Rational Numbers
- Introduction to Rational Numbers
- Examples of Rational Numbers
- Classification of Rational Numbers
- Types of Rational Numbers
- Decimals in the Form of Rational Numbers
- Operations on Rational Numbers
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction to Rational Numbers
Any number that can be written in the form of `p/q` (fraction) is called rational number. Here `p` and `q` are the integers and `q` can never be zero `(``q`` ≠ 0)`.
A rational number and a fraction are the same, except for the fact that a rational number can have a negative numerator or denominator, while fractions can only have positive integers in the numerator and denominator.
Note: `q` can never be zero but `p` can be zero.
For example: `0/2` is a rational number but `2/0` is not a rational number.
Examples of Rational Numbers
Here are some examples of rational numbers:
- `1` can be a rational number since we can write it in the form of `p/q` `(1/1=1)`.
- `1/2` is a rational number since it is already in the form of a fraction.
- `23` is a rational number since `23` can be written as `23/1`.
- `0` is a rational number since `0` can be written as `0/1`.
- `5/3` is also in the form of `p/q`, therefore it is a rational number.
- `-0.5` is a rational number since `-0.5` can be written as `- 1/2`.
- `\sqrt 4` is a rational number, since `\sqrt4 = 2`, and `2` can be written as a fraction `(2/1)`.
- `1.3\overline{45}` is a rational number, since it can be written as `74/55`.
- `2.66666…` is a rational number, since it can be written as `8/3`.
Classification of the Rational Numbers
Let’s look at the diagram given below:
The given diagram represents the following points:
- Natural numbers (`N`) are the smaller part of the whole numbers (`W`).
- Whole numbers (`W`) are the smaller part of the integers (`Z`).
- Integers (`Z`) are the smaller part of the rational numbers (`Q`).
Thus, we can conclude that “The rational numbers (`Q`) contains all the integers (`Z`), whole numbers (`W`) and the natural numbers (`N`).”
Types of Rational Numbers
There are two types of rational numbers, which are given below:
- Positive rational numbers:
When a rational number has both numerator and denominator positive or negative, it is called a positive rational number.
For example: `2/3, (-2)/(-3), (-12)/(-5), 5/4`
- Negative rational numbers:
When a rational number has either a numerator or denominator negative, it is called a negative rational number.
For example: `(-3)/5, 2/(-7), - 5/7`
Decimals in the Form of Rational Numbers
Since decimal numbers can be written in the form of fractions with denominators `10, 100, 1000,` and so on, we can conclude that decimals are also rational numbers.
Note: Decimals that cannot be expressed in the form of fractions will not be classified as rational numbers (`Q`).
Example: Non-terminating decimals `(\sqrt 2)`
Operations on Rational Numbers
`1`. Addition: There are two cases for adding rational numbers. Let’s discuss them.
- When denominators are equal for the rational numbers.
In this case, we’ll add the numerator directly and keep the denominator the same.
For example: `2/5 + 7/5`
- When the denominators are unequal for the rational numbers.
To perform the calculation with unequal denominators, we need to follow some steps:
Step `1`: Find the LCM of the unequal denominators.
Step `2`: Convert the given rational number into an equivalent rational number with the LCM as the new denominator.
Step `3`: After obtaining the same denominator, simply add up the numerator while keeping the LCM as the common denominator.
For example: `3/5 + 2/3`
Addition with different denominators
\[\color{#38761d}{\frac{3}{5}} + \color{#fb8500}{\frac{2}{3}}\]
Step `1`: `\text{LCM}(\color{#38761d}5,\color{#fb8500}3) = \color{#6F2DBD}15`
Step `2`: Find the equivalent rational numbers with denominator as `15`
\[\color{#38761d}{\frac{3}{5}}=\frac{\color{#38761d}{9}}{\color{#6F2DBD}{15}}\text{ and }\color{#fb8500}{\frac{2}{3}} =\frac{\color{#fb8500}{10}}{\color{#6F2DBD}{15}}\]
Step `3`: Add the numerators while keeping the denominator same
\[\frac{\color{#38761d}{9}}{\color{#6F2DBD}{15}}+\frac{\color{#fb8500}{10}}{\color{#6F2DBD}{15}}=\color{#6F2DBD}{\frac{19}{15}}\]
`2`. Subtraction: The rules are the same for subtraction as for addition.
For example: `7/4-5/4` and `2/3-1/4`
Subtraction with different denominators
\[\color{#38761d}{\frac{2}{3}} - \color{#fb8500}{\frac{1}{4}}\]
Step `1`: `\text{LCM}(\color{#38761d}3,\color{#fb8500}4) = \color{#6F2DBD}12`
Step `2`: Find the equivalent rational numbers with denominator as `12`
\[\color{#38761d}{\frac{2}{3}}=\frac{\color{#38761d}{8}}{\color{#6F2DBD}{12}}\text{ and }\color{#fb8500}{\frac{1}{4}} =\frac{\color{#fb8500}{3}}{\color{#6F2DBD}{12}}\]
Step `3`: Subtract the numerators while keeping the denominator same
\[\frac{\color{#38761d}{8}}{\color{#6F2DBD}{12}}-\frac{\color{#fb8500}{3}}{\color{#6F2DBD}{12}}=\color{#6F2DBD}{\frac{5}{12}}\]
`3`. Multiplication: To multiply two rational numbers, we‘ll multiply the numerator by the numerator, and the denominator by the denominator.
For example: `2/3\times 1/5`
`4`. Division: To divide two rational numbers, we need to follow some steps as given below.
Step `1`: Find the reciprocal of the second rational number.
Step `2`: Then multiply the first rational number by the reciprocal of the second rational number.
For example: `2/3\div 1/5`
Solved Examples
Example `1`: Subtract: `7/6 - 5/6`
Solution: `7/6-5/6=(7-5)/6=2/6=1/3`
Example `2`: Multiply: `5/6\times 7/2`
Solution: `5/6\times 7/2=(5\times 7)/(6\times 2)=(35)/(12)`
Example `3`: Divide: `2/5\div 7/3`
Solution: `2/5\div 7/3=2/5\times 3/7=(2\times 3)/(5\times 7)=6/(35)`
Example `4`: Write `0.2` as a rational number.
Solution: `0.2=2/(10)`
Example `5`: Which one is rational out of `1.34` and `1.341592….`
Solution: `1.34` can be written as `\frac{134}{100}` which is in fraction form. Therefore `1.34` is a rational number. On the other hand, `1.341592….` is a non-terminating non-repeating decimal number that can not be expressed as a fraction. Thus it is not a rational number.
Example `6`: Why `2/0` is not a rational number?
Solution: If you divide any integer with zero `(0)` this will give you an infinite value and we can not define the infinite value. Thus `2/0` is not a rational number.
Practice Problems
Q`1`. Which one of the following is not a rational number?
- `23`
- `- 67`
- `9.0005`
- `0.92456…`
Answer: d
Q`2`: Identify the rational number from the following options:
- \( \sqrt{3} \)
- \( \frac{5}{7} \)
- \( \pi \)
- `-\sqrt{5}`
Answer: b
Q`3`. \( \frac{2}{3} + \left(-\frac{5}{6}\right) =?\)
- \(\frac{1}{6}\)
- \(\frac{-7}{18}\)
- \(\frac{-1}{6}\)
- \(\frac{-1}{3}\)
Answer: c
Q`4`. \( \frac{4}{5} \times \left(\frac{3}{8}\right)=? \)
- \(\frac{3}{10}\)
- \(\frac{15}{40}\)
- \(\frac{12}{25}\)
- \(\frac{5}{8}\)
Answer: a
Q`5`. Find a rational number between \( -1 \) and \( 0 \).
- \(-\frac{1}{2}\)
- \(\frac{1}{2}\)
- \(-\frac{5}{2}\)
- \(-\frac{4}{3}\)
Answer: a
Q`6`. Divide \( \frac{5}{6} \) by \( \frac{2}{3} \).
- \( \frac{5}{9} \)
- \( \frac{5}{4} \)
- \( \frac{3}{4} \)
- \( \frac{1}{2} \)
Answer: b
Frequently Asked Questions
Q`1`: What is a rational number?
Answer: A rational number is any number that can be expressed as the quotient or fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\) is not equal to zero.
Q`2`. How do you add or subtract rational numbers?
Answer: To add or subtract rational numbers, find a common denominator and then perform the operation on the numerators while keeping the denominator constant.
Q`3`. Are integers also rational numbers?
Answer: Yes, integers are rational numbers. Every integer \(n\) can be expressed as the fraction \(\frac{n}{1}\).
Q`4`. What is the difference between a rational number and an irrational number?
Answer: Rational numbers can be expressed as fractions, while irrational numbers cannot. Irrational numbers have non-repeating, non-terminating decimal expansions.