 # 3.03 Significant figures

Lesson

There are often situations, typically involving measurement, where it is necessary and practical to round the values that have been obtained.

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.

For example,

• If $95446$95446 people attend a football game, the media may report this figure as $95000$95000. It's an easier number for the public to remember and no real meaning is lost in using it.
• If a chef wants to divide $500$500 g of pasta amongst $6$6 people, each portion should be $83.33333$83.33333... g. It is not possible to measure to this degree of accuracy, so the chef may choose instead to weigh each portion as $80$80 g, depending on the precision of the measuring device.

Significant figures

Significant figures are the digits considered to be significant in reporting a measurement, regardless of the location of the decimal point.

### Deciding which digits are significant

Before we begin rounding to a certain number of significant figures, we have to know which digits in a number are considered significant (shown below in red).

Starting from the left of the number, the first significant digit is the first non-zero digit.

1. All non-zero digits are significant. 2. Zeros that lie between non-zero digits are significant. 3. Zeros before the first non-zero digit are called 'leading zeros'.
In all cases, leading zeros are not significant. 4. Zeros at the end of a number are called 'trailing zeros'.
In a whole number, trailing zeros may or may not be significant. It will depend on the context of the question. 5. Trailing zeros in a decimal number are significant. ### Rounding using significant figures

Rounding to a certain number of significant figures follows the same rounding rules as rounding to a certain number of decimal places (see the 'Rules for rounding' panel below).

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.

#### Worked examples

##### Example 1

Write $95476$95476 to $3$3 significant figures.

Think: All digits in this number are significant because they are all non-zero.

Do: Starting from the left, the first three significant figures are $954$954, but the next digit, $7$7, is greater than $5$5, so we round up to $955$955. We replace the last two digits with zeros.

 $95476$95476 $=$= $95500$95500 ($3$3 s.f.)

Reflect: To write $95476$95476 to $2$2 significant figures, we would keep the first two significant digits $95$95. There is no need to round up, as the next digit, $4$4, is less than $5$5. We replace the remaining three digits with zeros.

 $95476$95476 $=$= $95000$95000 ($2$2 s.f.)

If we had to write $95476$95476 to $1$1 significant figure, notice that the first digit is $9$9 and because the next digit is $5$5, we need to round the $9$9 up to $10$10 and replace the remaining four digits with zeros.

 $95476$95476 $=$= $100000$100000 ($1$1 s.f.)

#### Example 2

Write $0.0003785$0.0003785 to $2$2 significant figures.

Think: Leading zeros are not significant, but we keep them in our answer.

Do: Starting from the left, the first two significant figures are $37$37. Because the next digit, $8$8, is greater than $5$5, we round up to $38$38 and remove the last $2$2 digits of the number.

 $0.0003785$0.0003785 $=$= $0.00038$0.00038 ($2$2 s.f.)

Rules for rounding
1. Locate the digit at the place where the number is to be rounded (the last significant digit).
2. Check the next digit after it.
1. 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).
2. 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).

#### Practice questions

##### Question 1

How many significant figures are there in the number $84.00$84.00?

1. One

A

Two

B

Three

C

Four

D

One

A

Two

B

Three

C

Four

D

##### Question 2

Round off $461585$461585 to three significant figures.

1. $\editable{}$

##### Question 3

Round off $0.060070047$0.060070047 to four significant figures.

### Outcomes

#### MS11-3

solves problems involving quantity measurement, including accuracy and the choice of relevant units