9.08 Significant figures
Introduction
The number of digits it takes to work out the precision of a number is referred to as the significant figure, regardless of the location of the decimal point.
Ideas
Significant figures
Let's look at measuring our height. We might measure ourselves with a measuring tape to the nearest centimetre. We may measure a height of 105 cm, which has three digits to tell us the precision of our measurement. The number of digits that tell us our level of precision is the number of significant figures. Here there are three digits and three significant figures.
We could also measure ourselves using kilometres, where our height would be 0.001\,05. We are still measuring ourselves to the nearest centimetre. As we haven't added any extra precision, the number of significant figures does not change. The leading zeros do not affect the number of significant figures because it is all about how precise the number is.
By converting our height to millimetres to get 1050 mm doesn't necessarily add to our precision. We are not sure whether this is accurate to the nearest millimetre. As such, the number of significant figures is 3 or 4.
However if we measured our height again to the nearest millimetre, and found that we were exactly 1050 mm. Now we are measuring to the nearest millimetre, thus we know that we have more precision, and so 4 significant figures.
It doesn't matter if we now convert this back to kilometres for 0.001\,050 km. We still know we are measuring to the nearest millimetre otherwise we wouldn't have bothered including that final zero. So 0.001\,050 km has 4 significant figures.
Significant figures collect the information required to have a measurement to a specific precision.
All non-zero digits are significant.
Any zero between non-zero digits are always significant.
Trailing zeros after a decimal points tells us another level of precision and therefore are significant.
Trailing zeros before a decimal point are not significant unless other information is available.
Examples
Example 1
A value has been rounded to give 54\,100.
What is the smallest number of significant figures that 54\,100 could have?
What is the largest number of significant figures that 54\,100 could have?
Example 2
How many significant figures are in 0.063?
Significant figures collect the information required to have a measurement to a specific precision.
All non-zero digits are significant.
Any zero between non-zero digits are always significant.
Trailing zeros after a decimal points tells us another level of precision and therefore are significant.
Trailing zeros before a decimal point are not significant unless other information is available.
Round to significant figures
There are often situations, typically involving measurement, where it is necessary and practical to round the values that have been obtained. For example, if asked to quantify the number of grains in this sandbox, giving the precise number is not practical, as such we would probably give the figure to 3 significant figures.
Although we are familiar with rounding values to a certain number of decimal places, rounding using significant figures can be applied to all numbers, whether or not they have a decimal point.
Examples
Example 3
Round 323\,385.794\,850 to 3 significant figures.
Example 4
A number has been rounded to two significant figures. If the rounded number is 8700, what is the largest possible integer that could have been the original number?
Rules for rounding:
Locate the digit at the place where the number is to be rounded (the last significant digit).
Check the next digit after it.
If the next digit is less than 5, we round down. This means that the last significant digit stays the same and the rest of the digits are removed, or replaced by zeros (in the case of whole numbers).
If the next digit is 5 or more, we round up. This means the last significant digit is increased by 1 and the rest of the digits are removed, or replaced by zeros (in the case of whole numbers).
Multiply and divide numbers to different significant figures
Consider that we need to find the floor area of a bathroom in square metres with the idea of tiling it. The room is 6.3 metres by 3.06 metres. We can get the area by multiplying the dimensions of the room, but we then need to determine the precision of that multiplied area. 6.3 \text{ m} \times 3.06 \text{ m}=19.278 \text{ m}^2
As we only had 2 and 3 significant figures in our measurements, the answer can only have 2 significant figures. If we only wanted to know the area of the bathroom then we would round the number to 19 \text{ m}^2. But, we want to tile the bathroom so we need to work out how many tiles we need. Suppose each tile has an area of 0.78 \text{ m}^2. We need to divide our non-rounded measurement by the area of the tile.\frac{19.278}{0.78}=24.7153846154
But we still have a lot of digits in our answer given that our measurements only had 2,\,3 and 2 significant figures. We now need to round this to 2 significant figures, to get 25 \, (2 s.f.).
To avoid any errors, we should round our final answer to the lowest number of significant figures used in the original question.
The type of rounding should always be indicated next to any value that has been rounded. For example, if a value has been rounded to 3 significant figures, we would write (3 s.f.). next to the value.
Examples
Example 5
Calculate 1.4\times 1.41 and give your answer to an appropriate number of significant figures.
To avoid any errors, we should round our final answer to the lowest number of significant figures used in the original question.
The type of rounding should always be indicated next to any value that has been rounded. For example, if a value has been rounded to 3 significant figures, we would write (3 s.f.). next to the value.