precise one):
13.6 is known only to the tenths place; its hundredths digit is unknown, so the sum is known only to the tenths place too. Notice that this rule compares decimal places, not numbers of significant figures.
Subtracting two nearly equal numbers loses significant figures: 25.67 g − 25.31 g = 0.36 g. Both measurements have 4 significant figures, but the difference has only 2.
Exact numbers don’t limit the answer
Two kinds of numbers have no uncertainty at all. These exact numbers准确数 act as if they had infinitely many significant figures, so they never limit the answer:
- Counted numbers: 3 nails, 6 apples.
- Defined relationships: 1 m = 100 cm, 1 L = 1000 mL, 1 h = 60 min, and 1 in = 2.54 cm.
If one nail has a mass of 2.453 g, three nails have a mass of 2.453 g × 3 = 7.359 g, with 4 significant figures; the 3 doesn’t count as one significant figure.
Rounding, and when to do it
Once you know which digit to keep, round normally: if the first digit you drop is less than 5, round down; if it’s 5 or more, round up.
In a multi-step calculation, don’t round the intermediate results. Carry an extra digit or two and round once, at the end. Compare two ways to work out 2.36 × 1.5 + 4.12:
- Rounding at the end: 2.36 × 1.5 = 3.54 (reliable only to the tenths place), and 3.54 + 4.12 = 7.66, which rounds to 7.7.
- Rounding too early: 3.54 becomes 3.5, and 3.5 + 4.12 = 7.62, which rounds to 7.6.
Rounding too early lets the error pile up.
If a rounded whole number ends in zero, write it in scientific notation so the significant figures are clear: 3.0 × 820 = 2460, which to 2 significant figures is .
Enter two measurements and pick an operation to see how the answer should be rounded. Try checking “exact” too:
Example 1: Calculating a density
A metal sample has a mass of 15.42 g and a volume of 6.3 mL. What is its density?
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Write the formula and substitute. Density is mass divided by volume; keep all the calculator digits for now:
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Decide how many figures to keep. For multiplication and division, follow the number with the fewest significant figures: 15.42 has 4, but 6.3 has only 2, so keep 2.
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Round. The third digit is 4, which is less than 5, so round down:
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A common mistake. Writing all of 2.4476. Extra digits don’t make the answer more precise; the volume was measured only to 0.1 mL, so the result can’t be good to 0.0001 g/mL.
Example 2: Subtract, then divide
An empty beaker has a mass of 47.92 g. With some salt added, the total is 48.37 g. The salt is dissolved to make 25.00 mL of solution. How many grams of salt are in each milliliter?
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Subtract first, using decimal places. For addition and subtraction, look at decimal places: both masses are known to 0.01 g, so the difference is too:
0.45 has only 2 significant figures (the leading zero doesn’t count).
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Then divide, using significant figures. 0.45 has 2 and 25.00 has 4, so keep 2:
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Think about it. The balance and the volumetric flask are both precise, yet the answer has only 2 significant figures, because so little salt was weighed that subtracting two close numbers lost digits. Weighing more salt, or using a more sensitive balance, would give a more precise answer.
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A common mistake. Reasoning that “47.92 and 48.37 both have 4 significant figures,” so the difference does too. Subtraction follows decimal places, not significant figures.
- 2.50 × 3.1
Multiplication: 3.1 has only 2, so 7.75 rounds to 7.8.
- 12.0 ÷ 4.000
Division: 12.0 has 3, so the answer is written 3.00.
- The total mass of 6 apples at 0.152 kg each
6 is a counted, exact number, so keep the 3 of 0.152: 0.912 kg.
- 0.0048 × 1.25
Leading zeros don't count, so 0.0048 has 2; the answer is 0.0060.
- 9.87 + 0.2
Addition: 0.2 is known only to the tenths place, so 10.07 rounds to 10.1, which has 3 significant figures.
- 25.00 − 24.9
Subtraction: 24.9 is known to the tenths place, so 0.10 rounds to 0.1, leaving just 1 significant figure.
- 1.000 × 3.0
Multiplication: 3.0 has 2, so the answer is written 3.0.
- 100.0 ÷ 2.50
2.50 has 3, so the answer is written 40.0.
- 8.314 × 298
298 has 3, so 2477.572 rounds to .
Common mistakes
Mistake 1: “Adding and subtracting use the significant-figure count too.” They use decimal places. 100.0 + 0.123 = 100.123, which rounds to the tenths place, 100.1. That’s 4 significant figures, more than the 3 in 0.123.
Mistake 2: “Round after every step.” Rounding intermediate results lets errors pile up. Carry an extra digit or two and round once, at the end.
Mistake 3: “The 3 in ‘3 nails’ has 1 significant figure, so the answer can keep only 1.” Counted numbers and defined relationships are exact and never limit the answer.
Quick check
A rectangle is 4.25 cm long and 2.1 cm wide. How should its area be reported?
Multiplication: 2.1 has only 2 significant figures, so 8.925 rounds to 8.9 cm².
How should 13.6 g + 2.05 g + 0.112 g be reported?
Addition: 13.6 is known only to the tenths place, so 15.762 rounds to 15.8 g.
Summary
- Multiplying or dividing: the answer has as many significant figures as the measurement with the fewest.
- Adding or subtracting: round to the decimal place of the least precise measurement; subtracting nearly equal numbers loses significant figures.
- Counted numbers and defined relationships are exact. Round normally, only at the end, and use scientific notation when trailing zeros would be ambiguous.
On to AP (for later)
AP calculations expect answers reported to a precision that matches the data, and lab questions often ask how precisely an instrument can be read or how many significant figures a result should have. Logarithms follow their own rule: the number of decimal places in a pH equals the number of significant figures in the concentration, which you’ll meet in lesson 2.12.3.