Lesson

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.

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$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 $0.00105$0.00105 km. 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$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$3 or $4$4.

However if we measured our height again to the nearest millimetre, and found that we were exactly $1050$1050 mm. Now we are measuring to the nearest millimetre, thus we know that we have more precision, and so $4$4significant figures.

It doesn't matter if we now convert this back to kilometres for $0.001050$0.001050 km. We still know we are measuring to the nearest millimetre otherwise we wouldn't have bothered including that final zero. So $0.001050$0.001050 km has $4$4 significant figures.

Rules of 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. For example we know that this figure is exact.

How many significant places are in $0.004304830$0.004304830?

**Think:** We need to include everything that tell us something about the level of precision. There are zeros to both the left and right of the nonzero digits, and we need to work out if any of them are significant.

**Do: **Locate the first and last nonzero digit. Count all places from first to last.

Ignore zeros to the left of the first nonzero digit.

If there are zeros after the decimal point, continue to count all the zeros.

We can now count the ticks to determine there are $7$7 significant figures in $0.004304830$0.004304830.

**Reflect: ** Working out how many significant figures a number has is quantifying the precision of that number.

How many significant places are in $410403400$410403400?

**Think:** There are some zeros in the number, so we have to determine if any of them are significant.

**Do: **Locate the first and last nonzero digit. Count all places from first to last.

Ignore zeros to the left of the first nonzero digit. There are no zeros to the left, so that is easy!

If there are trailing zeros and no decimal point, then the zeros may or may not be significant.

The number of significant figures for $410403400$410403400 is $7$7.

**Reflect:** This number may have been rounded to the nearest hundred, ten, or unit. We don't know, so without context we assume that there is only $7$7. If we **knew** the answer had not been rounded, then the number of significant figures could be $9$9.

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$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.

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$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$5 or more, we round up.**

This means the last significant digit is increased by $1$1 and the rest of the digits are removed, or replaced by zeros (in the case of whole numbers).

Round $217597527$217597527 to $3$3 significant figures.

**Think:** Remember we count significant figures from the leftmost nonzero digit and round the last digit as we would a decimal number.

**Do: ** Locate the first nonzero digit and count $3$3 digits. Noting that the last of these may change when rounded.

Round the third significant figure, by considering the fourth digit. If the fourth digit is greater or equal to 5, then we need to round the third significant figure up. We then add the same number of zeros as there were other digits.

So, $217597527$217597527 rounded to $3$3 significant figures is $218000000$218000000 ($3$3 s.f.).

**Reflect:** Another way to ask this question could be ''Round $217597527$217597527 to the nearest million."

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$6.3 metres by $3.06$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$6.3 m $\times$×$3.06$3.06 m$=$=$19.278$19.278 m^{2}

As we only had $2$2 and $3$3 significant figures in our measurement, the answer can only have $2$2 significant figures! If we only wanted to know the area of the bathroom then we would round the number to $19$19 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$0.78 m^{2}. We need to divide our **non-rounded** measurement by the area of the tile.

$\frac{19.278}{0.78}=24.7153846154$19.2780.78=24.7153846154

But we still have a lot of digits in our answer for starting out with $2$2, $3$3 and $2$2 significant figures in our measurements. We now need to round this to $2$2 significant figures, to get $25$25 ($2$2 s.f.).

Careful!

We cannot acquire extra precision just because we multiply two things together. multiplying metres by centimetres doesn't tell us anything about millimetres!

To avoid this error, we must make sure we round answers to the lowest number of significant figures used in the question.

Important!

Typically, we only round the final answer in a calculation. 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$3 significant figures, we would write ($3$3 s.f.) next to the value.

Calculate the area of a room that is $8.4$8.4 m long by $3.28$3.28 m wide.

**Think:** We need to multiply the length by width to calculate the area and then round to an appropriate number of significant figures.

**Do:** Calculate the area of the room by multiplying the length of the room by the width.

$8.4$8.4 m$\times$×$3.28$3.28 m$=$=$27.552$27.552 m^{2}

The measurements are to 2 and 3 significant figures, therefore the answer must be 2 significant figures.

$8.4$8.4 m$\times$×$3.28$3.28 m$=$=$28$28 m^{2} ($2$2 s.f.)

**Reflect:** We cannot acquire extra precision just because we multiply two things together. Multiplying metres by centimetres doesn't tell us anything about the millimetres!

How many significant figures could be in $54100$54100?

Select all options that apply.

$3$3 significant figures

A$4$4 significant figures

B$5$5 significant figures

C$6$6 significant figures

D$3$3 significant figures

A$4$4 significant figures

B$5$5 significant figures

C$6$6 significant figures

D

Express the fraction $\frac{3}{49}$349 as a decimal to four significant figures.

A number has been rounded to two significant figures. If the rounded number is $8700$8700, what is the largest possible integer that could have been the original number?

Calculate $1.4\times1.41$1.4×1.41 and give your answer to an appropriate number of significant figures.