Types of Numbers
Integers are whole numbers (both positive and negative, including
zero). Natural numbers are positive integers.
A rational number is
a number which can be written as a fraction where
numerator and denominator are integers (where the top and bottom of the fraction
are whole numbers). For example 1/2, 4, 1.75 (=7/4).
Irrational
numbers are numbers which cannot be written as fractions, such as pi and Ö2. In decimal form these numbers go on
forever and the same pattern of digits are not repeated.
Square
numbers are numbers which can be obtained by multiplying another number by
itself. E.g. 36 is a square number because it is 6 x 6 .
Surds are
numbers left written as Ön , where n is
positive but not a square number. E.g. Ö2 (see 'surds').
Prime numbers are numbers above 1
which cannot be divided by anything (other than 1 and itself) to give an
integer. The first 8 prime numbers are: 2, 3, 5, 7, 11, 13, 17,
19.
Real numbers are all the numbers which you will have come
across (i.e. all the rational and irrational numbers). All real numbers can be
written in decimal form (such as 3.165).
Prime Factor Decomposition
An important fact is that any number can be written as the product
(multiplication) of prime numbers in one way. For example, 20 = 5 x 2 x 2 . This
is the only way of writing 20 as the product of prime numbers. Writing a number
in this way is called prime factor decomposition.
Example
Find the prime factor decomposition of 36.
We look at 36 and try to find numbers which we can divide it by. We can see
that it divides by 2. 36 = 18 x 2
2 is a prime number, but 18 isn't. So we need to split 18 up into prime
numbers. We can also divide 18 by 2. 18 = 9 x 2 and so 36 = 18 x 2 = 9 x
2 x 2
But we haven't finished, because 9 is not a prime number. We know that 9
divides by 3. 9 = 3 x 3.
Hence 36 = 9 x 2 x 2 = 3 x 3 x 2 x 2.
This is the answer, because both 2 and 3 are prime numbers.
LCM and HCF
The lowest common multiple (LCM) of two or more numbers is the smallest
number into which they evenly divide. For example, the LCM of 2, 3, 4, 6 and 9
is 36. The highest common factor (HCF) of two or more numbers is the highest
number which will divide into them both. Therefore the HCF of 6 and 9 is 3.
Approximations
If the side of a square field is given as 90m, correct to the nearest
10m: The smallest value the actual length could be is 85m (since this is the
lowest value which, to the nearest 10m, would be rounded up to 90m). The largest
value is 95m. Using inequalities, 85£ length <95.
Sometimes you will be
asked the upper and lower bounds of the area. The area will be smallest when the
side of the square is 85m. In this case, the area will be 7725m². The largest
possible area is 9025m² (when the length of the sides are 95m).
BODMAS (/BIDMAS)
When simplifying an expression such as 3 + 4 × 5  4(3 + 2), remember to work
it out in the following order: brackets, of (/indices),
division, multiplication, addition, subtraction.
So do the thing in the brackets first, then any division, followed by
multiplication and so on. The above is: 3 + 20  4 × 5 = 3 + 20  20 = 3 .
You mustn't just work out the sum in the order that it is
written down.
Copyright © Matthew Pinkney 2003
