The significant figures of a measurement are the digits known reliably plus one estimated digit. Four rules decide which digits count. Trailing zeros without a decimal point are ambiguous.
An ordinary ruler is marked in millimetres. Measuring a pencil with it, you might read 14.3 cm. The 1 and the 4 are certain. The 3 is an estimate between two millimetre marks; it might really be 2 or 4. Writing 14.3000 cm would claim a precision of 0.0001 cm, which the ruler cannot give.
The significant figures of a measurement are the digits that are known reliably, plus the first digit that is estimated. Their number shows the precision of the measurement.
One student reports a table’s length as 1.2 m. Another reports 1.200 m. Do the two results claim the same precision?
No. 1.2 m has two significant figures. It claims the length is known to about the nearest 0.1 m: roughly between 1.15 m and 1.25 m.
1.200 m has four significant figures. It claims the length is known to the nearest 0.001 m: roughly between 1.1995 m and 1.2005 m. That needs an instrument marked in millimetres.
The trailing zeros in 1.200 are a statement about precision. Writing them when the instrument cannot support them is an unsupported claim.
How many significant figures does 0.0042 have, and why do the leading zeros not count?
Two: the 4 and the 2.
The leading zeros only show the position of the decimal point. Written in scientific notation, 0.0042 = 4.2 × 10⁻³, and the zeros are gone. The measured information is in 4.2. A significant digit stays when a number is rewritten this way; a placeholder zero does not.
How many significant figures does 100 have, written with no decimal point? The number alone does not say. It could be a rough value known to one figure, or a careful one known to three.
The usual convention is to treat such trailing zeros as not significant. To show that they are, the precision must be stated another way:
The same digits, three different precisions. Scientific notation states the number of significant figures without ambiguity, which is why measured values are often written that way.
In scientific notation, a number is written as a value from 1 up to (but not including) 10, multiplied by a power of ten. Every digit written in the first part is significant. There are no placeholder zeros.
1.00 × 10² has three significant figures. 4.2 × 10⁻³ has two. 6.02 × 10²³ has three. Changing the unit does not change the count: 4.2 × 10⁻³ m = 4.2 mm, still two significant figures.
1The significant figures of a measurement are…
2How many significant figures are in 0.0042? Answer: ____.
3How many significant figures does each number have?
4By the usual convention, how many significant figures does 100 (no decimal point) have?
5Why are leading zeros never significant?
6Why are measured values often written in scientific notation?
Next: how many significant figures a calculated result should keep.
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