Chapter 9 Decimal places, significant figures, and rounding
When making calculations, it is important that any numbers reported are communicated with the appropriate accuracy and precision. This means reporting numbers with the correct number of digits. This chapter focuses on correctly interpreting the decimal places and significant figures of a number, and correctly rounding.
9.1 Decimal places and significant figures
A higher number of digits communicates a greater level of accuracy. For example, the number 2.718 expresses a higher precision than 2.7 does. Reporting 2.718 implies that we know the value is somewhere between 2.7175 and 2.1785. But reporting 2.7 only implies that we know the value is somewhere between 2.65 and 2.75 (Sokal & Rohlf, 1995). These numbers therefore have a different number of decimal places and a different number of significant figures. Decimal places and significant figures are related, but not the same.
Decimal places are conceptually easier to understand. These are just the number of digits to the right of the decimal point. For example, 2.718 has 3 decimal places, and 2.7 has 1 decimal place.
Significant figures are a bit more challenging. These are the number of digits that you need to infer the accuracy of a value (Rahman, 1968). For example, the number 2.718 has 4 significant figures and 2.7 has 2 significant figures. This sounds straightforward, but it can get confusing when numbers start or end with zeros. For example, the number 0.045 has only 2 significant figures because the first two zeros only serve as placeholders (note that if this were a measurement of 0.045 m, then we could express the exact same value as 45 mm, so the zeros are not really necessary to indicate measurement accuracy). In contrast, the measurement 0.045000 has 5 significant figures because the last 3 zeros indicate a higher degree of accuracy than just 0.045 would (i.e., we know the value is somewhere between 0.044995 and 0.045005, not just 0.0445 and 0.0455). Lastly, the measurement 4500 has only 2 significant figures because the last 2 zeros are only serving as a placeholder to indicate magnitude, not accuracy (if we wanted to represent 4500 with 4 significant figures, we could use scientific notation and express it as \(4.500 \times 10^3\)).
Table 9.1 shows some examples of numbers, their decimal places, and their significant figures.
| Number | Decimal Places | Significant Figures |
|---|---|---|
| 3.14159 | 5 | 6 |
| 0.0333 | 4 | 3 |
| 1250 | 0 | 3 |
| 50000.0 | 1 | 6 |
| 0.12 | 2 | 2 |
| 1000000 | 0 | 1 |
It is a good idea to double-check that the values in these tables make sense. Make sure you are confident that you can report numbers to a given number of decimal places or significant figures.
9.2 Rounding
Often if you want to report a number to a specific number of decimals or significant figures, you will need to round the number. Rounding reduces the number of significant figures in a number, which might be necessary if a number that we calculate has more significant figures than we are justified in expressing. There are different rules for rounding numbers, but this book will follow Sokal & Rohlf (1995). When rounding to the nearest decimal, the last decimal written should not be changed if the number that immediately follows is 0, 1, 2, 3, or 4. If the number that immediately follows is 5, 6, 7, 8, or 9, then the last decimal written should be increased by 1.
For example, if we wanted to round the number 3.141593 to 2 significant figures, then we would write it as 3.1 because the digit that immediately follows (i.e., the third digit) is 4. If we wanted to round the number to 5 significant figures, then we would write it as 3.1416 because the digit that immediately follows is 9. And if we wanted to round 3.141593 to 4 significant figures, then we would write it as 3.142 because the digit that immediately follows is 5. Note that this does not just apply for decimals. If we wanted to round 1253 to 3 significant figures, then we would round by writing it as 1250.
Table 9.2 shows some examples of numbers rounded to a given significant figure.
| Original Number | Significant Figures | Rounded Number |
|---|---|---|
| 23.2439 | 4 | 23.24 |
| 10.235 | 4 | 10.24 |
| 102.39 | 2 | 100 |
| 5.3955 | 3 | 5.40 |
| 37.449 | 3 | 37.4 |
| 0.00345 | 2 | 0.0035 |