Chapters
Definition of Rational Numbers
Rational numbers are of the form
with the special case that 
because division by 
is meaningless (undefined)
where 
is any integer and 
is any integer other than
with
Recall that the Integers 
are the positive and negative whole numbers along with .
Division of 2 Rational Numbers
One should be familiar with the process of multiplying 2 fractions
before moving forward with the lesson.
When we want to divide one Rational number by another 
we need to simplify the expression by converting it into an equivalent one that involves only multiplication.
Multiplication by the Reciprocal of the Denominator
In the compound fraction
the Rational number is the denominator, meaning the denominator is itself a fraction.
This complicated form of a compound fraction that occurs in a division problem can be simplified by 'flipping' the fraction in the denominator
becomes 
When we 'flip' a fraction, we call it taking the reciprocal of that fraction
the reciprocal of 
is 
We then multiply the new 'flipped' fraction by the original numerator, the fraction
This rids us of the denominator in the complicated looking compound fraction and turns the problem into an equivalent one involving only multiplication.
Example
Divide 
by 
First we flip the bottom fraction 
Then we multiply that by the top fraction
Example
Divide 
by
Example
Divide 
by 
Notice that we could have just multiplied the 
together and gotten the same answer
Example
Divide by 
What is the problem asking for?
Basically it's asking how many 
are in or how many of the bottom fraction are in the top fraction.
We know that
and now we can ask how many are in 
?
It should be plain to see that there are four in the fraction , just like there are three 
in 
.
Rational Numbers and Division
The Rational Numbers 
are the first set of numbers that allow us to introduce and use the operation of division. This essentially means that the Rationals are the first set of numbers that let us divide one integer by another integer and have that number always exist as another Rational.
Just as the introduction of the Integers lets us extend our Algebraic operations to include subtraction (alongside addition and multiplication) by allowing the use of negative whole numbers, the introduction of the Rational Numbers does the same for division.
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Operations with Integers
If we add, subtract or multiply any integer by one or more other integers, that number will also always be an integer.
and 
are integers and their sum 
is also an integer
and 
are integers and their difference 
is also an integer
, and 
are all integers and their product 
is also an integer
Division Counterexample with Integers
and 
are both integers, but whether we divide 
by 
or divide 
by 
neither answer is also an integer.
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