August 31, 2020
Definition of a Rational Number
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 .
Definition of Multiplication of 2 Rational Numbers
The multiplication of 2 Rational numbers is performed the same way as other forms of multiplication.
With Rational numbers, we multiply the numerators together and the denominators together to form a new Rational number
The order that we multiply 2 Rational numbers does not matter.
This is a consequence of the operation of multiplication. It is the same for Natural Numbers and Integers.
Also take note that we can form 2 fractions that are entirely different than the ones that we started with by using this property
Multiplication of 3 or more Rational Numbers and the Associative Property
The property of commutativity extends to the multiplication of 3 or more Rational numbers. It also does not matter which 2 Rational numbers we multiply first, we will always get the same product.
This is called the Associative Property of Multiplication
We may encounter a problem where we need to multiply a sum of Rational numbers by another Rational number. We can just multiply each part of the sum by the number on the outside and then perform the sum.
This process is called distribution and the property is called the Distributive Property
Multiplicative Inverse and Identity
The inverse of a Rational number is
the inverse of is
If we multiply a Rational number by its inverse, the product is
This is the Multiplicative Inverse of a Rational number.
is the Multiplicative Identity of any rational number. This means the product of a Rational number and is just .
There are many different ways to form a Rational number that is equivalent to
We can sometimes pull out a common factor of a multiplication problem to simplify it