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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