It may seem pretty obvious to be looking at numbers when studying maths, but in this section we’re going to go over some of the basics before we get into the nitty-gritty of GCSE Mathematics.
The first thing we’re going to focus on in the size of a number and how to order them. This sounds straightforward I know, but it’s not always so simple when looking at decimals, percentages and fractions.
When ordering numbers there are also symbols that help us display this easily. For example if a number is greater than, or bigger than another we would use this symbol: >.
For example 20 > 15. Simple.
Alternatively if you have a number that is less than, or smaller than another we would use the symbol: <.
For example 15 < 20.
Now let’s look at ordering numbers, we’re going to start by looking at negatives.
So the rule stands that the larger the number part of a negative number, the smaller the number actually is (so, -17 < -1).
Negative numbers are always smaller than positive numbers.
You might also be asked to order fractions in size. And the easiest way to do this is to make all the fractions have a common denominator.
Let’s take this example: The easiest way to compare the size of these fractions is to transform them into have a common denominator, in this case 40: 3/8 3/4 1/5 1/2 1/5 3/8 1/2 3/4
And that’s the basics of ordering done. But just remember that if you have to order a mix of decimals or fractions, turn them all into decimal numbers to make for easier comparison!
Next to look at are factors.
The factors of a number are any numbers that divide into it exactly – so whole numbers (integers) only please, no decimals!
Let’s look at an example:
The factors of 8 are 1, 2, 4 and 8.
For larger numbers it is sometimes easier to pi the factors by writing them as multiplications.
For example, 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6. So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Nice and straightforward I think.
I’m sure by this stage you’re very good at adding, but just in case a recap is needed, here it is:
Adding a negative number is the same as subtracting it, and subtracting a negative number is the same as adding. As always, let’s see some examples:
4 + -6 = -2 and -4 – -8 = 4.
So what about when we get decimals, and have to round long decimals to three significant figures for example. Well, the rule is quite simple, if the fourth decimal is five or above you round up, if it’s below five you round down.
Here are the examples:
Calculate to three significant figures: -0.70833. The third significant is the 8 (0.008). This is because the decider is a 3 so we don’t round up.
So far so good! Next we’re going to look at some examples of estimation which you may be asked about in the non-calculator section of an exam.
For example if you were asked to estimate: 98.7/10.23 you should approximate 98.7 to 100 and 10.23 to 10.
Therefore the sum becomes 100/10 and easy-peasy the answer is 10.
Finally, we’re going to look at upper and lower boundaries.
Remember what we said earlier about rounding up and down, well boundaries follow the same principle.
Don’t worry here comes the example to clarify:
If we’re told that a piece of string is 25cm long to the nearest cm, then the range of possible lengths depend on the upper and lower boundaries.
It must be at least 14.5cm long to round up to 15cm. But it must be less than 15.5cm, to round down to 15cm. If it was 15.5cm or more we would round up to 16cm.
Since 15.5cm is the upper limit we call this the ‘upper bound’.
So we say that 14.5cm is the lower bound and 15.5cm is the upper bound for the length of the piece of string.
And that’s the basics of numbers cracked!