Num1

Numbers In mathematics there are lots of different types of numbers. One of the simplest ways of categorising numbers is whether they are odd or even. As you continue to study GCSE Mathematics you will come across a wide range of different types of numbers. Here are some types of numbers;

1) Square Numbers: The first five square numbers are 1, 4, 9, 16 and 25. These are square numbers because they are the products when you multiply the first five numbers by themselves, for example;

(1 × 1 )

(2 × 2 )

(3 × 3)

(4 × 4)

(5 × 5 )

1

4

9

16

25

2) Cube Numbers: The first five cube numbers are 1, 8, 27, 64 and 125. These are called cube numbers because these numbers are produced by cubing (multiplying a number by itself three times) a number. This can be illustrated by drawing out cubes of various dimensions and finding their volumes or by drawing out a table like we did with the square numbers;

(1 × 1 × 1 ) ( 2 × 2 × 2 ) ( 3 × 3 × 3 ) ( 4 × 4 × 4 ) ( 5 × 5 × 5 ) 1

3 units

8

27

64

125

It is often useful to draw out and calculate the volume of a cube. In the diagram on the left you can see that 3 cubed

3 units

3 units

will you a value of 27 (3x3x3=27).

1 © Ciarán McCormick 2008

3) Triangle Numbers: The first five triangle numbers are 1, 3, 6, 10 and 15. When calculating triangle numbers it is often useful to picture snooker balls in a triangle. If you were to use the balls to form a triangle how many balls would you need and how would they look? If we were to draw out the first 5 terms it might look like this; +2 1

+3 3

+4

+5

6

10

15

4) Prime Numbers: Prime numbers are only divisible by themselves and one. The first ten prime numbers are; 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 Things to note are; •

1 is not a prime number

•

apart from 2 and 5 all prime numbers end in 1, 3, 7 or 9

•

however, not all numbers ending in these 4 digits are prime just think of 21, 33, 27 and 39.

So how do you find out if a number is prime or not? There is a very simple method to find out if a number is a prime number.

1) It must end in 1, 3, 7 or 9. 2) It won’t divide by any prime number below the value of its own square root.

Example One Using the method above decide whether or not 313 is a prime number. •

First – does it end in a 1, 3, 7 or 9 – 313 – yes so lets move on to the next part of our method

2 © Ciarán McCormick 2008

•

Second – we find the square root of 313 using a calculator (if this question appears in a non-calculator paper it is ok to estimate the square root).

313 = 17.692 •

Third – list all the prime numbers less than this square root (you can exclude 2 and 5 straight away) - 3, 7, 11, 13 and 17.

•

•

Fourth – divide all of these primes into the original number;

313 ÷ 3 = 104.33

313 ÷ 7 = 44.71

313 ÷ 13 = 24.08

313 ÷ 17 = 18.41

313 ÷ 11 = 28.45

Since none of these numbers divide into 313 without a remainder then it is a prime number.

3 © Ciarán McCormick 2008