GCSE Maths is Easy What's in a number

It's not only people who are "useless" at maths that make errors. The answers to many calculations with numbers have to be approximated otherwise they get silly - such as the examination candidate who wrote:

What they meant to say is that 18 pupils will get 60% or above in their exam (and the other 0.456763788 of a pupil will be scraped up off the floor and placed in a clinical waste bag).

If a number has to be approximated then we say that it's got a percentage error where the % error is calculated by:

Don't think of the error as being something which is necessarily wrong - think of it more as the degree of accuracy of a number.

So, in the case of the silly student, the percentage error can be calculated as:

The error = The accurate value - The approximate value
= 18.456763788 - 18
= 0.456763788

Stop it now, because this is getting silly. The answer above is no good to anyone, it's nonsense to try and give a percentage error to 20 zillion places of accuracy.

Instead we need to shorten our answer to something more tidy and easier to understand (unless of course you are a computer controlled robot from planet Zorg).

There are 2 ways of tidying up your answers: one is to give the answer to e.g. 3 significant figures and the other is to give the answer to e.g. 2 decimal places.

These are not as you might imagine, captains of industry, senior politicians, teachers etc.

In exam questions you will often be asked to leave your answer in a certain way.

eg. 'give your answer to 3 significant figures' (this is sometimes shortened to 3 s.f. or 3 sig. fig.)

EXAMPLE:- The actual value of p is 3.1415927 Put this value to 3 significant figures.

All this means is:- You only count the FIRST 3 NON-ZERO digits in the number

You decide if you need to 'round up' or 'round down'
3.1415927
| | |
These are the first 3 NON-ZERO digits in this number, so forget the rest. The only thing left to decide is whether the answer to 3 significant figures should be 3.14 or 3.15.

The original number is obviously closer to 3.14 so that's our answer.

EXAMPLE:- The mass of an atom is 0.000547612mg. Give this mass to 3 sig fig.

Same again here-

0.000547612
          | | |
          These are the first NON ZERO digits in this number, so forget the rest. Now you just need to           decide whether it's 0.000547 or 0.000548 You should be able to see that 0.000548 is closer to
          the original number.

DECIMAL ROUNDING

What if you see a question which says "Write your answer to 2 decimal places" and after bouts of profuse sweating you work out your answer to be 16.879 ?

You have been told that there should only be 2 numbers after the decimal point.
So, you have to decide whether it's 16.87 or 16.88.

You should be able to see that it's 16.88

If you're ever in doubt, imagine it's a time for some athletics event, and you have to give the time to the nearest hundredth of a second (or whatever).

EXAMPLE:- Put 123.545 to 2 decimal places.

This is the only slightly tricky situation. Is it closer to 123.54 or 123.55 ?

123.545
            |
             If this number (the one that 'causes' the rounding) is 5 OR ABOVE then you ROUND UP              otherwise you ROUND DOWN.
             So the answer that we want is 123.55

EXAMPLE:- Round 0.02735 to 2 decimal places. Answer is 0.03

EXAMPLE:- Round 0.02735 to 3 decimal places. Answer is 0.027

EXAMPLE:- Round 0.02735 to 1 decimal place. Answer is 0.0

BACK TO ACCURACY

As we have seen, numbers are not always exact. Sometimes numbers are given which are just an average or correct to a certain degree of accuracy. This means the number provided would have a minimum (it can't be any less than this) and a maximum (it can't be any more than this) possible value. The possible values between minimum and maximum are called a range (to remember this think of a mountain range where you have minimum heights at the bottom of mountains and maximum heights at the top of mountains).

This is shown below:
2.4 given to 1 decimal place means:
The minimum is 2.35 (round this to 1 decimal place and you get 2.4)
The maximum is just under 2.45 (i.e. 2.4499999999...) as this rounded to 1 decimal place is also 2.4

10m given to the nearest metre means:
The minimum is 9.5m as this rounded to the nearest metre is 10m.
The maximum is anything up to (but not including ) 10.5m.

1.38 given to 3 significant figures means: The range is 1.375 up to 1.385.