Rule 1 Addition of two Numbers with the same sign The result is the sum of the numbers with the given sign ( the sign of the two numbers ) .
When we add two plus numbers we get a plus number Ex : 3+6=9.
When we add two minus numbers we get a minus number Ex : -5 + - 6 = - 11 .
Example 5 + 6 + 8 = 19 . Example -5 + - 8 + - 6 = - 19 .
Rule 2 : Addition of two numbers with different signs the result is the difference between the numbers with the sign of the" bigger number " .
Ex 6 + - 9 = - 3 : Ex 19 + -5 = 14 . Ex -15 + 13 = - 2 .
Note Ex : 14 - 9 is the same as 14 + - 9 = 5 . and - 7 - 5 is really -7 + - 5 = - 12
Example simplify - 6 + 9 - 7 + 3 + 6 - 8 = - 6 - 7 - 8 + 9 + 3 + 6 = - 21 + 18 = - 3 .
Note we add the minus numbers ,we add the plus numbers then tidy it up .
Rule 3 : Multiplication and signs :
The basic rules for multiplication and signs are as follows
(Minus ) x ( Minus ) = Plus . ( Minus ) x ( Plus ) = Minus , (Plus ) x (Plus ) = Plus .
This can be expressed as follows that (a) when we multiply two numbers with the same sign we get plus ( b) when we multiply two numbers with difference signs we get a minus .
Example : - 7 x - 5 = + 35 , -7 x 5 = -35 . Example - 7. x - 4 x 3 = 28 x 3 = 84 .
Example : - 5 x - 6 x - 2 = 30 . x. - 2 = - 60 . Note if we are asked to multiply three or more numbers remember you can only multiply two numbers at a time .
Example - 4 x -3 x 5 x -2 = 12 x 5 x - 2 = 60 . x - 2 = - 120 .
Letters are often used in place of numbers , letters are often referred to as variables or unknowns . Rules 1,2,3 above apply to all calculations with letters , we have to remember one further rule which applies to letters that is we can only add things together that are the same ie we can only add x's to x's and y's to y's etc . Note also 5a means 5 times a .
Example 5a - 7a = - 2a , Example - 3x + 7x = 4x, Example 5a + 6b - 4a - 11b ,= 5a - 4a + 6b - 11b = a - 5b .
Multiplying letters by letters :
Rule 5 : Brackets :
(a) A number immediatly outside a bracket means that everything
inside the bracket gets multiplied by that number
Ex
Ex 
(b)Minus sign outside
a bracket will change all the signs inside the brackets when the brackets are
removed : ex 
(c)Minus number outside a bracket does two things (i) everything inside the bracket gets multiplied by the number and (ii) all the signs change
]
Brackets by brackets :
Multiply everything in the second bracket by everything in
the first bracket and tidy it up .
Mathematical Words :
The following words appear frequently in Mathematics Exam Papers .
(1) Expression :
This is a collection of letters and numbers : eg (4x + 3y)(5x-7y) : 7x2+4xy-6 .
Words associated with expressions.
Simplify : Means get rid of the brackets and tidy it up :
ex Simplify
Factors : The factors of of an expression are two or more things which when multiplied together give that expression
.Example the factors of 14 are 7 and 2 because 7 x 2 =14
Factorise : means find the factors of : Example factorise 
(2) Equation :
This is an expression which contains an equals sign : All equations contain an equals sign .
7x- 5y = 10 : 4x2-6x+5 = 0, 3x+2y = 4
4x- 5y = 10.
Word associated with equations
Solve : Means find the value(s) of the letter that makes the equation true .
Example
Solve 
Types of equation (1) Linear equation one variable Example 3x-4=10
(2)Linear equations 2
variables 
these
are called simultaneous equations
(3)Quadratic equations
: Equations of the form 
are
called quadratic equations:
the solutions of a quadratic equation are called the roots of the equation , all quadratic equations have two roots!
is
a quadratic equation
Inequality : This is an expression which contains
one of the following symbols 
Word associated with Inequalities Solve;
Example Solve 
Verify : this means show that the information you are given is true
Verify
if 
Calculate means work
out ( using maths) the value of : Example Calculate 
Evaluate : (normally applies to a given expression) we are asked to simplify and find the value of .
Example
Evaluate 
Words associated with fractions :
Numerator : the top half of the fraction, Denominator :the bottom half of the fraction.
Common denominator : this is the smallest number that the bottom halves of two or more fractions will divide.
Example the common denominator
of 
is
30 as it is the smallest number that can be divided by 2,3, and 5 .
Lowest common multiple (LCM) is another name for the smallest number that a given set of numbers will divide .
Highest Common factor (HCF) : this is the biggest factor a set of numbers have in common
example the highest common factor of 15,25 and 30 is 5 : the HCF of 8,12,and 28 is 4 .
Words Associated with graphs :
Plot this means place on a coordinate plane some given points .
Estimate (from your graph ) read answers from your graph .
(1) Natural Numbers Symbol N This is the set of positive whole numbers .
N =(0,1,2,3,4,.....)
On the
numbers line we use" dots " to indicate Natural numbers 
(2) Integers Symbol Z This is the Set of positive and negative whole numbers Z={-3,-2,-10,1,2,3}
on the
number line we show the integers as "dots"
(3)Rational Numbers Symbol Q : This is the set of numbers that can be written as a fraction
Most numbers can be written
as a fraction eg 
In general you will not be asked to put rational numbers on the number line .
(4)Irrational Numbers Symbol Ir : Numbers that cannot be written as a fraction :
The only numbers that cannot
be written as a fraction are numbers such as 
(a
non repeating non terminating decimal) or surds such as 
(5)Real numbers Symbol R : This is the set which includes all of the above types
of numbers On the number line we use a thick line to show real numbers 
(6)Complex
numbers : These are numbers of the form x+yi where x and y are Real
numbers and ( i ) =
complex numbers are plotted on an Argand diagram.
Inequality symbols
In general read all information from left to right
>"
is greater than ", 
x
is greater than y . <"
is less than ",
a is less than b
"is
less than or equal to" 
x
is less than or equal to k .:
"is greater than or equal to"
p
is greater than or equal to h .
Compound
inequalities 
read
from the middle to the left then from the middle to the right
the above inequality reads " x is greater than or equal to a and x is less than or equal to b "
Symbols and Sets
:
"is an element of "belongs to a particular set 
is
not an element of .
. 
.
Union Au B means the set of all the elements of A and B with none
repeated
intersection
, A 
B
means the set of all the elements that A and B have in common .
"is
a subset of " :A
B means that A is a subset of B , A contains some or all of the elements of
B but no other elements .
means is not a subset .
The cardinal number of a set , this indicates the number of
elements in the set .
More Symbols :
means
is perpendicular to : A 
B
means A and B meet at right angles .
//
means is parallel to . 
means
implies 
:
abc means angle abc . 
also
means angle abc
means
the size of the angle , The symbol 
in
general means the size of or the measure of or the lenght of
, but another meaning is the absolute value of meaning the positive
value of something
Function
notation :
A function is a rule which is shown as an expression in x Ex
this
states that f is a function which turns x into x + 4 ,(4 is added to every x
) so f(3) = 3+ 4, f(7) = 7 + 4 . The way functions are shown can vary the same
function can be shown as follows 
The
first R is the domain of the function these are the numbers that can
be used in the function ,the second R is the Codomain of the function
these are the numbers that can come out of the function . Functions are often
referred as maps and are said to Map (connect up ) the elements of the Domain
with those of the Codomain . But the most common notation is f(x) equals .