The Mathematics of Physics

Learning Objective

• Apply knowledge of significant figures to scientific calculations

Key Points

• Significant figures are any non-zero digits or trapped zeros. They do not include leading or trailing zeros.
• When going between decimal and scientific notation, maintain the same number of significant figures.
• The final answer in a multiplication or division problem should contain the same number of significant figures as the original number with the fewest significant figures.
• In addition and subtraction, the final answer should contain the same number of decimal places as the original number with the fewest number of decimal places.

Term

Significant Figures: The digits that carry meaning in a number and contribute to its precision.

Significant figures of a number are digits which contribute to the precision of that number. Numbers that do not contribute any precision and should not be counted as a significant number are:

• leading or trailing zeros (those are place holders)
• digits that are introduced by calculations that give the number more precision than the original data allows.

Rules For Determining If a Number Is Significant or Not

• All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4, and 5).
• Zeros appearing between two non-zero digits (trapped zeros) are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1, and 2.
• Leading zeros (zeros before non-zero numbers) are not significant. For example, 0.00052 has two significant figures: 5 and 2.
• Trailing zeros (zeros after non-zero numbers) in a number without a decimal are generally not significant (see below for more details). For example, 400 has only one significant figure (4). The trailing zeros do not count as significant.
• Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0, and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers. For example, if a measurement that is precise to four decimal places (0.0001) is given as 12.23, then the measurement might be understood as having only two decimal places of precision available. Stating the result as 12.2300 makes it clear that the measurement is precise to four decimal places (in this case, six significant figures).
• The number 0 has one significant figure. Therefore, any zeros after the decimal point are also significant. Example: 0.00 has three significant figures.
• Any numbers in scientific notation are considered significant. For example, 4.300 x 10^-4 has 4 significant figures.

Conventions Addressing Significant Figures

The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:

• A bar may be placed over the last significant figure, showing that any trailing zeros following this are insignificant. For example, 1300 with a bar placed over the first 0 would have three significant figures (with the bar indicating that the number is precise to the nearest ten).
• The last significant figure of a number may be underlined; for example, “2000” has two significant figures.
• A decimal point may be placed after the number. For example “100.” indicates specifically that three significant figures are meant.
• In the combination of a number and a unit of measurement the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg, it is much clearer.

When converting from decimal form to scientific notation, always maintain the same number of significant figures. For example, 0.00012 has two significant figures, therefore the correct scientific notation for this number would be 1.2 x 10-⁴.

When multiplying and dividing numbers, the number of significant figures used is determined by the original number with the smallest amount of significant figures. When adding and subtracting, the final number should be rounded to the decimal point of the least precise number.