Rational Numbers Basics: Learn Types, Daily Life Examples, MCQs & Free Worksheets!🚀

Published: 13 Dec 2024

When we hear the term “number,” various types may come to mind—integers, decimals, fractions, and more. One important category among these is Rational Number. But what exactly is a rational number? This blog will explore rational numbers in detail, explaining their types and providing clear examples to make the concept easy to understand.

Table of Content

Introduction to Rational Numbers
What is a Rational Number?
1. Meaning of Rational Number
2. How to Write Rational Numbers
Definition of Rational Numbers
🌟Types of Rational Numbers🌟

Introduction to Rational Numbers

In mathematics, when one integer is divided by another, the result may or may not be an integer.

The Journey: Integers to Rational Numbers

Dividing $-14$ by $2$ gives the integer $-7$.
Dividing $-15$ by $2$ gives $\frac{-15}{2}$, which cannot be simplified further into an integer.

Numbers like $\frac{-15}{2}$, which can be expressed as the ratio of two integers, mathematicians introduced the concept of Rational Numbers.

Previous Knowledge Check

1. Natural Numbers:
Counting numbers starting from 1.
Example: $ 1, 2, 3, 4, \dots $

2. Whole Numbers:
Natural numbers including 0.
Example: $ 0, 1, 2, 3, \dots $

3. Even Numbers:
Exactly divisible by 2 without a remainder.
Example: $ 2, 4, 6, 8, \dots $

4. Odd Numbers:
Not divisible by 2.
Example: $ 1, 3, 5, 7, \dots $

5. Prime Numbers:
Greater than 1, divisible only by 1 and itself.
Example: $ 2, 3, 5, 7, 11, \dots $

6. Composite Numbers:
Greater than 1, with more than two factors.
Example: $ 4, 6, 8, 9, 10, \dots $

7. Integers:
Integers are the collection of whole and negative numbers without decimals or fractions.

What is a Rational Number?

Rational numbers are real numbers that can be written as a fraction, where both the numerator and denominator are integers, and the denominator is not zero. This includes whole numbers, integers, fractions, and decimals that either terminate or repeat.

Meaning of Rational Number

The word “rational” comes from the Greek word “ratio,” meaning a relationship between two numbers. Rational numbers are like fractions that represent these ratios, where both the numerator and denominator are integers.

How to Write Rational Numbers

A rational number can be written by dividing one integer by another integer. All numbers in the rational number set can be expressed as a ratio of two integers.

Rational Numbers and Integers

The set of rational numbers is a superset of the set of integers because every integer can be written as a rational number.

Definition of Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers in the form $\frac{p}{q}$, where $q \neq 0$.
Mathematically:
$ \text{Rational number} = \frac{p}{q} $
$ \text{ where } p, q \in \mathbb{Z} \text{ and } q \neq 0. $

Set of Rational Numbers

The set of rational numbers includes natural numbers, whole numbers, integers, fractions, and both terminating and repeating decimals.
\[
Q = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z}, \; q \neq 0 \right\}
\]

Rational Numbers and the Symbol Q

A rational number is denoted by the symbol $Q$, which comes from the word “Quotient.”

Examples of Rational Numbers:

Fractions: $ \dfrac{2}{3}, \quad \dfrac{-5}{8} $
Integers: $ -4 $ (which can be written as $ \frac{-4}{1} $)
Terminating decimals: $ 0.5, \quad 2.75 $
Repeating decimals: $ 0.\overline{3}, \quad 1.\overline{6} $

Daily Life Examples of Rational Numbers

Fractions
When you drink half a glass of water, you consume $\dfrac{1}{2}$ of the glass.
If you cut a cake into 4 equal pieces and eat one, you have eaten $\frac{1}{4}$ of the cake.
Integers
A building with 5 floors above and 2 floors underground can be represented as +5 and -2 respectively.
If the temperature is -3°C, it is a rational number because it can be written as $\dfrac{-3}{1}$.
Terminating Decimals
The price of a product like $4.75 is a terminating decimal because it can be written as $\dfrac{475}{100}$.
A 2.5-liter bottle of water represents a rational number since it can be expressed as $\dfrac{25}{10}$.
Repeating Decimals
If a car consumes $ \dfrac{1}{3} $ liters of fuel per kilometer, it means the fuel efficiency is $0.333\ldots$ or $ 0.\overline{3} $ liters per km.
If a cyclist rides $ \frac{10}{3} $ km in an hour, the speed is 3.333… km/h.

Positive and Negative Rational Numbers

Positive Rational Numbers	Negative Rational Numbers
The numerator and denominator both have the same sign.	The numerator and denominator have opposite signs.
Positive rational numbers are always greater than 0.	Negative rational numbers are always less than 0.
General Form: $ \dfrac{p}{q}$, $ \dfrac{-p}{-q}$	General Form: $ \dfrac{-p}{q}$, $ \dfrac{p}{-q}$
Examples: $ \dfrac{11}{7}$, $ \dfrac{-11}{-7}$	Examples: $ \dfrac{-11}{7}$, $ \dfrac{11}{-7}$

Standard form of a Rational Number

A rational number is in standard form if:

The numerator and denominator have no common factors other than 1.
The denominator is positive.
Example:
$\frac{15}{-35}=-\frac{3}{7} $
Thus, $-\frac{3}{7} $ is in standard form because it has no common factor and the denominator is also positive.

🌟Types of Rational Numbers🌟

There are 5 types of rational numbers you see every day. Let’s take a quick look at each one and explore them with examples!

Fractions in the form of $\frac{p}{q}$, where $p$ & $q$ are integers and $q \neq 0$, are rational numbers.
Examples:
$ \frac{8}{-3}, -\frac{6}{11} $
Natural, Whole numbers and integers can be written as rational numbers with a denominator of $1$.
Examples:
$5$: Expressed as $\frac{5}{1}$
$-6$: Expressed as $\frac{-6}{1}$
Zero can be written as a rational number with any non-zero denominator.
Examples:
$0$: Expressed as $\frac{0}{1}, \frac{0}{7}, \frac{0}{-5}$
Perfect roots of numbers are rational numbers if they simplify to integers.
Examples:
$\sqrt{25} = 5 = \frac{5}{1}$
$\sqrt[3]{27} = 3 = \frac{3}{1}$
$\sqrt[4]{256} = 4 = \frac{4}{1}$
$\sqrt[6]{64} = 2 = \frac{2}{1}$
Terminating and repeating decimals can also be written as rational numbers.
Examples:
$0.111 $
$-0.19 $
$3.333\ldots $

Mixed Fractions and Conversion

Mixed Fraction
A mixed fraction is a combination of a whole number and a proper fraction.
Conversion to Improper Fraction:
To convert a mixed fraction to an improper fraction, Multiply the whole number by the denominator of the fraction and then add to the numerator.
Example:
Convert $ 3 \frac{2}{5} $ to an improper fraction:
$ 3 \frac{2}{5} = \frac{(3 \times 5) + 2}{5} $
$ 3 \frac{2}{5} = \frac{15 + 2}{5} $
$ 3 \frac{2}{5} = \frac{17}{5} $
Thus, $ 3 \frac{2}{5} = \frac{17}{5} $.

Here are simple answers to the questions:

What happens when you divide one number by another, and the result is not a whole number? Could such numbers belong to a special category?
Yes! Such numbers belong to a category called rational numbers, which can be written as fractions.
Can a number be both a fraction and an integer at the same time? If so, what would you call it?
Yes! For example, $4$ can be written as $\frac{4}{1}$ , making it a rational number and an integer at the same time.
Are all decimals rational numbers, or do some decimals hide in a different category of numbers?
Not all decimals are rational numbers. Decimals that terminate or repeat are rational, but decimals that go on forever without repeating belong to irrational numbers.
What makes a rational number “rational”? Is it simply about being able to write it as a fraction, or is there more to it?
A rational number is called “rational” because it represents a ratio of two integers (numerator and denominator). It’s all about the fraction!

These answers keep the explanations simple and to the point. Let me know if you’d like to adjust them further!

Fractions
When you drink half a glass of water, you consume $\dfrac{1}{2}$ of the glass.
If you cut a cake into 4 equal pieces and eat one, you have eaten $\frac{1}{4}$ of the cake.
Integers
A building with 5 floors above and 2 floors underground can be represented as +5 and -2 respectively.
If the temperature is -3°C, it is a rational number because it can be written as $\dfrac{-3}{1}$.
Terminating Decimals
The price of a product like $4.75 is a terminating decimal because it can be written as $\dfrac{475}{100}$.
A 2.5-liter bottle of water represents a rational number since it can be expressed as $\dfrac{25}{10}$.
Repeating Decimals
If a car consumes $ \dfrac{1}{3} $ liters of fuel per kilometer, it means the fuel efficiency is $0.333\ldots$ or $ 0.\overline{3} $ liters per km.
If a cyclist rides $ \frac{10}{3} $ km in an hour, the speed is 3.333… km/h.