Real Numbers

Real Numbers
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If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set notation.
These are also called the natural numbers.
{0, 1, 2, 3, 4, . . . }
If we include 0 we have the set of whole numbers.
{ …, -3, -2, -1, 0,1, 2, 3, . . . }
Include the opposites of the whole numbers and you have the set of integers.
Whole numbers are a subset of integers and counting numbers are a subset of whole numbers.
If we express a new set of numbers as the quotient of two integers, we have the set of rational numbers
This means to divide one integer by another or “make a fraction”
There are numbers that cannot be expressed as the quotient of two integers. These are called irrational numbers.
irrational
numbers
The rational numbers combined with the irrational numbers make up the set of real numbers.
Translating English to Maths
ORDER OF OPERATIONS
When there is more than one symbol of operation in an expression, it is agreed to complete the operations in a certain order. A mnemonic to help you remember this order is below.
B I M D A S
rackets
ndices
ultiplication
ivision
ddition
ubtraction
Do any simplifying possible inside of brackets starting with innermost brackets and working out
Apply Indices

Complete multiplication and division from left to right

Complete addition and subtraction from left to right
The operations of both addition and multiplication are commutative
When adding, you can “commute” or trade the terms places
When multiplying, you can “commute” or trade the factors places
When adding, you can “associate” and add any terms first and then add the other term.
When multiplying, you can “associate” and multiply any factors first and then multiply the other factor.
The operations of both addition and multiplication are associative
The operation of multiplication distributes over addition
The distributive property also holds for a factor that is multiplied on the left.
A positive times a negative is
NEGATIVE
A negative times a positive is
NEGATIVE
The negative of a negative
POSITIVE
CAUTION: Remember that the value for a and/or b could also be positive or negative.
A positive divided by a negative or
A negative divided by a positive is
NEGATIVE
A negative divided by a negative is