Approximation and Errors

Hong Kong

Stage 1 - Stage 3

Lesson

In 2013 I attended the Liverpool friendly football game at the MCG in Melbourne, Australia. The crowd size was the second largest on record ever, reaching $95446$95446. When I saw a friend of mine soon after the game, she said a newspaper article had reported $95000$95000 crowd size and asked me what the atmosphere was like.

Who is right here? Is $95000$95000 accurately reported? Well yes it is if we consider significant figures. In fact $95400$95400, and $100000$100000 are also correct depending on the level of accuracy you want to use.

Well yes, all the digits in numbers have a particular role to play. They tell us the value of the number, the size of the number or individual place values. Zeros hold the place value in numbers so that we can read and understand them.

The concept of significant figures is about representing a number with a certain level of accuracy.

Digits that are significant are:

- all nonzero digits
- zeros appearing between two nonzero digits (holding the place value for us)
- trailing zeros in a number containing a decimal point.
- all digits in a number using scientific notation ($a\times10^n$
`a`×10`n`)

Here is a demonstration of the first three points:

So let's look at the crowd size at the football game I attended.

$95446$95446

- to **1** significant figure would be $100000$100000

*(the first significant figure in the number is* $9$9*, so we are rounding to the nearest ten thousand)*

- to **2** significant figures would be $95000$95000

* (the second significant figure in the number is* $5$5*, so we are rounding to the nearest thousand)*

- to **3** significant figures would be $95400$95400

*(the third significant figure in the number is* $4$4*, so we are rounding to the nearest hundred)*

- to **4** significant figures would be $95450$95450

* (the fourth significant figure in the number is* $4$4*, so we are rounding to the nearest ten)*

- to **5** significant figures would be $95446$95446

* (the fifth significant figure in the number is* $6$6*, so we are rounding to the nearest one)*

So for numbers larger than $0$0, using significant figures is about learning how to round a number.

The width of a human hair is known to be about $0.023$0.023 cm.

Let's see what this measurement becomes if we round to significant figures:

- to **1** significant figures it would be $0.02$0.02

* (the first significant figure is* $2$2, *so we are rounding to the nearest hundredth)*

- to **2** significant figures it would be $0.023$0.023

*(the second significant figure is* $3$3, *so we are rounding to the nearest thousandth. See how in this case the leading zeros do not count!)*

**Write**: the following numbers to the indicated level of significant figures

**Think**: Remember the rules to identify significant figures,

- all nonzero digits
- zeros appearing between two nonzero digits (holding the place value for us)
- trailing zeros in a number containing a decimal point.
- all digits in a number using scientific notation ($a\times10^n$
`a`×10`n`)

**Do**:

$10432$10432 to $3$3 significant figures = $10400$10400 The zero between the $1$1 and the $4$4 is counted as significant.

$1.040052$1.040052 to $3$3 significant figures = $1.04$1.04 The zero between the $1$1 and $4$4 is counted as significant.

$6.53126\times10^7$6.53126×107 to $4$4 significant figures = $6.531\times10^7$6.531×107 We only need $4$4 of the digits.

$6.00002\times10^8$6.00002×108 to $3$3 significant figures = $6.00\times10^8$6.00×108 See how we have to write the zeros here - all the digits count as significant in scientific notation, even the zeros.

How many significant figures are there in the number $108486$108486?

Three

AFour

BFive

CSix

D

Round off $461585$461585 to three significant figures.

$\editable{}$

Round off $0.006037736$0.006037736 to two significant figures.