Significant Figures and Decimal Places: The Difference

The two things being counted

Decimal places are counted from a fixed landmark: the decimal point. Two decimal places means two digits to the right of it, whatever those digits are and however large the number is. It measures precision in absolute terms - to the nearest hundredth - and a hundredth is the same size of step whether the number is 3 or 3,000.

Significant figures are counted from a moving landmark: the first non-zero digit. This measures precision relative to the number itself, which is why it survives a change of units. A length of 0.0432 m and one of 43.2 mm are the same measurement stated to 3 significant figures. To 4 decimal places they are not remotely comparable, because the decimal point has moved and the figures have not.

• 0.02397 - the significant figures are 2, 3, 9 and 7. The two zeros hold the place and do not count.
• 4562 - the significant figures are 4, 5, 6 and 2.
• 0.0240 - four digits written after the point, three of them significant: the final zero counts because it was written on purpose.

One number, rounded both ways

The table below rounds two numbers to each accuracy in turn. Reading down the columns is more instructive than reading across.

Two things are visible in it. The first is that the two methods sometimes agree by accident - 47.4 is both 1 decimal place and 3 significant figures - which is why a student who has been getting right answers may still not know which method they used, and will be caught out by the next number.

The second is what happens at the small end. Rounding 0.02397 to 2 decimal places gives 0.02 and throws away almost the whole measurement, while 2 significant figures gives 0.024 and keeps it. An answer of 0.00 is the clearest possible signal that the wrong instrument was picked up: the arithmetic is correct, and every piece of information has been removed.

• 0.02397 → 1 dp gives 0.0, and 1 sf gives 0.02
• 0.02397 → 2 dp gives 0.02, and 2 sf gives 0.024
• 0.02397 → 3 dp gives 0.024, and 3 sf gives 0.0240
• 47.362 → 1 dp gives 47.4, and 1 sf gives 50
• 47.362 → 2 dp gives 47.36, and 2 sf gives 47
• 47.362 → 3 dp gives 47.362, and 3 sf gives 47.4

The zeros, which is where the errors are

Almost every mistake in this topic is a mistake about a zero. The four rules below settle all of them.

The rounding error that actually costs marks is dropping a place-holder. 4562 to 2 significant figures is 4600, not 46. Those zeros are doing a job - holding the 4 and the 6 in the thousands and hundreds columns - and removing them changes the number by a factor of a hundred.

The opposite error is dropping a zero created by a carry. 9.97 to 1 decimal place is 10.0, not 10. Writing 10 claims an accuracy to the nearest whole number, which is not what was asked for, and the trailing zero is the only thing on the page that says otherwise.

• Leading zeros are never significant: 0.00456 has 3 significant figures.
• Zeros between significant digits always are: 4.06 has 3, and 1002 has 4.
• Trailing zeros after the decimal point are significant, because nobody writes them by accident: 0.4500 has 4.
• Trailing zeros in a whole number are ambiguous: 4500 might be 2, 3 or 4 significant figures, and only the context or standard form settles it - 4.5 × 10³ is unambiguously 2.

How to tell which one the question wants

Read the instruction and do what it says: to 2 decimal places, to 3 significant figures, to the nearest whole number, to 1 decimal place. When one of those appears there is nothing to decide, and choosing the other one is a wasted mark on a question you could otherwise do.

Where the question says nothing, context decides. Money goes to 2 decimal places. Angles in degrees conventionally go to 1 decimal place. Lengths and areas that come out of trigonometry or Pythagoras conventionally go to 3 significant figures, and Cambridge IGCSE papers state that requirement in the rubric for any answer that is not exact.

One habit removes most of the risk: write the unrounded value on the page, then round it on the next line. A reader seeing 11.1202 followed by 11.1 can see both the method and the rounding, and if the rounding is wrong the working above it is still there. A reader seeing 11.1 alone can see neither.