Numbers
Definition of numbers
Sets of numbers can be described in different ways:
 natural numbers: 1, 2, 3, 4, 5, ...
 positive numbers: +1, +2, +3, +4, ...
 negative numbers: 1, 2, 3, 4, ...
 square numbers: 1, 4, 9, 16, 25, 36, ...
 triangle numbers: 1, 3, 6, 10, 15, 21, ...
Multiples
The multiples of a number are the products of the multiplication tables. e.g.  Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, ...
 Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, ...
The lowest common multiple (LCM) is the lowest
multiple which is common to all of the given
numbers.
e.g. Common multiples of 3 and 4 are 12, 24, 36, ...
The lowest common multiple is 12. Factors
The factors of a number are the natural numbers
which divide exactly into that number (i.e. without a
remainder). e.g.  Factors of 8 are 1, 2, 4 and 8.
 Factors of 12 are 1, 2, 3, 4, 6 and 12.
The highest common factor (HCF) is the highest
factor which is common to all of the given numbers. e.g.  Common factors of 8 and 12 are 1, 2 and 4.
 The highest common factor is 4.
Prime numbers
A prime number is a natural number with exactly two factors (i.e. 1 and itself).
The following numbers have exactly two factors so
are prime numbers.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
e.g. 21 can be written as 3 x 7 where 3 and 7 are
prime factors.
60 can be written as 2 x 2 x 3 x 5 where 2, 3 and
5 are prime factors.
The prime factors of a number can be found by
successively rewriting the number as a product of
prime numbers in increasing order (i.e. 2, 3, 5, 7, 11,
13, 17, ... etc.). e.g. 84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7Squares
Square numbers are numbers which have been multiplied by themselves.
e.g. The square of 8 is 8 x 8 = 64 and
64 is a square number.
Cubes
Cube numbers are numbers which have been
multiplied by themselves then multiplied by
themselves again.
e.g. The cube of 5 is 5 x 5 x 5 = 125 and
125 is a cube number. Square roots
The square root of a number such as 36 is the
number which when squared equals 36 i.e. 6
(because 6 x 6 = 36).
The sign √ is used to denote the square root. √36 = 6 Cube roots
The cube root of a number such as 27 is the number
which when cubed equals 27 i.e. 3 (because 3 x 3 x 3
= 27).
The sign ^{3}√ is used to denote the cube root. ^{3}√27 = 3 Reciprocals
The reciprocal of any number can be found by converting the number to a fraction and turning the fraction upsidedown. The reciprocal of 2/3 is 3/2 and the reciprocal of 10 is 1/10. Directed numbers
A directed number is one which has a + or – sign
attached to it.
When adding or subtracting directed numbers,
remember that signs written next to each other can
be replaced by a single sign as follows: + + is the same as + +  is the same as   + is the same as    is the same as + eg. (1) + (2) = 12 = 3 (+2)  (3) = +2+3 = +5
To multiply or divide directed numbers, include the
sign according to the following rules:  if the signs are the same, the answer is positive
 if the signs are opposite, the answer is negative
eg. (8) x (+2) = 16 (+12) / (4) = =3 (2) / (5) = +2/5 (5)^{2} = +25
Positive, negative and zero indices
When multiplying a number by itself you can use the
following shorthand. 7 x 7 = 7^{2} 7 x 7 x 7 = 7^{3} Multiplying indices
You can multiply numbers with indices as shown, by adding the powers: 7^{4} x 7^{6} = 7^{10}
Dividing indices
You can divide numbers with indices as shown, by subtracting the powers: 5^{6} / 5^{4} = 5^{2} Negative powers8^{4} / 8^{6} = 8^{46} = 8^{2} 8^{2} = 1/8^{2} in general, a^{m} = 1/a^{m} eg 3^{2} = 1/3^{2} = 1/9 2^{4} = 1/2^{4} = 1/16 Zero powers5^{0} = 1 in general, a^{0} = 1 Any number raised to the power of zero is equal to 1 Significant figures
Any number can be rounded off to a given number of
significant figures (written s.f.) using the following rules.
 Count along to the number of significant figures
required.
 Look at the next significant digit.
 If it is smaller than 5, leave the ‘significant’ digits as they are.
 If it is 5 or greater, add 1 to the last of the
‘significant’ digits.
 Restore the number to its correct size by filling
with zeros if necessary.
e.g. Round 547.36 to 4, 3, 2, 1 significant figures.
547.36 = 547.4 (4 s.f.) 547.36 = 547 (3 s.f.) 547.36 = 550 (2 s.f.) 547.36 = 500 (1 s.f.) Decimal places
Any number can be rounded to a given number of
decimal places (written d.p.) using the following rules.  Count along to the number of the decimal places
required.
 Look at the digit in the next decimal place.
 If it is smaller than 5, leave the preceding digits
(the digits before it) as they are.
 If it is 5 or greater, add 1 to the preceding
digit.
 Restore the number by replacing any numbers to
the left of the decimal point.
e.g. Round 19.3461 to 4, 3, 2, 1 decimal places.
19.3461 = 19.3461 (4 d.p.) 19.3461 = 19.346 (3 d.p.) 19.3461 = 19.35 (2 d.p.) 19.3461 = 19.3 (1 d.p.) Multiplying decimals
To multiply two decimals without using a calculator:
 ignore the decimal points and multiply the numbers
 add the number of digits after the decimal point in
the numbers in the question
 position the decimal point so that the number of
digits after the decimal point in the answer is the
same as the total number of decimal places in the
question.
e.g. Calculate 1.67 x 5.3
167 x 53 = 8851 (Ignoring the decimal points and multiplying the numbers.)
The number of digits after the decimal point in
the numbers = 2 + 1 = 3.
1.67 x 5.3 = 8.851 (Replacing the decimal point
so that the number of digits
after the decimal point in
the answer is 3.)
It is helpful to check that the answer is approximately
correct i.e. 1.67 x 5.3 is approximately 2 x 5 = 10 so
the answer of 8.851 looks correct. Dividing decimals
You can use the idea of equivalent fractions to divide
decimals.
e.g. Work out 0.00308 ÷ 0.00014 0.00308 ÷ 0.00014 = 0.00308/0.00014 = 308/14 (Multiplying top and
bottom by 100 000
to obtain an
equivalent fraction.)
Now divide 308 ÷ 14
So 0.00308 ÷ 0.00014 = 22 Estimation and approximation
It is useful to check your work by approximating your
answers to make sure that they are reasonable.
Estimation and approximation questions are popular
questions on the examination syllabus. You will
usually be required to give an estimation by rounding
numbers to 1 (or 2) significant figures. eg. Estimate the value of (6.98 x (10.16)^{2})/(9.992 x √50)
Rounding the figures to 1 significant figure and
approximating √50 as 7: (6.98 x (10.16)^{2})/(9.992 x √50) > (7 x 10^{2})/(10 x 7) = 700/70 = 10
A calculator gives an answer of 10.197 777 so that
the answer is quite a good approximation. Imperial/Metric units
In number work, it is common to be asked to convert between imperial and metric units. In
particular, the following conversions may be tested in
the examination.
Fractions
The top part of a fraction is called the numerator and the bottom part is called the denominator. Equivalent fractions
Equivalent fractions are fractions which are equal in
value to each other. The following fractions are all
equivalent to 1/2.
1/2 = 2/4 = 3/6 = 5/10 = ...
Equivalent fractions can be found by multiplying or
dividing the numerator and denominator by the same
number. One number as a fraction of another
To find one number as a fraction of another, write the
numbers in the form of a fraction.
e.g. Write 4mm as a fraction of 8cm.
First ensure that the units are the same. Remember 8cm = 80mm 4mm as a fraction of 80mm = 4/80 = 1/20 so 4mm is 1/20 of 8cm
Addition and subtraction
Before adding (or subtracting) fractions, ensure that
they have the same denominator. eg 7/8  1/5 = 35/40  8/40 (writing both fractions with a denominator of 40) = 27/40
To find the common denominator of two numbers,
find their lowest common multiple or LCM.
The LCM of 8 and 5 is 40 Multiplication of fractions
To multiply fractions, multiply the numerators and
multiply the denominators. eg. 4/7 x 2/11 = (4 x 2)/(7 x 11) = 8/77 eg. 1/1/5 x 6/2/3 = 6/5 x 20/3 = (6 x 20)/(5 x 3) = 120/15 = 8 (converting to top heavy fractions. Multiplying the numerators and multiplying the denominators) Division of fractions
To divide one fraction by another, multiply the first
fraction by the reciprocal of the second fraction. eg. 3/7 ÷ 1/7 = 3/7 x 7/1 = 3 eg. 4/4/5 ÷ 1/1/15 = 24/5 ÷ 16/15 = 24/5 x 15/16 = (3 x 3)/(1 x 2) = 9/2 = 4/1/2 Fractions to decimals
A fraction can be changed to a decimal by carrying
out the division. eg. change 3/8 to a decimal 3/8 = 3 ÷ 8 = 0.375 eg. change 4/15 to a decimal. 4/15 = 4 ÷ 15 = 0.266666...
The decimal 0.266 666 6... carries on infinitely and is
called a recurring decimal. You can write the recurring decimal 0.266 666 6... as
0.26.. The dot over the 6 means that the number
carries on infinitely. If a group of numbers carries on infinitely, 2 dots can be used to show the repeating numbers.
Decimals to fractions
A decimal can be changed to a fraction by
considering place value.
e.g. Change 0.58 to a fraction. 0.58 = 58/100 = 29/50
Percentages
Percentages are fractions with a denominator of 100. 1% means 1 out of 100 or 1/100 25% means 25 out of 100 or 25/100 = 1/4 (in lowest term)
Percentages to fractions
To change a percentage to a fraction, divide by 100. eg. Change 65% to a fraction 65% = 65/100 = 13/20 eg. Change 33/1/2% to a fraction. 33/1/2% = (33/1/2)/100 = 67/100 Fractions to percentages
To change a fraction to a percentage, multiply by
100.
e.g. Change 1/4 to a percentage. 1/4 = 1/4 x 100% = 25% Percentages to decimals
To change a percentage to a decimal, divide by 100. e.g. Change 65% to a decimal. 65% = 65 ÷ 100 = 0.65 Decimals to percentages
To change a decimal to a percentage, multiply by
100.
e.g. Change 0.005 to a percentage. 0.005 = 0.005 x 100% = 0.5%
To compare and order percentages, fractions and
decimals, convert them all to percentages. Percentage changeTo work out the percentage change, work out the
change and use the formula: where change might be increase, decrease, profit,
loss, error, etc.
Percentage of an amount
To find the percentage of an amount, find 1% of the
amount and then the required amount.
e.g. An investment of £72 increases by 12%. What is
the new amount of the investment?
1% of £72 = £72/100 12% of £72 = 12 x £0.72 = £8.64
The new amount is £72 + £8.64 = £80.64
An alternative method uses the fact that after a 12%
increase, the new amount is 100% of the original
amount + 12% of the original amount or 112% of the
original amount.
The new value of the investment is 112% of £72
1% of £72 = £0.72
112% of £72 = 112 ¥ £0.72 = £80.64
Similarly, a decrease of 12% = 100% of the original
amount – 12% of the original amount or 88% of the
original amount. Reverse percentages
To find the original amount after a percentage
change, use reverse percentages.
e.g. A television is advertised at £335.75 after a price
reduction of 15%. What was the original price?
£335.75 represents 85% of the original price
(100% – 15%)
So 85% of the original price = £335.75
1% of the original price = £335.75/85 = £3.95
85
100% of the original price = 100 x £3.95 = £395
The original price of the television was £395.
e.g. A telephone bill costs £101.05 including VAT at
17.5%. What is the cost of the bill without the VAT?
£101.05 represents 117.5% of the bill
(100% + 17.5%)
117.5% of the bill = £101.05 1% of the bill = £101.05/117.5 = £0.86
100% of the bill =100 ¥ £0.86 = £86
The telephone bill was £86 without the VAT.
You should check the answer by working the
numbers back the other way. Ratio and proportion
A ratio allows one quantity to be compared to another quantity in a similar way to fractions.
e.g. In a box there are 12 lemons and 16 oranges.
The ratio of lemons to oranges is 12 to 16,
written as 12 : 16.
The order is important in ratios as the ratio of
oranges to lemons is 16 to 12 or 16 : 12.
Equivalent ratios
Equivalent ratios are ratios which are equal to each
other. The following ratios are all equivalent to 2 : 5.
2 : 5 = 4 : 10
= 6 : 15
= 8 : 20 =...
Equal ratios can be found by multiplying or dividing
both sides of the ratio by the same number.
e.g. Express the ratio 40p to £2 in its simplest form.
You must ensure that the units are the same. (Remember £2 = 200p.)
The ratio is 40 : 200 = 1 : 5 in its simplest form. (Dividing both sides of the ratio by 40.)
e.g. Two lengths are in the ratio 4 : 5. If the first
length is 60 cm, what is the second length?
The ratio is 4 : 5 = 4 cm : 5 cm
= 1 cm : 5/4 cm (Writing as an equivalent ratio with 1 cm on the lefthand side.)
= 60 cm : 60 x 5/4 cm (Writing as an equivalent ratio with 60 cm on the lefthand side.) = 60cm : 75cm So the second length is 75 cm. Proportional parts
To share an amount into proportional parts, add up
the individual parts and divide the amount by this
number to find the value of one part.
e.g. £50 is to be divided between two sisters in the
ratio 3 : 2. How much does each get?
Number of parts = 3 + 2
= 5
Value of each part = £50 ÷ 5
= £10
The two sisters receive £30 (3 parts at £10 each)
and £20 (2 parts at £10 each).
Check that the amounts add up correctly
(i.e. £30 + £20 = £50). Standard form
Standard form is a short way of writing very large and
very small numbers. Standard form numbers are
always written as:
A x 10n where A lies between 1 and 10 and n is a natural
number. Very large numbers
e.g. Write 267 000 000 in standard form.
Write down 267 000 000 then place the decimal point so A lies between 1 and 10.
To find n, count the ‘power of 10’.
Here, n = 8 so 267000000 = 2.67 x 108 Very small numbers
e.g. Write 0.000 000 231 in standard form.
Write down 0.000 000 321 then place the decimal point so A lies between 1 and 10
To find n in 0.000 000 321, count the ‘power of
10’.
Here, n = 7 so 0.000000321 = 3.21 x 10–7 Adding and subtracting
To add (or subtract) numbers in standard form when
the powers are the same you can proceed as follows. eg. (4.8 x 10^{11}) + (3.1 x 10^{11}) = (4.8 + 3.1) x 10^{11} = 7.9 x 10^{11} eg. (4.63 x 10^{2})  (2.7 x 102) = (4.63 2.7) x 10^{2} = 1.93 x 10^{2} To add (or subtract) numbers in standard form when
the powers are not the same, convert the numbers to
ordinary form.
e.g. (8.42 x 106) + (6 x 107)

