**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 t**wo numbers with the** same sign
** we get **plu**s ( 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) : 7x^{2}+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 : 4x^{2}-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 righ**t

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 i**s 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 .