 Calculate decimal approximations for irrational numbers

Lesson

What is an irrational number?

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.

Doing calculations with exact forms vs approximations

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

Example

Question 1

$\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.

To Round or Not to Round?

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!

Example

Question 2 Here $a=10$a=10, $c=45$c=45. Let's calculate $b$b.

I know $a^2+b^2=c^2$a2+b2=c2

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

Remember only to round at the end of the workings!

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

Example

Question 3

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!

Question 4

Indicate if the following is true or false:

1. $\sqrt{530}$530 is an exact value.

True

A

False

B

True

A

False

B
2. My calculator states that $\sqrt{530}$530 is $23.021728866$23.021728866. Is this still exact?

Yes

A

No

B

Yes

A

No

B