Working with significant figures

Significant figures record how precisely something was measured. A ruler marked in millimetres can honestly report 4.2 cm, but not 4.23178 cm. The extra digits would claim a precision the ruler never had, so they are dropped.

This is why your chemistry and physics teachers mark them. Reporting too many digits overstates what your equipment could actually detect, which is a harder error to forgive than bad arithmetic.

The four rules for counting significant figures

Work left to right and decide which digits carry real information.

• Every non-zero digit counts. So 4527 has four.
• Zeros between non-zero digits count. So 4007 has four.
• Leading zeros never count, because they only position the decimal point. So 0.0052 has two.
• Trailing zeros count only when the number has a decimal point. So 1.20 has three, and 4.5200 has five.

How many significant figures does 1200 have?

Two, by the strict reading of rule four. There is no decimal point, so the trailing zeros are treated as placeholders rather than measurements.

This answer is genuinely unsatisfying, because 1200 might well have been measured to the nearest unit. The notation simply cannot tell you. Writing 1200. with a trailing point signals four. Writing it as 1.200 × 10³ signals four beyond any doubt, which is why scientific notation is the usual fix.

If a question hands you a number like 1200 and asks for three significant figures in the answer, take the number at face value and follow the instruction. The ambiguity is the question setter’s problem, not yours.

How to round to a set number of significant figures

Count that many significant digits from the left, then look at the next digit along. If it is 5 or more, round the last kept digit up. If it is 4 or less, leave it alone. Then keep any zeros you need to hold the decimal point in place.

Rounding 0.0045678 to three significant figures gives 0.00457, because the first significant digit is the 4 and the digit after the 5 is a 7. Rounding 12345 to three gives 12300, where the two zeros are placeholders, not significant digits.

Round once, at the end. Rounding at every intermediate step compounds the error, which is how two students with identical method get different final answers.

Significant figures in multiplication and division

The answer carries as many significant figures as the least precise number you started with. Count the sig figs in each input, find the smallest count, and round your answer to that.

So 4.56 × 1.4 = 6.384 on a calculator. The 1.4 has only two significant figures, so the answer is 6.4.

Significant figures in addition and subtraction

Here the rule changes, and this catches people out. You match decimal places, not significant figures.

So 12.11 + 1.2 = 13.31 on a calculator. The 1.2 is known only to one decimal place, so the answer is 13.3. Notice that 13.3 has three significant figures even though one input had two, which is exactly why the two rules have to be kept apart.

Exact numbers do not limit your answer

Some numbers carry infinite precision, so they never set the limit.

• Counted whole objects. Exactly 8 beakers is exact, not one significant figure.
• Defined conversions. There are exactly 100 cm in a metre and exactly 12 inches in a foot.
• Fractions in a formula, such as the 2 in the diameter of a circle.

Measured conversions are different. 1 inch = 2.54 cm is exact by definition, but 1 kg ≈ 2.205 lb is a rounded measurement and does limit your answer.

Mistakes that cost marks

• Counting leading zeros. In 0.00340 the answer is three, not five.
• Using the multiplication rule on a sum. Addition matches decimal places.
• Rounding at every step instead of once at the end.
• Dropping trailing zeros that carry meaning. If the answer is 2.50, writing 2.5 loses a significant figure.
• Treating a counted quantity as one significant figure and wrecking an otherwise correct calculation.

Other tools for science homework

• Unit converter — switch between metric and imperial before you round
• Quadratic formula calculator — solve for x and see the steps
• Percentage calculator — percentage error and percentage change