Rational Numbers

Introduction

We know that for two give integers, m and n, their sum m + n, product m x n and the difference m – n are always integers. However, it may not always be possible for a given integer to exactly divide another integer. In other words, the result of the division of an integer by a non-zero integer may or may not be an integer.  For example, when 9 is divided by 4, the result is not an integer. In fact, $\frac{9}{4}$ is a fraction. Thus we need to extend the system of integers so that it may also be possible to divide any given integer by any other given integer different from zero. That is where we get the concept of rational numbers. Let us learn what are rational numbers and their importance and uses in Mathematics.

What are Rational Numbers?

Rational numbers are A number of the form $\frac{p}{q}$ or a number that can be expressed in the form of $\frac{p}{q}$, where p and q are integers and q ≠ 0 is called a rational number.

In other words, a rational number is any number that can be expressed as the quotient of two integers with the condition that the divisor is not zero.

Examples of rational numbers include $-\frac{1}{7}, \,\frac{-5}{18}, \,\frac{11}{18}, \,\frac{-17}{9}$, etc.

In the rational number $\frac{p}{q}$, p is known as the numerator while q is known as the denominator.

Positive Rational Number – A rational number is said to be positive if its numerator and denominator are either both positive integers or both are negative integers. For example, $\frac{5}{7}$ and $\frac{-5}{-7}$ are both positive rational numbers.

Negative Rational Number – A rational number is said to be negative if either its numerator or its denominator is a negative integer. For example, $\frac{-5}{7}$ and $\frac{5}{-7}$ are both negative rational numbers.

Properties of Rational Numbers

The properties of rational numbers are:

  1. Every natural number is a rational number but a rational number may not be a natural number. This means that every natural number n can be written as $n=\frac{n}{1}$, such as we can write 5 as $\frac{5}{1}$, -9 as $\frac{-9}{1}$ and so on. However, every rational number cannot be written as a natural number such as $-\frac{-1}{7}, \,\frac{-5}{8}$ are rational numbers but they are not natural numbers.
  2. Zero is a rational number. This is because 0 can be written in the form of $\frac{0}{1}$ which is a rational number.
  3. Every integer is a rational number but a rational number need not be an integer. This is because every integer Z can be written in the form of $\frac{z}{q}$, such as $1=\frac{1}{1}\, -2=\frac{-2}{1}$, and so on. On the other hand, every rational number might not be an integer such as, $\frac{11}{18}, \,\frac{-17}{9}$, etc.
  4. Every fraction is a rational number while every rational number is not a fraction. For example, $\frac{11}{-18}$ is a rational number while $\frac{11}{-18}$ is not a fraction as it has a negative sign in the denominator.
  5. If $\frac{p}{q}$ is a rational number and m is an integer, then

$\frac{p}{q}=\frac{p\times m}{q\times m}$

In other words, a rational number remains unchanged if we multiply its numerator and denominator by the same non-zero integer.

6. If $\frac{p}{q}$ is a rational number and m is an integer, then

$\frac{p}{q}=\frac{p/m}{q/m}$

In other words, a rational number remains unchanged if we divide its numerator and denominator by the same non-zero integer.

Equivalent Rational Numbers

If $\frac{p}{q}$ is a rational number and m is a non-zero integer, then  $\frac{p\times m}{q\times m}$ is a rational number equivalent to $\frac{p}{q}$.

For example, the equivalent of the rational number $\frac{3}{4}$ will be

$\frac{3\times 2}{4\times 2}=\frac{6}{8}, \,\frac{3\times 3}{4\times 3}=\frac{9}{12}$ and so on.

Standard Form of a Rational Number

We know that a number of the form $\frac{p}{q}$ or a number that can be expressed in the form of $\frac{p}{q}$, where p and q are integers and q ≠ 0 is called a rational number. So how do we define the standard form of a rational number?

Definition – A rational number $\frac{p}{q}$ is said to be in the standard form if q is positive and the integers p and q have no common divisor other than 1.

In order to express a given rational number in the standard form, the following steps should be followed:

Step 1 – Check whether the given number is in the form of $\frac{p}{q}$ . i.e. a rational number.

Step 2 – See whether the denominator of the rational number is positive or not. If it is negative, multiply or divide the numerator as well as the denominator by -1 so that the denominator becomes positive.

Step 3 – Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.

Step 4 – Divide the numerator and the denominator of the given rational number by the GCD (HCF) obtained in step III. The rational number so obtained is the standard form of the given rational number.

Let us understand the above steps with the help of some examples.

Example 1: Express the following rational numbers in standard form

a) $\frac{-8}{28}$

b) $\frac{-12}{30}$

Solution 

a) We have been given the rational number $\frac{-8}{28}$ and we need to express it in its standard form.

Let us find our answer using the above steps. We can see that the number is given to us in the form of $\frac{p}{q}$. Therefore, we can move to step 2.

The next step is to check whether the denominator of the rational number is positive or not. We can see that the number in the denominator is 28 which, is a positive number. In order to express it in standard form, we must divide its numerator and the denominator by the greatest common divisor of 8 and 28.

The greatest common divisor of 8 and 28 is 4.

Therefore,

Dividing the numerator and the denominator of $\frac{-8}{28}$ by 4, we get

$\frac{-8}{28}=\frac{-8/4}{28/4}=\frac{-2}{7}$ 

Hence, the standard form of $\frac{-8}{28}$ is $\frac{-2}{7}$

b) The number is given to us $\frac{-12}{-30}$

Again, we will find our answer using the above steps. We can see that the number is given to us in the form of $\frac{p}{q}$ . Therefore, we can move to step 2.

The next step is to check whether the denominator of the rational number is positive or not. We can see that the number in the denominator is -30 which, is a negative number.

Therefore, first, we will have to make the denominator positive. In order to do so, we will multiply the numerator as well as the denominator by -1.

Hence, multiplying the numerator and the denominator by -1 we get,

$\frac{p}{q}=\frac{(-12)\times (-1)}{(-30)\times(-1)}=\frac{12}{30}$ 

Now, we have a positive denominator; therefore let us move to the next step.

In order to express it in standard form, we must divide its numerator and the denominator by the greatest common divisor of 12 and 30.

The greatest common divisor of 12 and 30 is 6.

Therefore,

Dividing the numerator and the denominator of  $\frac{12}{30}$ by 6, we get

$\frac{12}{30}=\frac{12/6}{30/6}=\frac{2}{5}$

Hence, the standard form of $\frac{-12}{-30}$ is $\frac{2}{5}$

Operations on Rational Numbers

Let us now learn how to perform different mathematical operations on two or more rational numbers.

Addition of Rational Numbers

The addition of rational numbers is carried out in the same way as that of fractions. If two rational numbers are to be added we first express each one of them as rational numbers with a positive denominator. For addition purposes, we divide the rational numbers into two categories, namely,

  1. Rational Numbers with the same denominator
  2. Rational Numbers with different denominator

Rational Numbers with the same denominator

In order to add two rational numbers with the same denominator, we follow the following steps:

  1. Obtain the numerators of the two given rational numbers and their common denominator
  2. Add the numerators obtained in the first step.
  3. Write a rational number whose numerator is the sum obtained in the second step and the denominator is the common denominator of the given rational numbers.

Let us understand this through an example.

Example

Suppose we want to add the rational numbers $\frac{3}{5}$ and $\frac{13}{5}$

Solution

Here we can see that both the rational numbers have the same denominator, i.e. 5.

Therefore, we go by the above-defined steps.

We check the numerators of both the rational numbers. They are 3 and 13.

Then, we add these numerators and get 3 + 13 = 16.

Now, we write the sum of these rational numbers as $\frac{16}{5}$

Hence, $\frac{3}{5}+\frac{13}{5}=\frac{16}{5}$

Now, we have understood how to add two rational numbers having the same denominator. What if the denominators are different? Let us find out

Rational Numbers with different denominator

To find the sum of two rational numbers which do not have the same denominator, we will follow the following steps:

  1. Obtain the rational numbers and see whether their denominators are positive or not. If the denominator of one ( or both ) of the numbers is negative, rewrite it so that the denominator becomes positive.
  2. Obtain the denominators of the rational numbers in the first step.
  3. Find the LCM of the denominator obtained in the previous step.
  4. Express each one of the rational numbers in the first step so that the LCM obtained in step 2 becomes their common denominator.
  5. Write a rational number whose numerator is equal to the sum of the numerators of the rational numbers obtained in the fourth step and denominators as the LCM obtained in the third step.
  6. The rational number obtained in the fifth step is the required form.

Let us understand this through an example.

Example

Suppose we want to add $\frac{5}{12}$ and $\frac{3}{8}$

Solution

We have been given two rational numbers $\frac{5}{12}$ and $\frac{3}{8}$

Here we can clearly see that the denominators of the given numbers are positive. Also, the denominators are different.

So, we take the LCM of the denominators i.e. LCM of 12 and 8.

We know that the LCM of the numbers 12 and 8 will be 24.

So, now we express $\frac{5}{12}$ and $\frac{3}{8}$ into the forms in which both of them have the same denominator 24. We get

$\frac{5}{12}=\frac{5\times 2}{12\times2}=\frac{10}{24}$ 

$\frac{3}{8}=\frac{3\times 3}{8\times3}=\frac{9}{24}$ 

Now, we can see that we have the denominators with the same denominator. So we add the numerators and get the rational numbers according to step 5. We now have,

$\frac{5}{12}+\frac{3}{8}=\frac{10}{24}+\frac{9}{24}=\frac{19}{24}$ 

 Hence,  $\frac{5}{12}+\frac{3}{8}=\frac{19}{24}$

Subtraction of Rational Numbers

If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers, then subtracting $\frac{c}{d}$ from $\frac{a}{b}$ means adding inverse (negative) of  $\frac{c}{d}$ to $\frac{a}{b}$. The subtraction of $\frac{c}{d}$ from $\frac{a}{b}$ is written as $\frac{a}{b}-\frac{c}{d}$

Thus, we have

$\frac{a}{b}-\frac{c}{d}=\frac{a}{b}+(\frac{-c}{d})$  [because the additive inverse of $\frac{c}{d}$ is $\frac{-c}{d}$ ]

Let us understand this through an example.

Example

Subtract $\frac{3}{4}$ from $\frac{5}{6}$

Solution

The additive inverse of $\frac{3}{4}$ is $\frac{-3}{4}$

Therefore,

To find the value of $\frac{5}{6}-\frac{3}{4}$ 

We first make the denominators the same.

LCM of 6 and 4 is 12.

Therefore

$\frac{5}{6}=\frac{5\times 2}{6\times2}=\frac{10}{12}$ and

$\frac{3}{4}=\frac{3\times 3}{4\times3}=\frac{9}{12}$ 

Therefore,

$\frac{5}{6}-\frac{3}{4}=\frac{10}{12}-\frac{9}{12}=\frac{10-9}{12}=\frac{1}{12}$

Hence, $\frac{5}{6}-\frac{3}{4}=\frac{1}{12}$

Multiplication of Rational Numbers

Multiplication of rational numbers is similar to the multiplication of fractions. To obtain the product of two or more rational numbers, we multiply the numerators with the numerators and the denominator with the denominators. The following steps are involved in multiplying two or more rational numbers:

  1. First, we need to multiply all the numerators.
  2. Next, we need to multiply all the denominators
  3. Lastly, we need to simplify the rational number, if required

Therefore, we can say that

Product of rational numbers  = $\frac{Product\,of\,the\,Numerators}{Product\,of\,the\,denominators}$

Thus if $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers then,

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

Let us understand this through an example.

Example

Suppose we want to find the value of $\frac{3}{4}\times \frac{5}{7}$

Solution

We have to find the value of $\frac{3}{4}\times \frac{5}{7}$

According to the given formula for multiplication we have,

$\frac{3}{4}\times \frac{5}{7}=\frac{3\times 5}{4\times 7}=\frac{15}{28}$

Hence, $\frac{3}{4}\times \frac{5}{7}=\frac{15}{28}$

Reciprocal of a Non-zero Rational Number

What do we mean by reciprocal of a rational number? Let us find out.

For every, non-zero rational number $\frac{a}{b}$, there exists a rational number $\frac{b}{a}$ such that

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

The rational number $\frac{b}{a}$ is called the multiplicative inverse or reciprocal of $\frac{a}{b}$ and is denoted by $(\frac{a}{b})^{-1}$

Division of Rational Numbers

If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers such that $\frac{c}{d}\neq 0$ then the result of dividing $\frac{a}{b}$ by $\frac{c}{d}$ is the rational number obtained on multiplying $\frac{a}{b}$ by the reciprocal of $\frac{c}{d}$.

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

Let us understand it through an example

Example

Suppose we want to find the value of $\frac{3}{5} / \frac{4}{25}$

First, we will find the reciprocal of $\frac{4}{25}$ which is $\frac{24}{4}$

Therefore, $\frac{3}{5} / \frac{4}{25}=\frac{3}{5}\times\frac{25}{4}=\frac{15}{4}$

Hence, $\frac{3}{5} / \frac{4}{25}=\frac{15}{4}$

Solved Examples

Example 1    

The product of two rational numbers is $\frac{-28}{81}$. If one of the numbers is $\frac{14}{27}$, find the other.

Solution              

We have been given that the product of two rational numbers is $$ and one of the numbers is $$.

Let the other number be “p”. Therefore, we will have

$p\times\frac{14}{27}=\frac{-28}{81}$

$p=\frac{-28}{81} /\frac{14}{27}$

$p=\frac{-28}{81}\times\frac{27}{14}=\frac{-2}{3}$

Hence, the other number is $\frac{-2}{3}$

Example 2    

The sum of two rational numbers is $\frac{-3}{5}$. If one of the numbers is $\frac{-9}{20}$, find the other.

Solution              

We have been given that the sum of two rational numbers is $\frac{-3}{5}$ and one of the numbers is $\frac{-9}{20}$.

Let the other number be “p”.

Therefore,

$p+\frac{-9}{20}=\frac{-3}{5}$

⇒ $p=\frac{-3}{5}-(\frac{-9}{20})$

⇒ $p=\frac{-3}{5}+\frac{9}{20}$

⇒ $p=\frac{(-3)\times4+(9\times1)}{20}$

⇒ $p=\frac{(-12)+9}{20}$

⇒ $p=\frac{-3}{20}$

Hence, the other number is $\frac{-3}{20}$

Remember

  1. A number of the form $\frac{p}{q}$ or a number that can be expressed in the form of $\frac{p}{q}$, where p and q are integers and q ≠ 0 is called a rational number.
  2. If $\frac{p}{q}$ is a rational number and m is a non-zero integer, then $\frac{p\times m}{q\times n}$ is a rational number equivalent to $\frac{p}{q}$.
  3. A rational number $\frac{p}{q}$ is said to be in the standard form if q is positive and the integers p and q have no common divisor other than 1.
  4. The addition of rational numbers is carried out in the same way as that of fractions.
  5. If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers, then subtracting $\frac{c}{d}$ from $\frac{a}{b}$ means adding inverse ( negative ) of  $\frac{c}{d}$ to $\frac{a}{b}$. The subtraction of $\frac{c}{d}$ from $\frac{a}{b}$ is written as $\frac{a}{b}-\frac{c}{d}$
  6. To obtain the product of two or more rational numbers, we multiply the numerators with the numerators and the denominator with the denominators.
  7. Product of rational numbers  = $\frac{Product\, of\, the\, Numerators}{Product\,of\,the\,denominators}$
  8. The rational number $\frac{b}{a}$ is called the multiplicative inverse or reciprocal of $\frac{a}{b}$ and is denoted by $(\frac{a}{b})^{-1}$
  9. If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers such that $\frac{c}{d}\neq 0$ then the result of dividing $\frac{a}{b}$ by $\frac{c}{d}$ is the rational number obtained on multiplying $\frac{a}{b}$ by the reciprocal of $\frac{c}{d}$.
  10. $\frac{a}{b}/\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$