Significant figures
Use and rounding of significant figures
Shutterstock / Marquisphoto / Petrol price sign, showing prices for unleaded, mid-grade and ethanol fuel. The sign is mounted on the side of the roof of the petrol station drive-through. The prices are to the nearest thousandths.
The term significant figures, or significant digits, refers to the number of decimal places needed to give the required level of accuracy or precision. The more digits in a number, the surer we can be of its accuracy.
Determining significant figures
Significant figures are all about precision. Knowing which numbers are significant can help us present calculations, and round data, more accurately. Before we come to those principles, let’s first look at the rules we should follow in determining significant figures.
All of the digits in a number, except for zero, are always significant:
235 has 3 significant figures
1.235 has 4 significant figures.
Any zeroes between nonzero digits are significant:
2035 has 4 significant figures
1.0235 has 5 significant figures.
Zeroes to the left of the first nonzero digits are not significant. They are placeholders to show the position of the decimal point:
0.0235 has 3 significant figures
0.02 has 1 significant figure.
If zeroes are placed after a decimal point and they have no nonzero numbers after them, then those zeroes are significant:
0.20 has 2 significant figures
23.500 has 5 significant figures.
For numbers ending in zeroes that are not to the right of the decimal point, the zeroes may or may not be significant:
230 may have 2 or 3 significant figures.
A measurement of 230 m has 3 significant figures if the measurement was made to the nearest unit (e.g. measured exactly 230 rather than 229 or 231).
A measurement of 230 m has 2 significant figures if measured to the nearest ten (e.g. measured as closer to 230 than 220 or 240).
When numbers are written using scientific notation, the number of significant figures is determined by the coefficient and not by the number of exponential digits.
Using significant figures in calculations
The overriding rule is that a calculated result should only be as accurate as the least accurate measurement within the calculation.
When you are adding and subtracting, the answer should have the same number of decimal places as the limiting value (the value that has the fewest decimal places).
2.3 + 2.53 = 4.83
The number in the operation with the fewest decimal places is 2.3 (1 decimal place), so the result should have only 1 decimal place = 4.8.
When you are multiplying and dividing, the number of significant figures in the answer should match the number of significant figures in the limiting value (the value that has the fewest significant figures).
The number in the operation with the fewest significant figures is 2.3 (2 significant figures), so the result should have only 2 significant figures = 5.8.
Rounding significant figures
In the calculations above, the final answers needed to be rounded to the correct number of significant digits. There are rules we adhere to in rounding numbers:
If the digit to the right of the one being retained is greater than 5, increase the retained digit by 1:
6.89 is rounded to 6.9.
If the digit to the right of the one being retained is less than 5, leave the retained digit as is:
6.81 is rounded to 6.8.
Rounding when the digit to the right of the one being retained equals 5 has special rules, to avoid bias in rounding. These rules mean that half the time the number is rounded up, and half the time it is rounded down.
If the digit to the right of the one being retained equals 5 (and there are no nonzero digits after it), round the retained digit so that it will be even:
6.85 is rounded to 6.8
6.35 is rounded to 6.4.
If the digit to the right of the one being retained equals 5 (and there are nonzero digits after it), increase the retained digit by 1:
6.45003 is rounded to 6.5.
(It is rounded up because the digits after 6.45 indicate that the number is larger than 6.45, and can’t be smaller, so it has to be rounded up.)
