Calculate complex mathematical expressions and automatically apply significant figures rules. Perfect for scientific calculations, laboratory work, and educational purposes.
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Divide the numerator by the denominator:
Leading zeros (0.00234) are placeholders, not significant figures.
1.23e4 has 3 sig figs, not 5. The exponent doesn't add digits.
Keep extra digits during calculation, round only the final result.
2.3 + 4.567 = 6.9
1.23 * 4.5 = 5.5
sqrt(16.0) = 4.00
log(100) = 2.0
(2.5 + 3.7) / 2.0 = 3.1
1.5e3 * 2.0e-2 = 3.0e1
Significant figures (sig figs) are the meaningful digits in a number that indicate the precision of a measurement or calculation result.
I lost 3 points on a chemistry lab report because I wrote 2.50 instead of 2.5. My professor circled it and wrote "sig figs matter." I had no idea what she meant. The numbers are the same, right?
They're not. 2.5 has two significant figures. 2.50 has three. That trailing zero claims you measured to the hundredths place — that your instrument was precise enough to distinguish 2.50 from 2.49 or 2.51. If your thermometer only reads to the tenths, writing 2.50 is a lie about your precision.
Significant figures aren't about math. They're about honesty. Every digit you write is a claim about how precisely you measured something.
Counting sig figs confuses people because zeros play by different rules depending on where they sit. Here's the complete ruleset:
| Rule | Example | Sig Figs | Why |
|---|---|---|---|
| All non-zero digits count | 1234 | 4 | Always |
| Zeros between non-zeros count | 1002 | 4 | "Captive" zeros are significant |
| Leading zeros don't count | 0.0050 | 2 | They're just placeholders |
| Trailing zeros after decimal count | 2.50 | 3 | They claim measured precision |
| Trailing zeros without decimal are ambiguous | 1500 | 2, 3, or 4? | Could be exact or rounded |
That last row — trailing zeros without a decimal point — is the source of 90% of sig fig arguments. Is 1500 measured to the nearest unit (4 sig figs) or the nearest hundred (2 sig figs)? Without context, you can't tell. That's why scientific notation exists: is unambiguously 4 sig figs. is 2.
How many significant figures in 0.0050? Two. The leading zeros (0.00) are just placeholders — they tell you the decimal position, not the precision. The 5 and the trailing 0 are the significant digits.
Think of it this way: 0.0050 grams is the same as 5.0 milligrams. Changing units doesn't change precision. Both have 2 sig figs. The leading zeros in 0.0050 are an artifact of the unit choice, not a measurement.
Quick test: write the number in scientific notation. Whatever digits remain are your significant figures. 0.0050 → → 2 sig figs. 100.0 → → 4 sig figs.
Multiplication and division: your answer gets the same number of sig figs as the input with the fewest sig figs.
Example: 4.56 × 1.4 = 6.384 on your calculator. But 4.56 has 3 sig figs and 1.4 has 2. Your answer gets 2: 6.4.
Addition and subtraction: your answer gets the same number of decimal places as the input with the fewest decimal places.
Example: 12.11 + 18.0 + 1.013 = 31.123 on your calculator. But 18.0 has only 1 decimal place. Your answer: 31.1.
Different rules for different operations. That's the part that catches people. Multiplication counts sig figs; addition counts decimal places. Mix them up and your density calculation or log problem will have the wrong precision.
The Mars Climate Orbiter crashed in 1999 because one team used pounds of force and another used newtons. That's a unit error, not a sig fig error — but the underlying principle is the same: precision and accuracy in numbers have real consequences.
In engineering, reporting a measurement as 2.50 mm when your caliper only reads to 0.1 mm implies false precision. A machinist might try to hit that tolerance and waste time chasing a phantom hundredth of a millimeter. In pharmaceutical dosing, the difference between 0.5 mg and 0.50 mg signals different levels of measurement confidence.
Sig figs aren't pedantry. They're a communication system. Every digit you write tells the reader how much you actually know.
Rounding to a set number of significant figures trips people up in three specific places: when the digit after the cut-off is exactly 5, when leading zeros don't count, and when you have to pad a whole number with zeros that aren't significant. One example of each.
Count from the first non-zero digit: 4 (1st), 5 (2nd), 5 (3rd), 1 (4th). The cut-off falls right after that 1.
The next digit is 4, which is below 5, so the 4th digit stays as it is — it does not round up. Everything after the cut-off is dropped.
Answer: 45.51
Common mistake: writing 45.52 by looking at the 7 at the end. Only the single digit immediately after the cut-off decides the rounding — 45.5147 → look at the 4, not the 47.
Leading zeros are never significant, so counting starts at the 6: 6 (1st), 8 (2nd). The cut-off is after the 8, and the next digit is 2 — below 5, so no round-up.
Answer: 0.0068
The three zeros stay because they hold the decimal place — they just don't count toward the two significant figures.
Counting from the left: 3 (1st), 6 (2nd). The next digit is 4, below 5, so the 6 stays. But you can't just write "36" — the number has to keep its magnitude, so the remaining places are filled with zeros.
Answer: 3600 (or 3.6 × 10³ to make it unambiguous)
Those two trailing zeros are placeholders, not significant digits. Scientific notation is the only way to write this without ambiguity, which is why lab reports prefer it.
Ambiguous without more context. It could be 2 (if measured to the nearest hundred), 3 (nearest ten), or 4 (exact count). To remove ambiguity, use scientific notation: 1.5 × 10³ (2 sig figs), 1.50 × 10³ (3 sig figs), or 1.500 × 10³ (4 sig figs). Adding a decimal point (1500.) also indicates 4 sig figs in some conventions.
No. Leading zeros are placeholders that indicate the decimal position, not measurement precision. 0.0050 has 2 sig figs (the 5 and the trailing 0). Converting to scientific notation makes this clear: 5.0 × 10⁻³ — only the 5 and 0 after the decimal are significant.
Count from the first non-zero digit to the desired number of sig figs, then round normally. For multiplication/division, match the fewest sig figs in your inputs. For addition/subtraction, match the fewest decimal places. When the digit to be dropped is exactly 5, round to the nearest even number (banker's rounding) to avoid systematic bias.
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