This lesson will look at what numbers are and how we use them as well as some buzz words like “integer”, “odd”, “even”, “multiple” and “factor”.
Integers are whole numbers. They are very useful to describe an entire thing.They can be positive, negative or 0 whole numbers.
Because numbers represent how much of something there is (known as magnitude) we can order them by how big or how small they are. For example we can order the numbers: $$ 1 , 6 , 2, 0, 3 $$ in ascending (getting bigger in size) order of how big they are: $$ 0,1,2,3,6 $$ descending means the numbers would be getting smaller in size: $$ 6,3,2,1,0 $$
Numbers can be changed by having certain operations done to them. It’s a bit like a patient being operated on in hospital; we need a surgeon (another number) and a certain tool (an operation symbol). You should already be able to do this. If you can’t visit here.
Unlike real operations (I hope), we can have many operations that appear together. So how do we choose what to do first and what not to do? Well there’s an order of operations that will help you out. A quick and easy way to remember the order of operations is BODMAS:
Bracket’s is what you do first and subtraction is what you do last. Let’s take this example: $$ 3(5 * 3 + 5) \div 2 $$
Whilst this looks really hard, if we follow the rules it’s a walk in the park:
$$ 3(5 * 3 + 5) \div 2 = 3(\color{red}{20}) \div 2 = \color{red}{3 * 20} \div 2 $$ $$ \color{red}{60} \div 2 = \color{red}{30} $$
Maths is super easy provided you obey the rules. Next up we need to study some buzz words. These words are meant to sound confusing and without a proper explanation of what they mean they are bizarre.
Firstly an even number is a number which divides by 2 to give a whole number. Let’s try it out on the numbers 5 and 4:
$$5 \div 2 = 2.5 \text{ and } 4 \div 2 = 2$$
Next an odd number is a number which isn’t even. In other words it divides by 2 to give a number which isn’t whole. 5 would be an example of an odd number.
Factors of a number are pairs of numbers we can multiply together to get the number. To find the factors of a number we can list the number and see if any other number multiplies to make it. If we can’t then we subtract 1 and try again. Let’s try and find the factors of 4:
$$ 4 * 1 = 4 $$ $$ 3 * \text{:(} $$ $$ 2 * 2 = 4 $$ $$ 1 * 4 = 4 $$
We can stop looking once a pair repeats itself. All of are factors will be the numbers listed, don’t repeat them again. It’s also important to keep in mind that the negative version of the number is also a factor (but you probably don’t have to write this in the exam). For example:
$$ 2 * 2 = 4, (-2) * (-2) = 4 $$
Multiples of a number are numbers that divide by an integer to give the original number. The multiples of a number are just it’s times table. For example the multiples of 5 would be:
$$\leftarrow -5, 0, 5, 10 \rightarrow $$ $$ $$ $$ 5 \color{red}{*-1} = -5 $$ $$ 5 \color{red}{*0} = 0 $$ $$ 5 \color{red}{*1} = 5 $$ $$ $$ $$ -5 \div 5 = \color{red}{-1}$$ $$ 0 \div 5 = \color{red}{0}$$ $$ 5 \div 5 = \color{red}{1}$$ $$ 10 \div 5 = \color{red}{2}$$
You can have infinitely many multiples (it’s a pattern that repeats forever on the number line since you can always add or subtract the number).
Common factors are factors which are common amongst all the numbers. A way to work out common factors is to write out the factors for each number and find the ones common in each list. For example, find the common factors of 12 and 30:
$$ 12: 1, 2, 3, 4, 6, 12 $$ $$ 30: 1, 2, 3, 5, 6, 10, 15 $$
Common multiples of a number are just multiples which are common amongst all the numbers instead of factors.
Prime numbers are numbers which can only be divided exactly only by 1, or itself. In other words prime numbers has only 1 pair of factors, 1 and the number itself.
The final buzz word to learn is what prime factors are. These are simply prime numbers which are also factors of the number. For example 2 and 3 are prime factors of 6:
$$ 2 * 3 = 6 $$
[BODMAS and ordering integers questions (10)]()