August 26, 2020
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
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
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.
First we flip the bottom fraction
Then we multiply that by the top fraction
Notice that we could have just multiplied the together and gotten the same answer
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.
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.