Definition of a Rational Number

Rational numbers are of the form

\frac{a}{b}  with the special case that  b\neq0

because division by 0 is meaningless (undefined)

where a is any integer and b is any integer other than 0

a, b\in\mathbb{Z}  with  b\neq0

Recall that the Integers \mathbb{Z} are the positive and negative whole numbers along with 0.

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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

\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b\times d}=\frac{ac}{bd}

Example

\frac{3}{4}\times\frac{5}{6}=\frac{3\times5}{4\times6}=\frac{15}{24}=\frac{5}{8}

Commutative Property

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.

\frac{a}{b}\times\frac{c}{d}=\frac{c}{d}\times\frac{a}{b}

Example

\frac{2}{5}\times\frac{3}{4}=\frac{2\times3}{5\times4}=\frac{3\times2}{4\times5}=\frac{3}{4}\times\frac{2}{5}

Also take note that we can form 2 fractions that are entirely different than the ones that we started with by using this property

\frac{2}{5}\times\frac{3}{4}=\frac{2\times3}{5\times4}=\frac{3\times2}{5\times4}=\frac{3}{5}\times\frac{2}{4}

or

\frac{2}{5}\times\frac{3}{4}=\frac{3}{5}\times\frac{2}{4}=\frac{6}{20}=\frac{3}{10}

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

\frac{a}{b}(\frac{c}{d}\times\frac{e}{f})=(\frac{a}{b}\times\frac{c}{d})\frac{e}{f}=\frac{c}{d}(\frac{a}{b}\times\frac{e}{f})=\frac{ace}{bdf}

Example

\frac{1}{2}(\frac{3}{7}\times\frac{5}{6})=\frac{1}{2}(\frac{15}{42})=\frac{15}{84}=\frac{5}{28}

(\frac{1}{2}\times\frac{3}{7})\frac{5}{6}=(\frac{3}{14})\frac{5}{6}=\frac{15}{84}=\frac{5}{28}

\frac{3}{7}(\frac{1}{2}\times\frac{5}{6})=\frac{3}{7}(\frac{5}{12})=\frac{15}{84}=\frac{5}{28}

Distributive Property

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

\frac{a}{b}(\frac{c}{d}+\frac{e}{f})=(\frac{a}{b})(\frac{c}{d})+(\frac{a}{b})(\frac{e}{f})

Example

\frac{2}{3}(\frac{1}{2}+\frac{3}{4})=((\frac{2}{3})(\frac{1}{2})+(\frac{2}{3})(\frac{3}{4}))=(\frac{2}{6}+\frac{6}{12})=\frac{4}{12}+\frac{6}{12}=\frac{10}{12}=\frac{5}{6}

Multiplicative Inverse and Identity

The inverse of a Rational number \frac{a}{b} is \frac{b}{a}

the inverse of \frac{2}{3} is \frac{3}{2}

If we multiply a Rational number by its inverse, the product is 1

\frac{a}{b}\times\frac{b}{a}=1

This is the Multiplicative Inverse of a Rational number.

Example

\frac{3}{4}\times\frac{4}{3}=\frac{3\times4}{4\times3}=\frac{12}{12}=1

1 is the Multiplicative Identity of any rational number. This means the product of a Rational number \frac{a}{b} and 1 is just \frac{a}{b}.

\frac{a}{b}\times1=\frac{a}{b}

There are many different ways to form a Rational number that is equivalent to 1

Examples

\frac{2}{2}=1   \frac{3}{3}=1    \frac{6}{6}=1    \frac{12}{12}=1

Common Factors

We can sometimes pull out a common factor of a multiplication problem to simplify it

Example

\frac{15}{16}+\frac{13}{12}=\frac{1}{4}(\frac{15}{4}+\frac{13}{3})=\frac{1}{4}(\frac{97}{12})=\frac{97}{48}

 

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