How many significant figures should my answer have?
Updated
The short answer
When you multiply or divide, round to the fewest significant figures among your measurements, and when you add or subtract, round to the fewest decimal places. The common slips are copying every digit the calculator shows and using the multiplication rule on a sum.
What it looks like
Here are two lines from a lab notebook, each rounded the wrong way and the right way.
| Calculation | Wrong | Right |
|---|---|---|
| 12.47 g ÷ 5.2 mL | 2.398 g/mL | 2.4 g/mL |
| 12.47 g + 5.2 g | 18 g | 17.7 g |
The calculator shows 2.398076923 for the first line. The volume, 5.2 mL, has two significant figures, so the density gets two: 2.4 g/mL. In the second line the rule changes. Adding cares about decimal places, and 5.2 g has one, so the sum is 17.7 g. Rounding the sum to two significant figures, as if it were a product, throws away the tenths digit, which both measurements pin down.
These are the rounding rules in OpenStax Chemistry 2e. Chemistry is one of the four subjects in alai, an iPad app that reads your handwriting as you work.
Why it happens
A calculator shows every digit it can, and cutting digits can feel like throwing information away. The extra digits claim a precision the measurements never had. A volume read as 5.2 mL is uncertain in its last digit, so a density to the thousandths place claims more than you measured.
The two rules look alike and track different things. Multiplying and dividing care about how many significant figures each number has. Adding and subtracting care about where each number's last digit sits.
Two more things cause trouble. Rounding in the middle of a problem lets small errors pile up, so keep a digit or two extra until the last line and round once. And counted numbers and defined conversions, like the 1000 in 1 km = 1000 m, are exact, so they never limit your answer.
How to catch it yourself
Before you round, find the number that sets the limit. For multiplying and dividing, ask:
Which measurement in your calculation has the fewest significant figures?
For adding and subtracting, look for the fewest decimal places instead.
Try these
- 4.56 cm × 1.4 cm
- 25.1 mL + 3.256 mL
- 0.0250 mol ÷ 0.150 L
Answers
- The calculator gives 6.384, which rounds to 6.4 cm², because 1.4 cm has two significant figures.
- The calculator gives 28.356, which rounds to 28.4 mL, because 25.1 mL has one decimal place.
- The calculator gives 0.1666…, which rounds to 0.167 mol/L. Both numbers have three significant figures, because zeros before the first nonzero digit don't count and the final zero in 0.0250 does.
How alai helps
alai is a notes app for iPad and the web, and Homework help takes the chemistry worksheet you already have, as a photo or a PDF. Its tutor, Ally, follows each step you write and asks a question when one goes wrong, and Ally remembers slips from past sessions for next time.