Rational Numbers

Rational numbers are an essential part of mathematics, allowing us to represent parts of a whole and compare values precisely. They help us solve everyday problems like dividing resources or understanding proportions. Exploring rational numbers builds critical thinking skills and deepens our understanding of how numbers connect to the world around us.

Table Of Contents

What are Rational Numbers ?
Is zero a rational number?
Are all integers rational numbers?
Are all fractions and decimals considered rational numbers?
What are the properties of rational numbers?
How do rational numbers differ from irrational numbers?
How do you find the reciprocal of a rational number?
Related Topics:
Boost Maths Marks with Gradeup
Useful links:

What are Rational Numbers ?

Ratural numbers ( $\mathbb{Q}$ ) rational number is any number that can be expressed in the form $\frac {p} {q}$ , where ${p}$ and ${q}$ are integers and ${q} \neq{0}$ . This includes integers, fractions, and terminating or repeating decimals.

Is zero a rational number?

Yes, zero is a rational number because it can be expressed as $\frac {0} {1}$ , which fits the definition of a rational number.

Are all integers rational numbers?

Yes, all integers are rational numbers because any integer ${n}$ can be written as $\frac {n} {1}$ , satisfying the form $\frac {p} {q}$ .

Are all fractions and decimals considered rational numbers?

A fraction is a rational number only if both the numerator and denominator are integers, and the denominator is not zero.

Example: $\frac{3}{4}$ (rational, integers involved), $\frac {-2}{5}$ (rational, integers involved)

Counterexample: $\frac {\sqrt 2}{3}$ (not rational because $\sqrt{2}$ is irrational).

A decimal is a rational number only if it is …

Terminating: Ends after a finite number of digits.

Example: ${0.,5} = \frac{1}{2}$ and ${0,25} = \frac {1}{4}$

Repeating: Has a repeating pattern.

Example: ${0,333...} = \frac {1}{3}$ and ${0,142857...} = \frac {1}{7}$ .

What are the properties of rational numbers?

Rational numbers ( $\mathbb{Q}$ ) have several mathematical properties:

Closure: The set of rational numbers is closed under addition, subtraction, multiplication, and division (except division by zero). This means that performing any of these operations between two rational numbers always results in another rational number.

Addition example: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Subtraction example: $\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$

Multiplication example: $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$

Division example: $\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4}$
Commutative Property: Rational numbers are commutative under addition and multiplication, meaning the order of the numbers does not affect the result.

Addition Example: $\frac{2}{5} + \frac{3}{5} = {1}$ and $\frac{3}{5} + \frac{2}{5} = {1}$

Multiplication Example: $\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$ and $\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$
Associative Property: Rational numbers are associative under addition and multiplication, meaning the grouping of numbers does not affect the result.

Addition Example: $({a} + {b})+{c} = {a} + ({b}+{c})$

Multiplication Example: $({a} \times {b}) \times {c} = {a} \times ({b} \times {c})$
Distributive Property: Multiplication distributes over addition for rational numbers. This means that multiplying a number by the sum of two numbers is the same as multiplying each number individually and then adding the results.

Example: ${a} \times ({b}+{c})$ = $({a}\times {b}) + ({a} \times {c})$ .
Inverse Property:
Additive Inverse: Every rational number has an opposite such that their sum is 0.
Multiplicative Inverse: Every non-zero rational number has a reciprocal such that their product is 1.

Additive Inverse Example: The additive inverse of $\frac {3}{4}$ is $-\frac {3}{4}$ because $\frac {3}{4} + (-\frac {3}{4}) = {0}$

Multiplicative Inverse Example: The multiplicative inverse of $\frac {2}{5}$ is $\frac {5}{2}$ because $\frac {2}{5} \times \frac {5}{2} = {1}$

How do rational numbers differ from irrational numbers?

Rational numbers can be written as exact fractions with integer numerators and non-zero integer denominators.
In contrast, irrational numbers cannot be expressed as such fractions; their decimal expansions are non-terminating and non-repeating.

Examples of irrational numbers include $\pi$ and $\sqrt{2}$ .

How do you find the reciprocal of a rational number?

The reciprocal of a rational number $\frac{a}{b}$ (where ${a}$ and ${b}$ are integers, and ${b} \neq {0}$ ) is $\frac {b}{a}$ . Essentially, you swap the numerator and the denominator.

Example: the reciprocal of $\frac{3}{4} = \frac {4}{3}$

Note that zero does not have a reciprocal, as division by zero is undefined.