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Chapter 1: Numbers and the number system
Number system
Definition:
Natural Numbers: basic counting numbers: ie.1, 2, 3....


Integers: all positive and negative natural numbers: ie. ....-4, -3, -2, -1, 0, 1, 2, 3....
Real Numbers: numbers that are integers and all numbers in between, includes decimals:
ie. ....2.1, 2.2, 2.3....

Rational Numbers: are real numbers as a ratio of two integers: ie. 5/4 = 1 25.

Irrational Numbers: are real numbers that cannot be written as a ratio of two integers: ie.
= 2 3606...... the decimal never terminates nor repeats.

Even Numbers: integers that are divisible by 2: ie. -4, -2, 0, 2, 4, 6....
Odd Numbers: integers that are not divisible by 2: ie. -3, -1, 1, 3, 5....
Factor: is a number that will divide exactly into another number with no remainder: ie. 6 is a


5
factor of 18 as it will go into 18 exactly 3 times.

Multiple: is a number which is formed by multiplying two of more factors: ie. 18 is formed by 6
times 3.

Prime Number: is a number which has 2 factors only, the number itself and one: ie. 7 is a
prime number with only two factors; 7 and 1.

Square Number: is a number formed by multiplying a number with itself: ie. 16 is formed by 4
times.
Approximation
Irrational numbers are approximated to a required number of decimal places, note that a decimal
figure ending in 5 or above is rounded up, or if ending in a 4 or below, is rounded down. ie. 2 435
rounded to 2 decimal numbers becomes 2 44, and 2 432 rounded to 2 decimal numbers becomes 2 43.
Example: 2 35423333...
Three decimal places: 2 35423333 = 2 354
Two decimal places:
2 35423333 = 2 35
One decimal place:
Whole numbers only:
2 35423333 = 2
2 35423333 = 2 4
Significant Figures
A number may be expressed as a given number of significant figures, ie. 24545 has 5 significant
figures, starting from the most significant (far left) and counting towards the right. If this is to be given
as 3 significant figures it would read 24500.
With small numbers the first figure counted can not be a zero, therefore 0 0056 has only 2 significant
figures (the 5 and the 6).
Fractions
Definition:
A fraction is a ratio of two integers, or one integer divided by another integer. Note: the fractions below
are shown with either with a bar or a slash separating the numerator from the denominator.




Numerator: is the upper integer on top of the line 2/.
Denominator: is the lower integer underneath the line /3.
Proper Fraction: where the upper integer is smaller than the lower integer, ie. 2/3 or .
Improper Fraction: where the upper integer is greater than the lower integer, ie 3/2
or


.
Mixed Number: is where a number is made up of an integer and a proper fraction, ie.
3/2 as a mixed number is: 1 .
Common Denominator: is necessary when adding or subtracting. The fractions have to
be of the same type; have the same value denominator, done by increasing the
numerator by the same vale as the increase on the denominator.
Decimals:
Conversions: A decimal figure is a representation of a fraction. However a decimal figure is not
always as accurate. This is because rounding may be done to limit the decimal's length.
= 75:100
Example: Fraction:
0 75
. Decimal 0 75. This decimal figure is accurate.
Example: Fraction: . Decimal 0 6666 for ever. The decimal figure has to be rounded and can never
be accurate. This is a recurring decimal.
Powers: 3 where 3 is the base and is the index of exponent.
A number multiplied by itself is its square ie. 3 x 3 is the same as 3 which equals 9. A number may be
multiplied by itself more than twice, that number is the power. ie. 3x3x3x3x3x3 is the same as 3 to
power 6 or 3 which equals 729.
Negative powers operate in a similar way but the answer is a denominator with 1 over. ie 3 to power -6
or 3 whichis the same as 1 3 .
The rules for powers when adding or subtracting are different from multiplying or dividing. Care must
be taken.
Example: Addition or subtraction: 3 + 3 = 9 + 9 = 18. It is not 6 nor 3 which give completly different
answers.
Example: Multiplication: only when the bases are the same, add the powers. 3 x 3 = 3 (9 x 9 = 81).
Example: Division: only when the bases are the same, subtract the powers. 3
81).
3 = 3 (729 9 =
Roots: The opposite operation to 3 = 9, is equal to 9 = 3 is equal to 9 .
When dealing with algebraic factors and terms it is simpler to convert a root into a power.
Example:
256 is the 4th root of 256, changing to a power is 256 . 4 x 4 x 4 x 4 = 256. 4 = 256.
Therefore 256 = 4.
Ratio and Scale:
Ratio: if two numbers A and B are in the ratio 1 2 it means that A is half the size of B or B is twice the
size of A. The rules for equivalent fractions and lowest terms also apply.
Example: A ratio 400 650. Using the rules for fractions (equivalent, or lowest terms) equals: 8 13. The
first number is 8/13 of the second number, and the second number is 13/8 of the first number.
Scale: a scale is a measurement that represents another value of measurement.
Example: A scaled model is 1 10 the original size. Therefore the model is 10 times smaller or the
original is 10 times greater. Every measurement on the model is scaled to 1/10 of the original size. ie.
a section measuring 20cm long on the model would be 2m long on the full size.
Proportion:
Two quantities may increase at the same rate, they are proportional, if they increase at opposite rates
they are inversely proportional.
Example: proportional: 10 cds cost £150 how much would 20 cds cost ? Answer: 10 150 what is 20
increase by a factor of 2, giving: 20 300. Therefore 20 cds cost £300.
Example: Inversely proportional: A driver travels to work at 30mph and it takes 30 minutes. If he cycled
at 10mph how long would it take ?
Answer: 30 30 what is 10 (inversely proportional) if one is decreased by a factor of 3 then the other is
increased by a factor 3, giving 10 90. Therefore time is 90 minutes.
Percentage Conversion:
Are comparisons between numbers. A percentage is a fraction with a particular number divided by
100.
To convert a fraction to a percentage multiply by 100. ie. x 100 = 100 2 = 50%.
To convert a percentage to a fraction divide by 100. ie. 50% equals 50 100 equals or half.
Equivalent fractions may be necessary to find lowest terms.
Example: Convert
to a percent: Multiply by 100. Equals 100 4 = 25%.
Increasing / Decreasing:
To increase an amount by a certain percentage, first calculate the percentage and add it to the original
figure.
Example: Increase £21:50 by 15%. First find 15% of £21:50. 21.5 x 15 100 = 322.5 100. This gives
£3 225 (two decimal places = £3 23). Then add to the original: £21:50 + £3:23 = £24:73.
To decrease an amount by a certain percentage, first calculate the percentage and subtract it from the
original figure.
Example: Decrease £21:5 by 15%. As for increasing 15% of 21 5 was 2 225. Subtract from 21 5 = 18
275 or £18:28.
Surds:
Basic surd manipulation
Surds are numbers left in 'square root form' (or 'cube root form' etc). They are therefore irrational numbers. The
reason we leave them as surds is because in decimal form they would go on forever and so this is a very clumsy
way of writing them. Leaving them as surds is more mathematically precise.
Addition and subtraction of surds:
7 - 7 = 27.
5
Note: 52 + 33 cannot be manipulated because the surds are different (one is 2 and one is 3).
Multiplication:
5 × 15 = 75 (= 15 × 5)
 × 3
= 53.
(1 + 3) × (2 - 8)
= 2 - 8 + 2 - 24
= 2 - 22 + 23 - 26
[The brackets are expanded as usual]
Rationalising the denominator:
It is untidy to have a fraction which has a surd denominator. This can be 'tidied up' by multiplying the top and
bottom of the fraction by a surd. This is known as rationalising the denominator, since surds are irrational
numbers and so you are changing the denominator from an irrational to a rational number.
Example:
Rationalise the denominator of:
1 2
1
a)
b)
1 2
2
a) Multiply the top and bottom of the fraction by 2. The top will become 2 and the bottom will become 2 (2
times 2 = 2).
b) In situations like this, look at the bottom of the fraction (the denominator) and change the sign (in this case
change the plus into minus). Now multiply the top and bottom of the fraction by this.
Therefore:
1 + 2 = (1 + 2)(1 + 2) = 1 + 2 + 2 + 22 = 3 + 32
1 - 2
(1 - 2)(1 + 2)
1 + 2 - 2 - 2
-1
= -3(1 + 2)
Chapter 2: Equations, Formulae and Identities
Rules:
An equation is a statement or mathematical expression which says one side is equal to the other side,
eg. 12 2 = 3 + 3. Usually equations are combined with algebra and contain at least one unknown, eg.
12 a = 3 + 3, and to solve the equation the unknown value has to be determined.
The basic rule to be observed is any operation done one one side of the equals sign has to be copied
on the other side. In doing this it may be necessary to include brackets around expressions which
have to be done first in the order of operations.
Consider the example above: 12 a = 3 + 3.
Multiplying both sides by 'a' removes the denominator: 12a a = 3 + 3 x a.
The left side of the statement is now 12, but the right side of the statement looks wrong.
The order of operations states multiply before sum, but this would give an incorrect answer.
In this case the 3 + 3 has to be preserved and is bracketed: (3 + 3)a.
Now to make 'a' the subject of the formula: divide both sides by (3 + 3), leaving:
12 (3 + 3) = a, or a = 2. Check by putting into the original equation: 12 2 = 3 + 3, correct.
Without brackets it would have been 12 = 3 + 3a, minus 3 from both sides giving 9 = 3a, dividing both
sides by 3 leaves: a = 3, which when put into the original equation gives: 12 3 = 3 + 3, which is
incorrect.
Always check your result by substitution.
Simple Equations:
Example: Solve the equation, 4x - 8 = 32.
Example: Solve the equation, 6 (8x + 4) = 48.
Example: Solve the equation, 6x - 4 = 18x - 12.
Example: Solve the equation: (p - 1) p = p (p - 3).
Simultaneous Equations: A simultaneous equation is one where there is more than one
unknown and more than one equation presented. These are normally done on two unknowns and two
lines of equations and normally solved by elimination, but an example is done solving by substitution.
Solution by Elimination: To solve by elimination, one of the unknowns has to be made equal to
the other and eliminated by addition or subtracting, leaving a simple equaltion to solve for the other
unknown.
Solution by Substitution: Done by making one of the unknowns the subject of one equation.
Example:
 4x - y  8
a) 
2x  y  10
b)
3x  6y  30

 6x - 2y  18
c)
 4x  2y - 12

- 9x  7y  32
Quadratic Equations: There are different methods to solving the unknown co-efficient in a
quadratic equation. The simplest method is factorisation, as shown in Algebra. This does not always
work when the unknown is not a whole number. The unknown in a quadratic equation may then be
solved by completing the square or by using the quadratic formula. Another way of solving is by
drawing a graph of the equation.
The general form for a quadratic is: ax + bx + c = 0. where a, b and c are real numbers. A quadratic
has two solutions for (in this case) ' x '. The two solutions may be equal. The solutions are called
roots.
Solution by Formula: A formula based on the completing the square method is used. This has to
be remembered.
The expression: ax + bx + c = 0, where
' a ' is the co-efficient of the 'x ' ;
' b ' is the co-efficient of the ' x ' ; and
' c ' is the number.
Formula:
By substituting the numbers for the letters in the formula and completing the calculation will give the
solution.
When factorising the method was to find two factors of the non ' x ' number that added or subtracted to
give the co-efficient of the ' x '. When this is not possible we can find the solution by formula.
Note: in the example a dot is used to signify a multiplication.
Factorising
Factorising is the reverse of expanding brackets, so it is, for example, putting 2x² + x - 3 into the form
(2x + 3)(x - 1). This is an important way of solving quadratic equations.
The first step of factorising an expression is to 'take out' any common factors which the terms have. So
if you were asked to factorise x² + x, since x goes into both terms, you would write x(x + 1) .
Factorising Quadratics
There is no simple method of factorising a quadratic expression, but with a little practise it becomes
easier. One systematic method, however, is as follows:
Example
Factorise 12y² - 20y + 3
= 12y² - 18y - 2y + 3
[here the 20y has been split up into two numbers whose multiple is 36. 36 was
chosen because this is the product of 12 and 3, the other two numbers].
The first two terms, 12y² and -18y both divide by 6y, so 'take out' this factor of 6y.
6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y² - 18y]
Now, make the last two expressions look like the expression in the bracket:
6y(2y - 3) -1(2y - 3)
The answer is (2y - 3)(6y - 1)
Example
Factorise x² + 2x - 8
We need to split the 2x into two numbers which multiply to give -8. This has to be 4 and -2.
x² + 4x - 2x - 8
x(x + 4) - 2x - 8
x(x + 4)- 2(x + 4)
(x + 4)(x - 2)
Example: Solve by formula x + x - 6 = 0.
x  3x  2  0
x  3, 2
Inequalities
a<b
a≤b
a≥b
a>b
means a is less than b (so b is greater than a)
means a is less than or equal to b (so b is greater than or equal to a)
means a is greater than or equal to b etc.
means a is greater than b etc.
If you have an inequality, you can add or subtract numbers from each side of the inequality, as with an
equation. You can also multiply or divide by a constant. However, if you multiply or divide by a
negative number, the inequality sign is reversed.
Example
Solve 3(x + 4) < 5x + 9
3x + 12 < 5x + 9
∴ -2x < -3
∴ x > 3/2 (note: sign reversed because we divided by -2)
Inequalities can be used to describe what range of values a variable can be.
E.g. 4 ≤ x < 10, means x is greater than or equal to 4 but less than 10.
Example
On the number line below show the solution to these inequalities.
-7 ≤ 2x - 3 < 3
This can be split into the two inequalities:
-7 ≤ 2x - 3 and 2x - 3 < 3
∴ -4 ≤ 2x and 2x < 6
∴ -2 ≤ x and x < 3
The circle is filled in at -2 because the first inequality specifies that x can equal -2, whereas x is less
than (and not equal to) 3 and so the circle is not filled in at 3.
Chapter 3: Sequences, Functions and Graphs
Number Sequences
In the sequence 2, 4, 6, 8, 10... there is an obvious pattern. Such sequences can be expressed in terms of the nth term
of the sequence. In this case, the nth term = 2n. To find the 1st term, put n = 1 into the formula, to find the 4th term,
replace the n's by 4's: 4th term = 2 × 4
= 8.
Example
What is the nth term of the sequence 2, 5, 10, 17, 26... ?
To find the answer, we experiment by considering some possibilities for the nth term and seeing how far away we are:
n
=
1
2
3
4
5
n²
=
1
4
9
16
25
n²+1
=
2
5
10
17
26
This is the required sequence, so the nth term is n² + 1. There is no easy way of working out the nth term of a
sequence, other than to try different possibilities.
Tips: if the sequence is going up in threes (e.g. 3, 6, 9, 12...), there will probably be a three in the formula, etc.
In many cases, square numbers will come up, so try squaring n, as above. Also, the triangular numbers formula often
comes up. This is n(n + 1)/2 .
Example
Find the nth term of the sequence: 2, 6, 12, 20, 30...
n
= 1 2
3
4
5
n(n + 1)/2
= 1 3
6
10
15
Clearly the required sequence is double the one we have found the nth term for, therefore the nth term of the required
sequence is 2n(n+1)/2 = n(n + 1).
The Fibonacci sequence
The Fibonacci sequence is an important sequence which is as follows: 1, 1, 2, 3, 5, 8, 13, 21, ... . The next term of this
well-known sequence is found by adding together the two previous terms.
Functions
A function is a rule which indicates an operation to perform.
e.g. if f(x) = x² + 3
f(2) = 2² + 3 = 7
(i.e. replace x with 2)
Inverse Functions
The inverse function of y = 2x is y = ½x . The inverse of a function does the opposite of the function.
To find the inverse of a function, follow the following procedures: let y = f(x). Swap all y's and x's .
Rearrange to give y = . This is the inverse function.
Example:
Find the inverse of f(x), where f(x) = 3x - 7
f(x) = 3x - 7
y = 3x - 7 (let f(x) = y)
x = 3y - 7 (swap x's and y's)
y=x+7
3
Combining Functions
If f(x) = 3x + 1 and g(x) = x² + 2
f(x) + g(x) = 7
3x + 1 + x² + 2 = 7
(x - 1)(x + 4) = 0
x = 1 or –4
Composite Functions
Composite functions are combinations of two or more functions.
f。g(x) (which is the same as f [g(x)] or fg(x), means do function g first, then function f.
Example:
If f(x) = x and g(x) = 2x + 1, find f。g(x).
f。g(x) = f(2x + 1) = 2x + 1
Example
If f(x) = x2 + 1 and g(x) = x – 3, find f。g(x).
f。g(x) = (x - 3) 2+1 = x2 – 6x + 10
Graph
Co-ordinates: To find the co-ordinates of a graph the axis have to be identified. The vertical axis is
known as the 'y' axis, and the horizontal axis is known as the 'x' axis. The
position of a point on a graph is defined by its 'x' co-ordinate first, and its 'y'
co-ordinate second. The point A is at x = 3, and y = 2, or A(3,2). When there
are more than one point on a graph they are connected by a straight line or a
curved line.
Example: A Farenheit to Centigrade chart. The formula for conversion
is F = C x +32. If a few conversions are made the rest will be found
after the straight line has been drawn.
When C = 0, F = 32.
When C = 20, F = 68.
When C = 35, F = 95.
First point: (32, 0), Second point: (68, 20), Third point (95, 35).
On completion of the chart it is possible to find other conversions simply by reading any value of C.
Follow the reading to the sloping straight line. Where it hits the line read down for the corresponding
value in F, ie 10C = 50F.
Gradients
Basically, gradient is another word for "slope". The higher the gradient of a graph at a point, the
steeper the line is at that point. A negative gradient means that the line slopes downwards.
Finding the gradient of a straight-line graph
It is often useful or necessary to find out what the gradient of a graph is. For a straight-line graph, pick
two points on the graph. The gradient of the line = (change in y-coordinate)/(change in x-coordinate) .
In this graph, the gradient = (change in y-coordinate)/(change in x-coordinate) = (8-6)/(10-6) = 2/4 =
1/2
We can, of course, use this to find the equation of the line. Since the line crosses the y-axis when y = 2,
the equation of this graph is y = ½x + 2 .
Finding the gradient of a curve
To find the gradient of a curve, you must draw an accurate sketch of the curve. At the point where you
need to know the gradient, draw a tangent to the curve. A tangent is a straight line which touches the
curve at one point only. You then find the gradient of this tangent.
Example
Find the gradient of the curve y = x² at the point (3, 9).
Gradient of tangent = (change in y)/(change in x)
= (9 - 5)/ (3 - 2.3)
= 5.71
Note: this method only gives an approximate answer. The better your graph is, the closer your answer
will be to the correct answer. If your graph is perfect, you should get an answer of 6 for the above
question.
Parallel Lines
Two lines are parallel if they have the same gradent.
Example
The lines y = 2x + 1 and y = 2x + 3 are parallel, because both have a gradient of 2.
Perpendicular Lines
Two lines are perpendicular if one is at right angles to another- in other words, if the two lines cross
and the angle between the lines is 90 degrees.
If two lines are perpendicular, then their gradients will multiply together to give -1.
Example
Find the equation of a line perpendicular to y = 3 - 5x.
This line has gradient -5. A perpendicular line will have to have a gradient of 1/5, because then (-5) ×
(1/5) = -1. Any line with gradient 1/5 will be perpendicular to our line, for example, y = (1/5)x.
Straight Lines:
Horizontal and Vertical:
1. The horizontal line bold, shows Y = 6. There are
no X co-ordinates.
2. The Vertical line bold, shows X = 4. There are no
Y co-ordinates.
Diagonals:
3. The diagonal line travels uphill from left to right and is Y = X.
The slope is 1, for each increased Y measurement it also
increased in X measurement.
4. The diagonal line travels downhill from left to right and is Y = -X.
The slope is -1.
Four Types of Graph:
Straight Line: y = mx + c.
Fig 1. y = 3x +3.
Positive: climb to the right.
Negative: climb to the left.
Quadratic: y = x . Fig 2.
Positive: smile shape.
Negative: upsidedown.
Fig 2a. y = x -5x -3.
Cubic: y = x . Fig 3.
Positive: climb to the right.
Negative: climb to the left.
Reciprocol Graphs: y = 1 x. Fig 4.
Symmetrical around the origin.
Positive: top right, bottom left quadrants.
Negative: top left, bottom right quadrants.
Intersecting Graphs
If we wish to know the coordinates of the point(s) where two graphs intersect, we solve the equations
simultaneously.
Solving Equations
Any equation can be solved by drawing a graph of the equation in question. The points where the
graph crosses the x-axis are the solutions. So if you asked to solve x² - 3 = 0 using a graph, draw the
graph of y = x² - 3 and the points where the graph crosses the x-axis are the solutions to the equation.
We can also sometimes use the graph of one equation to solve another.
Example
Draw the graph of y = x² - 3x + 5 .
Use this graph to solve 3x + 1 - x² = 0 and x² - 3x - 6 = 0
Answer:
1) Make a table of values for y = x² - 3x + 5 and draw the graph.
2) Make the equations you need to solve like the one you have the graph of.
So for 3x + 1 - x² = 0:
i) multiply both sides by -1 to get x² - 3x - 1 = 0
ii) add 6 to both sides: x² - 3x + 5 = 6
Now, the left hand side is our y above, so to solve the equation, we find the values of x when y = 6
(you should get two answers).
Try solving x² - 3x - 6 = 0 yourself using your graph of y = x² - 3x + 5. You should get a answers of
around -1.4 and 4.4 .
Simultaneous Equations: Graphical Solution: A simultaneous equation of two separate equations
may be solved by plotting their straight lines onto the same graph. The co-ordinates where the lines
cross is the solution for x and y. (A graphical representation of three or more simultaneous equations
would require a three dimentional or more graph).
Example:
y - 2x = 2.
y + x = 8.
Rearrange to standard: (y = mx + c)
y = 2x + 2.
y = -x + 8.
Sketch the Graph:
The intersection is at Y = 6 and X = 2.
Therefore: x = 2 and y = 6.
Chapter 4: Geometry
Angles
Lines AB and CD are parallel to one another (hence the » on the lines).
a and d are known as vertically opposite angles. Vertically opposite angles are equal. (b and c, e
and h, f and g are also vertically opposite).
g and c are corresponding angles. Corresponding angles are equal. (h and d, f and b, e and a are
also corresponding).
d and e are alternate angles. Alternate angles are equal. (c and f are also alternate). Alternate angles
form a 'Z' shape and are sometimes called 'Z angles'.
a and b are adjacent angles. Adjacent angles add up to 180 degrees. (d and c, c and a, d and b, f and
e, e and g, h and g, h and f are also adjacent).
d and f are interior angles. These add up to 180 degrees (e and c are also interior).
Any two angles that add up to 180 degrees are known as supplementary angles.
Polygons: A polygon is a plane figure with straight sides. The names of some polygons are: 3
sided...Triangle; 4 sided...quadrilateral; 5 sided...pentagon; 6 sided...hexagon; 7 sided...heptagon; 8
sided...octagon; 9 sided...nonegon; 10 sided...decagon.
Triangles: 3 sided polygon. These are known in 3 different categories:
Scalene; 3 sides are different length, 3 angles are different sizes, no reflection possible - no symmetry.
(Acute or Obtuse; possible Right Angled).
Isosceles; 2 sides are equal length, 2 angles are equal in size, there is a single line of symmetry.
(Acute or Obtuse; possible Right Angled).
Equilateral; 3 sides are equal length, 3 angles are equal in size (60 ), there are 3 lines of symmetry
from all 3 points. (Acute only, cannot be Right Angled).
All triangles have internal angles which total to 180 . If the triangle was to sit on a straight line the
external angle would be the sum of the opposite two sides (a straight line also has a total angle of
180).
Diagrams of triangles may show a bar across two or three sides to denote equal lengths, or letters in
the corners to denote the angles. If these letters are the same then it is to be assumed the angles are
the same size. The angles may be given a letter on the outside of each corner to indicate which angle
is being examined ie. angle PQR means angle Q is being examined.
Angle Sum of a Triangle
Using some of the above results, we can prove that the sum of the three angles inside any triangle
always add up to 180 degrees.
If we have a triangle, you can always draw two parallel lines like this:
Now, we know that alternate angles are equal. Therefore the two angles labelled x are equal. Also, the
two angles labelled y are equal.
We know that x, y and z together add up to 180 degrees, because these together is just the angle
around the straight line. So the three angles in the triangle must add up to 180 degrees.
Exterior Angles
The exterior angles of a shape are the angles you get if you extend the sides. The exterior angles of a
hexagon are shown:
A polygon is a shape with straight sides. All of the exterior angles of a polygon add up to 360°.
because if you put them all together they form the angle all the way round a point:
Therefore if you have a regular polygon (in other words, where all the sides are the same length and
all the angles are the same), each of the exterior angles will have size 360 ÷ the number of sides. So,
for example, each of the exterior angles of a hexagon are 360/6 = 60°.
Interior Angles
The interior angles of a shape are the angles inside it. If you know the size of an exterior angle, you
can work out the size of the interior angle next to it, because they will add up to 180° (since together
they are the angle on a straight line).
Exterior Angle of a Triangle
Angle x is an exterior angle of the triangle:
The exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices. In
other words, x = a + b in the diagram.
Quadrilaterals: 4 sided polygon. Known a square, rectangle, rhombus, parallelogram, trapezium,
kite. All these shapes have a pair or both pair of sides in parallel. If any of these shapes are split from
opposite corner it results in two triangles being formed of sum of angles 180 . Therefore the angles
sum of two triangles is 360 .
A square and a rectangle both have four angles of 90 . A rhombus has all four sides equal in length
and opposite corners of equal angle. A parallelogram also has opposite corners of equal angle. A
trapezium has only one pair of paralell sides and no assumption can be made of the angles. A kite has
its adjacent sides equal in length, therefore the two opposite angles where the different lengths meet
are equal.
Regular Polygons: A regular polygon is one where all the sides and angles are identical in length
and degree.The sum of the exterior angles for any sided polygon of any length is 360 . Each external
angle is 360 n (where n is the number of sides of any length). The sum of the interior angles is
calculated by the formula: 180n - 360. Each regular polygon has n internal isoscles triangles, (base is
the outside edge),and many lines of symmetry.
Example using a regular six sided hexagon:
Angle A + a = 180
Angle B + b = 180
Angle C + c = 180
Angle D + d = 180
Angle E + e = 180
Angle F + f = 180
Sum of internal and external angles = 1080 . 1080 - external = 720 . Each internal
angle = 720 n (720 6) = 120 and each external angle = 360 n (360 6) = 60 .
Circles: A shape designed of fixed points of equal distance from the centre, this distance is known as
the radius. The circumference is the perimeter. An arc is a section of the circumference. A chord is
a line joining two points on the circumference. The diameter is a chord which passes through the
centre. A segment is part of the circle that is separated by a line. A sector is part of the circle
separated by two lines of radius. A tangent is an outside line that just touches the perimeter at one
point only.
Properties and Rules:
(i) The angle of a triangle on the circumference which includes a line through the diameter is always
90 , (see (i) angle a at points A, B, C,) Right Angle.
(ii) An isosceles triangle is formed by 2 radii, the angles on the circumference are identical (see(ii)
angle a at points M and N). A perpendicular line from the base of the triangle always (i) cuts the line in
half and (ii) passes through the centre of the circle.
(iii) The opposide angle from a chord is twice the size at the centre than at the circumference (see (iii)
angle at point O is twice the angle at point P). Note angles b are identical.
(iv) The angles around the circumference are identical if they are from a triangle from the same chord
(see (iv) angle a is equal to angle a), and if a triangle is added to the opposite side of the chord, angle
b the two opposite angles a + b sum to 180 .
(v) The angle of a triangle on the circumference which includes a chord for the base is always the
same (see (v) angles a at points C and D).
(vi) A cyclic quadrilateral is one inside the circle in which all of its corners touch the circumference. The
opposite angles of a cyclic quadrilateral add up to 180 (see (vi) angles a and c add up to 180 ; also
angles b and d add up to 180 ). If a line is extended from a corner, that angle is equal to the opposite
angle (see (vi) angle e is equal to angle a).
(vii) The angle between a radius and a tangent is a right angle (see (vii) where angles at points M and
N are right angles). Tangents that meet at an external point are equal in length (see(vii) line MP is
equal to line NP).
(vii) A tangent (see (viii) tangent perpendicular to OP) and an adjoined chord PR have an identical
angle a at point P to the angle in the opposite segment a at point Q.
Pythagoras's Theorem
In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the
other two sides.
i.e.: c² = a² + b² in the following diagram:
Example
Find AC in the diagram below.
AB² + AC² = BC²
AC² = BC² - AB²
= 13² - 5²
= 169 - 25 = 144
AC = 12cm
Trigonometry
Right Angled Triangles: Trigonometry equations interconnect the angles and sides of right
angled triangles.
The longest side is always called the hypotenuse and is always opposite the right angle. The other two
sides of the triangle are named in relation to the subject angle. The first triangle in the diagram above
shows the subject angle in blue. These other sides make up the right angle and are named as
opposite to the subject angle, and adjacent to the subject angle. The second triangle above shows the
names of the sides when the subject angle is changed.
The sine of the angle = the length of the opposite side
the length of the hypotenuse
The cosine of the angle = the length of the adjacent side
the length of the hypotenuse
The tangent of the angle = the length of the opposite side
the length of the adjacent side
Bearings
A bearing is an angle, measured clockwise from the north direction. Below, the bearing of B from A is
025 degrees (note 3 figures are always given). The bearing of A from B is 205 degrees.
Example
A, B and C are three ships. The bearing of A from B is 045º. The bearing of C from A is 135º. If AB=
8km and AC= 6km, what is the bearing of B from C?
tanC = 8/6, so C = 53.13º
y = 180º - 135º = 45º (interior angles)
x = 360º - 53.13º - 45º (angles round a point)
= 262º (to the nearest whole number)
VECTORS.
Contents:
Notation & Representation: A vector is a quantity made up of magnitude (size) and direction.
Examples of vectors are: displacement, acceleration, force and momentum. The wind is an example
of a vector, it has distance and direction. A scalar is magnitude only, examples are distance, time,
speed, area, etc. To be able to operate with vectors a knowledge of Pythagorus' Theroem, and
Trigonometry is necessary.
A vector may be represented by an arrowed line on a diagram:
. The length of the
line represents the magnitude and the arrow represents the direction. The vector may be named:
,
the letters denoting the end points, the arrow denoting direction from A to B. The |AB| may be used to
represent the magnitude or scalar. The vector may also be named in the diagram using a small bold
typed letter:
. A small letter may also represent a vector when underlined: a, and the
magnitude represented by: |a| or a.
A Unit Vector is a vector with a magnitude of 1. A null vector is a vector with a magnitude of zero.
Displacement Vector: Is the movement from one position to another. A displacement vector can
be
represented by a column vector. The distance of x over the
distance of y. x and y are the components of the vector.
The magnitude |a| =
x2  y2 .
A position vector describes the movement from an origin to a point on a co-ordinate grid system. The
position of P is given in relation to the origin O.
The inverse of a vector a is -a, equal in magnitude but
opposite direction.
The inverse of a vector
is -
or
.
Equal or equivalent vectors are equal in magnitude and direction making them parallel.
Multiplication by a Scalar: Multiplying a vector a by a scalar increases the magnitude of the
vector by that scalar number.
1. vector a.
2. vector a times 2.
3. vector a times - 3.
When multiplying a column vector by a scalar, multiply each part by the scalar:
ie. a =
then -3a =
.
Addition & Subtraction: Vectors can be combined by addition and subtraction, the new vector
becomes the resultant. The resultant is usually marked with a double arrow.
Addition: a + b = c.
+
Subtraction: a - b = c.
-
=
.
=
.
Column Vectors: Addition and subtraction done as above, beware of the signs.
Example: p =
and q =
.
Find (i), p + q. Answer: p + q. =
+
=
.
Find (ii), q - p. Answer: q - p. =
-
=
.
Find (iii), 3p - 2q. Answer: 3p - 2q. =
-
=
.
Find (iv), |p|. Answer: |p|. = 5 + 2 . = 5 4.
Geometric Problems: Vectors may be used to show geometric results, usually done by finding an
expression for an unknown vector in terms of other given vectors. This is done by finding a pathway
through the diagram from the start point to the finish point. Always label the unknown vector with a
double arrow. The diagram shows M and N as halfway points.
= -c -b - a.
= -a + b + c. or -d.
= a - d.
Applied Example:
The width of the river is 50metres, a swimmer swims across at the speed of 2metres per second (m/s),
the current is 3m/s. Find the overall speed, the direction from the start point, and the distance the
swimmer swam to reach the far bank.
Answer: (i) Overall speed. The speed of the swimmer and current are needed.
Magnitude |resultant| =
Answer (ii) Direction. Using Trigonometry; Tan
2 2  32 = 3.6m/s.
= 56.3 .
Answer (iii) Distance. Using Trigonometry; cos 56.3 = 50
Therefore distance swam = 50
cos 56 3 . = 90 1 metres.
distance.
Transformations
A translation occurs when a shape is moved from one place to another. It is equivalent of picking up
the shape and putting it down somewhere else. Vectors are used to describe translations.
Reflections
A reflection is like placing a mirror on the page. When describing a reflection, you need to state the
line which the shape has been reflected in. The distance of each point of a shape from the line of
reflection will be the same as the distance of the reflected point from the line.
For example, below is a triangle that has been reflected in the line y = x (the length of the pink lines
should be the same on each side of the line y=x):
Rotations
When describing a rotation, the centre and angle of rotation are given. If you wish to use tracing
paper to help with rotations: draw the shape you wish to rotate onto the tracing paper and put this over
shape. Push the end of your pencil down onto the tracing paper, where the centre of rotation is and
turn the tracing paper through the appropriate angle (if you are not told whether the angle of rotation is
clockwise or anticlockwise, it would usually be anticlockwise). The resultant position of the shape on
the tracing paper is where the shape is rotated to.
Enlargements
Enlargements have a centre of enlargement and a scale factor.
1) Draw a line from the centre of enlargement to each vertex ('corner') of the shape you wish to
enlarge. Measure the lengths of each of these lines.
2) If the scale factor is 2, draw a line from the centre of enlargement, through each vertex, which is
twice as long as the length you measured. If the scale factor is 3, draw lines which are three times as
long. If the scale factor is 1/2, draw lines which are 1/2 as long, etc.
Example
The centre of enlargement is marked. Enlarge the triangle by a scale factor of 2.
Graphical Representation of Data
Bar Chart
A bar chart is a chart where the height of bars represents the frequency. The data is 'discrete' (discontinuous- unlike
histograms where the data is continuous). The bars should be separated by small gaps.
Pie Chart
A pie chart is a circle which is divided into a number of parts.
The pie chart above shows the TV viewing figures for the following TV programmes:
Eastenders, 15 million
Casualty, 10 million
Peak Practice, 5 million
The Bill, 8 million
Total number of viewers for the four programmes is 38 million. To work out the angle that 'Eastenders' will have in
the pie chart, we divide 15 by 38 and multiply by 360 (degrees). This is 142 degrees. So 142 degrees of the circle
represents Eastenders. Similarly, 95 degrees of the circle is Casualty, 47 degrees is Peak Practice and the remaining 76
degrees is The Bill.
Cumulative Frequency Graphs
Cumulative Frequency
This is the running total of the frequencies. On a graph, it can be represented by a cumulative
frequency polygon, where straight lines join up the points, or a cumulative frequency curve.
Example
Frequency:
Cumulative
Frequency:
4
4
6
10
(4 + 6)
3
13
(4 + 6 + 3)
2
15
(4 + 6 + 3 + 2)
6
21
(4 + 6 + 3 + 2 + 6)
4
25
(4 + 6 + 3 + 2 + 6 + 4)
The Median Value
The median of a group of numbers is the number in the middle, when the numbers are in order of
magnitude. For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6:
1, 2, 4, 6, 6, 7, 8
(6 is the middle value when the numbers are in order)
If you have n numbers in a group, the median is the (n + 1)/2 th value. For example, there are 7
numbers in the example above, so replace n by 7 and the median is the (7 + 1)/2 th value = 4th value.
The 4th value is 6.
When dealing with a cumulative frequency curve, "n" is the cumulative frequency (25 in the above
example). Therefore the median would be the 13th value. To find this, on the cumulative frequency
curve, find 13 on the y-axis (which should be labelled cumulative frequency). The corresponding 'x'
value is an estimation of the median.
Quartiles
If we divide a cumulative frequency curve into quarters, the value at the lower quarter is referred to as
the lower quartile, the value at the middle gives the median and the value at the upper quarter is the
upper quartile.
A set of numbers may be as follows: 8, 14, 15, 16, 17, 18, 19, 50. The mean of these numbers is
19.625 . However, the extremes in this set (8 and 50) distort this value. The interquartile range is a
method of measuring the spread of the middle 50% of the values and is useful since it ignore the
extreme values.
The lower quartile is (n+1)/4 the value (n is the cumulative frequency, i.e. 157 in this case) and the
upper quartile is the 3(n+1)/4 the value. The difference between these two is the interquartile range
(IQR).
In the above example, the upper quartile is the 118.5th value and the lower quartile is the 39.5th value.
If we draw a cumulative frequency curve, we see that the lower quartile, therefore, is about 17 and the
upper quartile is about 37. Therefore the IQR is 20 (bear in mind that this is a rough sketch- if you plot
the values on graph paper you will get a more accurate value).
Histograms
Histograms are similar to bar charts apart from the consideration of areas. In a bar chart, all of the bars are the same
width and the only thing that matters is the height of the bar. In a histogram, the area is the important thing.
Example
Draw a histogram for the following information.
Height (feet): Frequency Relative Frequency
0-2
0
0
2-4
1
1
4-5
4
8
5-6
8
16
6-8
2
2
(Ignore relative frequency for now). It is difficult to draw a bar chart for this information, because the class divisions
for the height are not the same. The height is grouped 0-2, 2-4 etc, but not all of the groups are the same size. For
example the 4-5 group is smaller than the 0-2 group.
When drawing a histogram, the y-axis is labelled 'relative frequency' or 'frequency density'. You must work out the
relative frequency before you can draw a histogram. To do this, first you must choose a standard width of the groups.
Some of the heights are grouped into 2s (0-2, 2-4, 6-8) and some into 1s (4-5, 5-6). Most are 2s, so we shall call the
standard width 2. To make the areas match, we must double the values for frequency which have a class division of 1
(since 1 is half of 2). Therefore the figures in the 4-5 and the 5-6 columns must be doubled. If any of the class
divisions were 4 (for example if there was a 8-12 group), these figures would be halved. This is because the area of
this 'bar' will be twice the standard width of 2 unless we half the frequency.
If you are having problems working out the height of each of the bars, you can use the formula
Area of bar = frequency x standard width
Statistical Measures
Mean
There are four types of average: mean, mode, median and range. The mean is what most people
mean when they say 'average'. It is found by adding up all of the numbers you have to find the mean of,
and dividing by the number of numbers. So the mean of 3, 5, 7, 3 and 5 is 23/5 = 4.6 .
Grouped Data
When you are given data which has been grouped, you can't work out the mean exactly because you
don't know what the values are exactly (you just know that they are between certain values). However,
we calculate an estimate of the mean with the formula: ∑fx / ∑f , where f is the frequency and x is the
midpoint of the group (∑ means 'the sum of').
Example
Work out an estimate for the mean height, when the heights of 23 people are given by the first two
columns of this table:
Height (cm)
Number of
People (f)
Midpoint (x)
fx
101-120
1
110.5
110.5
121-130
3
125.5
376.5
131-140
5
135.5
677.5
141-150
7
145.5
1018.5
151-160
4
155.5
622
161-170
2
165.5
331
171-190
1
180.5
180.5
In this example, the data is grouped. You couldn't find the mean the "normal way" (by adding up the
numbers and dividing by the number of numbers) because you don't know what the values are. You
know that three people have heights between 121 and 130cm, for example, but you don't know what
the heights are exactly. So we estimate the mean, using "∑fx / ∑f".
A good way of setting out your answer would be to add two columns to the table, as I have.
"Midpoint" means the midpoint of each of the groups. So the first entry is the middle of the group
101-120 = 110.5 .
Now,
∑fx (add up all of the values in the last column) = 3316.5
∑f = 23
So an estimate of the mean is 3316.5/23 = 144cm (3s.f.)
Mode
The mode is the number in a set of numbers which occurs the most. So the modal value of 5, 6, 3, 4, 5,
2, 5 and 3 is 5, because there are more 5s than any other number.
Range
The range is the largest number in a set minus the smallest number. So the range of 5, 7, 9 and 14 is
(14 - 5) = 9.
The Median Value
The median of a group of numbers is the number in the middle, when the numbers are in order of
magnitude. For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6:
1, 2, 4, 6, 6, 7, 8
(6 is the middle value when the numbers are in order)
If you have n numbers in a group, the median is the (n + 1)/2 th value. For example, there are 7
numbers in the example above, so replace n by 7 and the median is the (7 + 1)/2 th value = 4th value.
The 4th value is 6.
Probability
Introduction
Probability is the likelihood or chance of an event occurring.
Probabilit y 
the number of ways of achieving success
the total number of possible outcomes
For example, the probability of flipping a coin and it being heads is ½, because there is 1 way of
getting a head and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .
The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.
Single Events
Example
There are 6 beads in a bag, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a
yellow?
The probability is the number of yellows in the bag divided by the total number of balls, i.e. 2/6 = 1/3.
Example
There is a bag full of coloured balls, red, blue, green and orange. Balls are picked out and replaced.
John did this 1000 times and obtained the following results:
Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50
a) What is the probability of picking a green ball?
b) If there are 100 balls in the bag, how many of them are likely to be green?
a) For every 1000 balls picked out, 450 are green. Therefore P(green) = 450/1000 = 0.45
b) The experiment suggests that 450 out of 1000 balls are green. Therefore, out of 100 balls, 45 are
green (using ratios).
Multiple Events
Independent and Dependent Events
Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice
and consider the probability that both are 6's.
We call two events independent if the outcome of one of the events doesn't affect the outcome of
another. For example, if we throw two dice, the probability of getting a 6 on the second die is the same,
no matter what we get with the first one- it's still 1/6.
On the other hand, suppose we have a bag containing 2 red and 2 blue balls. If we pick 2 balls out of
the bag, the probability that the second is blue depends upon what the colour of the first ball picked
was. If the first ball was blue, there will be 1 blue and 2 red balls in the bag when we pick the second
ball. So the probability of getting a blue is 1/3. However, if the first ball was red, there will be 1 red and
2 blue balls left so the probability the second ball is blue is 2/3. When the probability of one event
depends on another, the events are dependent.
Possibility Spaces
When working out what the probability of two things happening is, a probability/ possibility space can
be drawn. For example, if you throw two dice, what is the probability that you will get: a) 8, b) 9, c)
either 8 or 9?
a) The black blobs indicate the ways of getting 8 (a 2 and a 6, a 3 and a 5, ...). There are 5 different
ways. The probability space shows us that when throwing 2 dice, there are 36 different possibilities
(36 squares). With 5 of these possibilities, you will get 8. Therefore P(8) = 5/36 .
b) The red blobs indicate the ways of getting 9. There are four ways, therefore P(9) = 4/36 = 1/9.
c) You will get an 8 or 9 in any of the 'blobbed' squares. There are 9 altogether, so P(8 or 9) = 9/36 =
1/4 .
Probability Trees
Another way of representing 2 or more events is on a probability tree.
Example
There are 3 balls in a bag: red, yellow and blue. One ball is picked out, and not replaced, and then
another ball is picked out.
The first ball can be red, yellow or blue. The probability is 1/3 for each of these. If a red ball is picked
out, there will be two balls left, a yellow and blue. The probability the second ball will be yellow is 1/2
and the probability the second ball will be blue is 1/2. The same logic can be applied to the cases of
when a yellow or blue ball is picked out first.
In this example, the question states that the ball is not replaced. If it was, the probability of picking a
red ball (etc.) the second time will be the same as the first (i.e. 1/3).
Example: A drawer has 6 blue socks and 10 red socks, what is the probability of:
(i) getting two blue socks; (ii) getting two red socks; (iii)one sock of each colour.
Answer: As the second sock event is dependant on the first sock event and is conditional, also the
events are mutually exclusive. Draw a probability tree diagram, and enter the probable outcomes.