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Scales in space and time
Scales in space and time

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Significant figures and decimal places

When using numbers with decimal places, zeros that are after the decimal point but precede non-zero numbers are not included in the count of significant figures. Such digits are just place-holders indicating number size, and they are not giving you information about how precisely you know the value. For example, 0.0034 has two significant figures, meaning you are confident it is closer to 0.0034 than it is to 0.0033 or 0.0035.

In contrast, when dealing with zeros that fall after the decimal point and follow non-zero numbers these digits are significant, because they are giving you information about the precision. So 0.00340 has three significant figures, meaning you are confident it is closer to 0.00340 than it is to 0.00339 or 0.00341.

This is a special case that only affects decimal numbers. So, 3000 has one significant figure, but 3000.0 has five significant figures because it indicates that it has been measured to one decimal place. Another benefit of using scientific notation is that it is an efficient way to show the number of significant figures. For example, 3000 would be written in scientific notation as 3 × 103 but 3000.0 would be written as 3.0000 × 103, where it is clear that the second number has more (five) significant figures.

You might already be familiar with the concepts of precision and significant figures. A selection of questions is provided below for you to test your knowledge.

  • If you multiply 3.01 by 2.1 (3.01 × 2.1) to how many significant figures should you report the answer?

  • The answer should be reported to two significant figures because 2.1 is the least precise number and is quoted to two significant figures.

  • What is the result of multiplying 3.01 by 2.1, to the correct level of precision?

  • 3.01 × 2.1 = 6.321, which to two significant figures is 6.3 (because 6.321 is nearer to 6.300 than it is to 6.400).

  • Express 2052 × 0.033 with the appropriate precision.

  • The exact answer is 67.716, which has five significant figures. However, 2052 is significant to four figures and 0.033 is significant to two figures. The answer should therefore not be quoted to more than two significant figures either, so 67.716 = 68 to 2 sig fig.

  • Express the number 9.2499 × 103 to two significant figures.

  • 9.2499 × 103 is expressed as 9.2 × 103 to two significant figures. (The first digit after the final significant figure in question is 4, which is less than 5 so the number is rounded down to 9.2 rather than up to 9.3.)