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OUR NUMBER SYSTEM
The natural (or counting) numbers are 1, 2, 3, 4, 5.
The whole numbers are the natural numbers together with 0.
The integers are the set of real numbers consisting of the natural numbers, their additive
inverses and zero. {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...}
The rational numbers are those numbers which can be expressed as a ratio between two
integers. For example, the fractions 1/3 and –1111/8 are both rational numbers.
All decimals which terminate are rational numbers (since 8.27 can be written as 827/100.)
Decimals which have a repeating pattern after some point are also rationals: for example,
0.083333333... = 1/12.
An irrational number is a number that cannot be written as a ratio (or fraction). In decimal
form, it never ends or repeats.
π = 3.14159265358979...
The real numbers is the set of numbers containing all of the rational numbers and all of the
irrational numbers. The real numbers are “all the numbers” on the number line. In later
years you will learn also about imaginary numbers.
Integers
The Integers include zero, the counting numbers, and the negative of the counting numbers,
to make a list of numbers that stretch in either direction indefinitely.
Absolute Value
How far a number is from zero. Example "6" is 6 away from zero, but "−6" is also 6 away
from zero.So the absolute value of 6 is 6, and the absolute value of −6 is also 6.
The symbol "|" is placed either side to mean "Absolute Value", so we write: |−6| =
How to Add and Subtract
Positive and Negative Numbers
Fractions
A fraction is a part of a whole
How to Add and Subtract
Positive and Negative Fractions
Pre-Algebra
Order of Operations
The rules that say which calculation comes first in an expression
They are:
• do everything inside parentheses first: ()
• then do exponents, like x2
• then do multiplication and division from left to right
• then do addition and subtraction from left to right
An Expression is a group of terms (the terms are separated by + or − signs) There is NO =
sign.
A Term is either a single number or a variable, or numbers and variables multiplied together.
Equation
An equation says that two things are equal. It will have an equals sign "=" like this:
x
+
2
=
6
That equation says: what is on the left (x + 2) is equal to what is on the right (6)
Parts of an Equation So people can talk about equations, there are names for different parts
Here we have an equation that says 4x − 7 equals 5, and all its parts:
A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.
A number on its own is called a Constant.
A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a
coefficient)
Like Terms
Like Terms are terms whose variables (and their exponents such as the 2 in x2) are the
same.
In other words, terms that are "like" each other. (Note: the coefficients can be different)
Example:
(1/3)xy2
-2xy2
6xy2
Are all like terms because the variables are all xy2
How to Solve
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to
end up with something like "x = 6".
But instead of saying "obviously x=6", use this neat step-by-step approach:

Work out what to remove to get "x = ..."

Remove it by doing the opposite (adding is the opposite of subtracting)

Do that to both sides
Here is an example:
To remove it, do
We want to
the opposite,
Do it to both
remove the "-2"
in this case add
sides:
Which is ...
Solved!
2:
Additive Inverse
.
What you add to a number to get zero.
The negative of a number.
Example:
The additive inverse of -2 is 2, because -2 + 2 = 0.
The additive inverse of +2 is -2 as well.
Lets look at multiplication.
8x
=
32
In English we say "eight times x equals thirty-two", meaning that 8 x's make 32.
Multiplicative Inverse
Another name for Reciprocal.
When you multiply a number by its "Multiplicative Inverse" you get 1.
Example: 8 × (1/8) = 1
So to solve the above equaton, we multiply both sides by 1/8 and get 1 x = 4
Let’s look at divison
x/3 =
Start with:
5
x/3 = 5
What we are aiming for is an answer like "x = ...", and the divide by 3 is in the way of that!
If we multiply by 3/1 we can cancel out the divide by 3 . 3/1 is the multiplicative inverse of 1/3
and will give us 1x which is what we need. Just remember to do the same thing to the other side.
So, let us have a go at multiplying by 3 on both
sides:
A little arithmetic (3/3 = 1 and 5×3 = 15) becomes:
x/3 ×3 = 5 ×3
1x = 15