Rounding to Significant Figures

You measure a reaction and your calculator shows 0.004050 g. Your lab manual requires the answer to three significant figures. The first instinct is to write 0.00405, but that has only two sig figs. This is where the rules for rounding significant figures stop being a classroom exercise and start being a real requirement that can lose you marks or invalidate a data point. You need the counting rules, the rounding procedure, and the traps that trip up students who know the theory but miss the details.

Not every zero counts. A zero is significant only when it sits between non-zero digits or when it appears after the decimal point and after a non-zero digit. Zeros that simply fix the decimal point and could be replaced by a power of ten are never significant. The table below shows every case.

Rule	Example	Sig Figs	Why
Non-zero digits always count	1.23	3	Every non-zero digit is significant by definition
Zeros between non-zero digits count	1002	4	The two zeros are sandwiched, so they carry information
Zeros after a decimal point and after a non-zero digit count	4.70	3	The zero after 7 says the measurement was read to that precision
Zeros before the first non-zero digit do not count	0.003	1	They only position the decimal point; 0.003 is 3 × 10⁻³
Zeros at the end of a number without a decimal point are ambiguous	1000	1 to 4	Without a decimal, you cannot tell if the zeros were measured or are placeholders

Apply the rules in order. Start from the first non-zero digit on the left and count every digit from there to the last digit on the right, using the zero rules above.4.00 has three sig figs (the two zeros after the decimal point and after a non-zero digit count). 300 has one sig fig if written without a decimal point; write it as 300. if you mean three. OpenStax Chemistry 2e uses exactly these examples in its section 1.5 treatment of sig fig counting.

For trailing zeros significant: the only way to signal that the zeros in 1 500 metres are measured is to write a decimal point: 1500. has four sig figs, 1500 has two (the zeros are ambiguous). Scientific notation solves this cleanly: 1.500 × 10³ has four sig figs, 1.5 × 10³ has two.

Identify the Last Digit and Apply the Rule

To round to a given number of significant figures, identify the last digit you want to keep and look at the digit immediately to its right. If that digit is less than 5, leave the kept digit unchanged. If it is greater than 5, increase the kept digit by one. If it is exactly 5 (or 5 followed only by zeros), apply a tie-breaking rule. NIST SP 811, the official US guide for SI measurements, states that when the digit to be discarded is exactly 5, or 5 followed only by zeros, you round to the nearest even digit. This is round half to even, also called bankers' rounding, and it matches the IEEE 754-2019 default for floating-point arithmetic.

Worked Examples and a Common Failure

Worked example: round 2.3456 to three significant figures. The first three digits are 2, 3, 4. The digit after the third is 5. Because the third digit (4) is even, you leave it alone. The result is 2.34. If the number were 2.3556, the third digit is 5, which is odd, so you round up to 2.36. If the digit after the third were 6 instead of 5, you would round up regardless: 2.346 to three sig figs is 2.35.

Common failure: rounding 1.499 to two significant figures. The first two digits are 1 and 4. The digit after them is 9. Since 9 > 5, you round the 4 up to 5, giving 1.5. Many students mistakenly stop at 1.4 because they think of the number as one and a half, but the rule looks at the single digit after the last kept digit, not at the overall magnitude.

When a measurement ends in zeros and has no decimal point, you cannot tell how many of those zeros are significant. The number 2000 could mean 2 × 10³ (one sig fig), 2.0 × 10³ (two sig figs), 2.00 × 10³ (three sig figs), or 2.000 × 10³ (four sig figs). The only way to resolve this is to write the number in scientific notation, which forces you to show exactly which digits are significant. This is the standard practice in NIST SP 811 for reporting SI measurements. If you see trailing zeros in a problem, assume they are ambiguous; your instructor likely expects you to state the ambiguity or to use the given context to decide.

Decimal places count digits after the decimal point. Significant figures count all digits from the first non-zero digit onward. The two rules produce different results when the number is small. Rounding 0.01499 to one decimal place gives 0.0. Rounding the same number to two significant figures gives 0.015. The first result has zero sig figs; the second has two. A student who confuses the two will lose the precision that the measurement actually had. OpenStax Chemistry 2e section 1.5 emphasises this distinction because mixing up the two rules is the most common error in introductory chemistry calculations.

Multiplication and Division: Fewest Significant Figures

Multiplication and division use the rule of the fewest significant figures. If you multiply 4.56 (three sig figs) by 1.2 (two sig figs), the result can have only two sig figs. 4.56 × 1.2 = 5.472, rounded to two sig figs gives 5.5. NIST SP 811 and OpenStax Chemistry 2e both state this rule.

Addition and Subtraction: Fewest Decimal Places

Addition and subtraction use the rule of the fewest decimal places. If you add 4.56 (two decimal places) and 1.2 (one decimal place), the result can have only one decimal place. 4.56 + 1.2 = 5.76, rounded to one decimal place gives 5.8. The number of sig figs in each term is irrelevant; only the decimal places matter.

The failure case: applying the multiplication rule to an addition problem. You measure 10.0 g (three sig figs) and add 0.1 g (one sig fig). The multiplication rule would suggest the answer has one sig fig, giving 10 g. The correct addition rule gives 10.1 g (one decimal place, three sig figs). The NIST SP 811 treatment of this distinction is authoritative: it warns that the two rules are not interchangeable.

Work through these. Answers follow each question.

• How many significant figures in 0.003405? Four. The leading zeros do not count; the 3, 4, 0 (between 4 and 5), and 5 do.
• Round 0.003405 to three significant figures. Keep the first three digits: 3, 4, 0. The digit after the third is 5. The third digit (0) is even, so leave it. Result: 0.00341. Write it as 3.41 × 10⁻³ to avoid ambiguity.
• Round 7.2500 to three significant figures. The first three digits are 7, 2, 5. The digit after them is 0 (it is 5 followed only by zeros). The third digit (5) is odd, so round up to 6. Result: 7.25.
• How many significant figures does 100. have? Three. The decimal point after the zeros makes them significant.
• Add 3.456 and 0.02, then round appropriately. 3.456 + 0.02 = 3.476. The term with fewer decimal places (0.02) has two. Round to two decimal places: 3.48.

No single rounding method is correct for every situation. The NIST SP 811 round-half-to-even rule is the standard for SI measurements because it prevents systematic drift when you average many rounded numbers. But your instructor, textbook, or programming language may use a different rule. The most common failure in significant figures is assuming that your calculator or spreadsheet applies the same tie-breaking rule that your exam expects. Check the method before you trust the result, especially for numbers that fall exactly on the halfway point.