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.

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.

**All non-zero digits are significant**.

**Zeros that lie between non-zero digits are significant**.

- Zeros before the first non-zero digit are called 'leading zeros'.

In all cases,**leading zeros are not significant**.

- 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.

**Trailing zeros in a decimal number are significant**.

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.

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.) |

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

- 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).

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

One

ATwo

BThree

CFour

DOne

ATwo

BThree

CFour

D

Round off $461585$461585 to three significant figures.

$\editable{}$

Round off $0.060070047$0.060070047 to four significant figures.

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