Definition of Rational Numbers

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.

The best Maths tutors available
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Ayush
5
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Ayush
5
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
First Lesson Free>

Division of 2 Rational Numbers

One should be familiar with the process of multiplying 2 fractions

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

before moving forward with the lesson.

When we want to divide one Rational number \frac{a}{b} by another \frac{c}{d}

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

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

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

the Rational number \frac{c}{d} 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

\frac{c}{d} becomes \frac{d}{c}

When we 'flip' a fraction, we call it taking the reciprocal of that fraction

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

We then multiply the new 'flipped' fraction \frac{d}{c} by the original numerator, the fraction \frac{a}{b}

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

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 \frac{3}{4} by \frac{1}{2}

\frac{\frac{3}{4}}{\frac{1}{2}}

First we flip the bottom fraction \frac{1}{2}\to\frac{2}{1}

Then we multiply that by the top fraction \frac{3}{4}

\frac{\frac{3}{4}}{\frac{1}{2}}=\frac{3}{4}\times\frac{2}{1}=\frac{(3)(2)}{(4)(1)}=\frac{6}{4}=\frac{3}{2}

Example

Divide \frac{5}{8} by \frac{2}{3}

\frac{\frac{5}{8}}{\frac{2}{3}}=\frac{5}{8}\times\frac{3}{2}=\frac{(5)(3)}{(8)(2)}=\frac{15}{16}

Example

Divide \frac{1}{3} by 3

\frac{\frac{1}{3}}{3}=\frac{\frac{1}{3}}{\frac{3}{1}}=\frac{1}{3}\times\frac{1}{3}=\frac{1}{9}

Notice that we could have just multiplied the 3's together and gotten the same answer

\frac{\frac{1}{3}}{3}=\frac{1}{3\times 3}=\frac{1}{9}

Example

Divide \frac{2}{3} by \frac{1}{6}

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

What is the problem asking for?

Basically it's asking how many \frac{1}{6}'s are in \frac{2}{3} or how many of the bottom fraction are in the top fraction.

We know that

\frac{2}{3}=\frac{4}{6}

and now we can ask how many \frac{1}{6}'s are in \frac{4}{6} ?

It should be plain to see that there are four \frac{1}{6}'s in the fraction \frac{4}{6}, just like there are three \frac{1}{3}'s in \frac{3}{3}=1.

Rational Numbers and Division

The Rational Numbers \mathbb{Q} 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.

13+9=22

13 and 9 are integers and their sum 22 is also an integer

1-3=-2

1 and -3 are integers and their difference -2 is also an integer

-5\times-3\times-4=-60

-5, -3 and -4 are all integers and their product -60 is also an integer

Division Counterexample with Integers

4 and 7 are both integers, but whether we divide a=4 by b=7

\frac{a}{b}=\frac{4}{7}

or divide a=7 by b=4

\frac{a}{b}=\frac{7}{4}

neither answer is also an integer.

 

Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 5.00/5 - 1 vote(s)
Loading...

Patrick