Significant Figures

1.3 Significant Figures

Every measurement carries some error. When you report a measured value, you must communicate how precise that measurement actually is. The standard way to do this is to write down all the digits you are sure of, plus the first digit that you are uncertain about. Those digits together — the reliable ones plus the first doubtful one — are called significant digits or significant figures.

Consider the period of a simple pendulum reported as 1.62 s. The digits 1 and 6 are certain; the digit 2 is uncertain. The measured value therefore has three significant figures. Similarly, a length reported as 287.5 cm has four significant figures — the digits 2, 8, and 7 are certain, while 5 is uncertain.

The number of significant figures in a measurement depends on the least count of the instrument used. A crucial point: changing the unit does not change the number of significant figures. This principle underlies all the rules that follow.

Rules for Determining Significant Figures

Rule 1: Non-zero digits are always significant

Every digit from 1 to 9 counts as significant, regardless of where it appears.

Rule 2: Zeros between non-zero digits are always significant

Take the length 2.308 cm. The digits 2, 3, 0, and 8 are all significant — that is four significant figures. Now write the same length in different units:

• 0.02308 m
• 23.08 mm
• 23080 µm

All these numbers have the same four significant figures (2, 3, 0, 8). The location of the decimal point is irrelevant.

Rule 3: Leading zeros in numbers less than 1 are not significant

In the number 0.002308, the zeros to the right of the decimal point but to the left of the first non-zero digit (2) are not significant. Only 2, 3, 0, and 8 count — again four significant figures.

Rule 4: Trailing zeros without a decimal point are not significant

The number 123 m can be written as 12300 cm or 123000 mm. In each case, only the digits 1, 2, and 3 are significant — three significant figures. The trailing zeros are not significant because there is no decimal point to indicate that they were measured.

Rule 5: Trailing zeros with a decimal point are significant

The numbers 3.500 and 0.06900 each have four significant figures. The trailing zeros after the decimal point tell you the precision of the measurement — they were actually measured and are therefore significant.

Rule 6: Use scientific notation to remove ambiguity

The best way to avoid confusion is to report every measurement in scientific notation: $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer (positive or negative).

For 4.700 m:

The power of 10 is irrelevant to the number of significant figures. All zeros appearing in the base number $a$ are significant. Each number above has four significant figures.

Order of Magnitude

When you only need an approximate idea of a quantity, you can round the number $a$ in scientific notation to 1 (if $a \leq 5$) or to 10 (if $5 < a \leq 10$). The quantity is then approximately $10^b$, and $b$ is called the order of magnitude.

For example, the diameter of the Earth is $1.28 \times 10^7$ m. Since $1.28 \leq 5$, the order of magnitude is 7. The diameter of a hydrogen atom is $1.06 \times 10^{-10}$ m, giving an order of magnitude of $-10$. The Earth's diameter is therefore 17 orders of magnitude larger than the hydrogen atom.

Summary of Rules for Numbers Not in Scientific Notation …