Surds

UK Secondary (7-11)

Calculate decimal approximations for irrational numbers

Lesson

A number is said to be irrational if you cannot write it as an exact fraction where both the numerator and denominator are integers.

You already know one irrational number : $\pi$π! Like all other irrational numbers, $\pi$π really goes on forever without repeating itself. We say therefore that it doesn't terminate, or repeat.

Another number that is famously irrational is $\sqrt{2}$√2. The square root of groups of other numbers are irrational.

First you may want to have a refresher on rounding.

Firstly, consider a third. This can be written as a fraction $\frac{1}{3}$13, and as a decimal we know it repeats itself as $0.333333$0.333333... forever. So if we want to do an exact calculation that includes $\frac{1}{3}$13, we should keep it as a fraction throughout the calculation. Or we could type $1\div3$1÷3 into a calculator, and it would show us around $8$8 or $9$9 digits on the screen. This is now an approximation. We know that $0.3333333333$0.3333333333is a good approximation of $\frac{1}{3}$13, but it has been rounded to fit on your calculator screen, so it is no longer *exact*.

$\frac{1}{3}$13of $30$30

If we rounded $\frac{1}{3}$13 to $2$2 decimal places before calculating, we would calculate:

$0.33\times30=9.9$0.33×30=9.9.

However, we know $\frac{1}{3}$13of $30$30 is actually $10$10.

So $9.9$9.9 is a good approximation, but it's not exact. If my friend owed me $\frac{1}{3}$13 of $\$30$$30, but only gave me $\$9.90$$9.90, I probably wouldn't be too upset. But I wouldn't have been given *exactly *what I was owed.

There will be times when exact answers are important, and times when an approximate answer will do.

When calculating with an irrational number, you have a choice whether you keep it in exact form, or round it. Often you will be told that you can use $\frac{22}{7}$227 or $3.14$3.14 as your value of $\pi$π. These are good approximations, but they are not exact.

During your time at school this will probably not be a choice you get to make. You may be asked to give your answer to a given amount of decimal places, or to leave your answer in terms of $\pi$π, or as a surd. However, groups of people use maths in their work and they may have a choice.

For example, engineers will often decide whether to use exact values or approximated values. This makes sense. If I asked you to cut me a piece of metal that is $\sqrt{2130}$√2130 cm long, you might have trouble measuring this. However, I could ask for a piece of metal to be cut that is $46.15$46.15 cm long, and you could do that for me, and it would be accurate enough for what I need.

Be careful! If a question asks you to calculate to a given amount of decimal places, you should keep exact values throughout your calculation and only round at the end!

Here $a=10$`a`=10, $c=45$`c`=45. Let's calculate $b$`b`.

I know $a^2+b^2=c^2$`a`2+`b`2=`c`2

Substituting in the given values

$a^2+b^2$a2+b2 |
$=$= | $c^2$c2 |

$10^2+b^2$102+b2 |
$=$= | $45^2$452 |

$100+b^2$100+b2 |
$=$= | $2025$2025 |

$b^2$b2 |
$=$= | $1925$1925 |

$b$b |
$=$= | $\sqrt{1925}$√1925 |

This answer is now in an exact form.

However, I could round it to $3$3 decimal places, in which case it will be

$b=43.875$`b`=43.875

It is important to remember to only round at the end of your workings, as you lose accuracy each time you round.

Evaluate $\sqrt{18}+\sqrt{41}$√18+√41 correct to $2$2 decimal places.

Answer this question using your calculator by typing in the full workings in one line. **Then **round the final answer.

This should be $10.6457649246$10.6457649246, which rounds to $10.65$10.65 to $2$2 decimal places.

Careful!

If you had rounded each part to two decimal places you would have calculated

$\sqrt{18}=4.2426406871$√18=4.2426406871, which rounds to $4.24$4.24

$\sqrt{41}=6.4031242374$√41=6.4031242374, which rounds to $6.40$6.40

$4.24+6.40=10.64$4.24+6.40=10.64

This answer still states $2$2 decimal places, but as you can see, it is $1$1 thousandth smaller. It's a small difference, but it will mean you are marked as incorrect!

Indicate if the following is true or false:

$\sqrt{530}$√530 is an exact value.

True

AFalse

BTrue

AFalse

BMy calculator states that $\sqrt{530}$√530 is $23.021728866$23.021728866. Is this still exact?

Yes

ANo

BYes

ANo

B