Enrichment · Passion Projects

Every digit counted,
and the reason why.

Type a measurement to see which digits are significant. Type a calculation and the page applies the rules for you, shows each step and rounds once, at the end. Then practice until you can do it without the page.

Open the calculatorWhat you get

The calculator

The example is a density lab: a metal cylinder weighed at 46.23 g, with its volume found by the rise of water from 17.2 mL to 22.4 mL.

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What you get

Enrichment · Passion ProjectsFree

What you finish with
The correct number of significant figures for any measurement, a calculated answer rounded the way your science class expects with the working written out, and a notebook of saved calculations you can print or hand in.
Who it is for
Students in chemistry, physics, biology and earth science from grade 8 through first-year college, homeschool families, and teachers who want a quick check for lab answers.
Format
Browser tool with four parts: read a number, calculate, practice and a notebook. Prints the notebook with the working; downloads a spreadsheet (.csv) and a file you can import on another device.
Access
Free. No account or sign-in. It runs in your web browser on a phone, tablet or computer. Everything you type stays in this browser and is saved there as you go; nothing is sent to Caveat.
Where it lives
The Passion Projects space in Caveat Enrichment.
Support
Questions, a correction or a request: team@caveatacademic.com.

How to use it

  1. Choose how your class treats a dropped 5: rounds up, or rounds to the even digit.
  2. On Read a number, type a measurement exactly as it was written, zeros included. Each digit is marked as counted or not, with the reason.
  3. On Calculate, type the whole expression. Mark counted or defined numbers as exact, and say how many zeros were measured in a number like 1200.
  4. Read the steps, then the answer. Label it with what was measured and save it to the notebook.
  5. Use Practice until your streak holds. The page checks the precision of your answer as well as its size.

Open the calculator

How the rules work

Significant figures record how well a number was measured.

Two rules, applied in order

Adding and subtracting are limited by position. The answer stops at the last decimal place that every value reaches: 12.11 + 18.0 + 1.013 is 31.123 on a calculator, but 18.0 is only known to tenths, so the sum is 31.1.

Multiplying and dividing are limited by count. The answer keeps as many significant figures as the value that has the fewest: 0.6238 × 6.6 is 4.1, because 6.6 has two.

In a longer calculation the page tracks both limits through every step but keeps the full unrounded value, and rounds once at the end. Rounding in the middle and then again at the end can move the last digit.

Exact numbers and the tie rule

Counted things and defined conversions (12 eggs, 100 cm in a metre) have no uncertainty. Mark them exact and they never limit the answer.

Textbooks differ on a dropped 5 with nothing after it. Most school courses round it up; OpenStax Chemistry and many laboratories round to the even digit so that ties do not push results upward. Choose the rule your class uses. The page shows when it mattered.

How it is checked

The arithmetic is done on whole-number fractions, never on binary decimals, so 0.1 + 0.2 is exactly 0.3 and a tie is a real tie. Before each release the engine is run against a second, independent implementation on 5,013 random calculations and must agree on every one.

end of the strip4.04.14.24.34.44.5cm4.274 and 2: read from the marks7: estimated between marks
Why the last digit is different. A ruler marked in tenths of a centimetre lets you read 4.2 with certainty and estimate one more digit between the marks. Those three digits, 4.27, are the significant figures: every digit you are sure of, plus one you estimated.

Sources

  1. Flowers, P., Neth, E. J., Robinson, W. R., Theopold, K., & Langley, R. (2019). Chemistry: Atoms First 2e, section 1.5, Measurement Uncertainty, Accuracy, and Precision. OpenStax, Rice University. Rules for counting significant figures, the addition and multiplication rules, and rounding a dropped 5 to the even digit.
  2. Taylor, B. N., & Thompson, A. (Eds.) (2008). Guide for the Use of the International System of Units (SI), NIST Special Publication 811, Appendix B.7, Rounding numbers. National Institute of Standards and Technology.
  3. Joint Committee for Guides in Metrology (2008). Evaluation of measurement data: Guide to the expression of uncertainty in measurement, JCGM 100:2008. Significant figures are a shorthand; a stated uncertainty says more.

Significant figures are a classroom shorthand for uncertainty. A laboratory report that needs real error analysis should state the uncertainty of each measurement. The example measurements are made up. Support: team@caveatacademic.com.