Numbers
Types of Numbers
Integers are whole numbers (both positive and negative, including zero). So they are ..., -2, -1, 0, 1, 2, .... So a negative integer is a negative whole number, such as -3, -10 or -23. 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 √2. 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 √n , where n is positive but not a square number. E.g. √2 (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).
A factor (or divisor) of a number is a number which will divide into your number exactly. So you can divide a number by one of its factors and you won't be left with a remainder. For example, 3 is a factor of 6 because you can divide 6 by 3 and you won't be left with a remainder (you get 2).
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 × 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 × 2
and so 36 = 18 × 2 = 9 × 2 × 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 × 2 × 2 = 3 × 3 × 2 × 2.
This is the answer, because both 2 and 3 are prime numbers.
Example
a and b are prime numbers, ab3 = 54. Find the values of a and b.
So ab3 is the prime factor decomposition of 54.
54 = 2 × 27 = 2 × 3 × 9 = 2 × 3 × 3 × 3 = 2 × 33
So a = 2 and b = 3.
LCM and HCF
The least (or 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.
