# Chapter 1 – Rational Numbers

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You might have come across several applications of rational numbers in your daily life without being aware of their importance and value in mathematics. In simpler words, rational numbers are any numbers that are involved in various mathematical applications, for example, addition, division, subtraction, and multiplication. You will also learn about different kinds of properties of rational numbers.

#### Natural Numbers

• They are the set numbers starting from 1 and ending at infinity.
• This set is denoted by  ‘N’.
Closure propertyAppliesDoesn’t applyAppliesDoesn’t apply
Commutative propertyAppliesDoesn’t applyAppliesDoesn’t apply
Associative propertyAppliesDoesn’t applyAppliesDoesn’t apply

#### Whole Numbers

• They are the set of numbers starting from 0 and ending at infinity.
• The set is denoted by a ‘W’.
Closure propertyAppliesDoesn’t applyAppliesDoesn’t apply
Commutative propertyAppliesDoesn’t applyAppliesDoesn’t apply
Associative propertyAppliesDoesn’t applyAppliesDoesn’t apply

#### Integers

• They are a set of natural numbers with the addition of their negatives.
• The set is denoted by a ‘Z’.
Closure propertyAppliesAppliesAppliesDoesn’t apply
Commutative propertyAppliesDoesn’t applyAppliesDoesn’t apply
Associative propertyAppliesDoesn’t applyAppliesDoesn’t apply

#### Rational Numbers

In this section, a proper introduction to rational numbers will be given to you. Rational numbers can be represented in a p/q form, where both p and q are integers and q is unequal to zero.

This set of numbers is denoted by ‘Q’.

For example- -6/8,  2/5, -5,and 4

Properties of Rational Numbers

Below are some critical properties of rational numbers:

1. Closure Property

For any two random rational numbers, say ‘a’ and ‘b’, a∗b=c∈Q.

• * is any binary operation (multiplication, addition, subtraction, and division)
• C is the product of the binary operation applied between a and b.
• All three numbers should belong to the set of rational numbers

Closure property pertains to rational numbers under addition.

For example:

3/5+ 1/2= (6+5)/10 = 11/10

• Subtraction

Closure property pertains to rational numbers under subtraction.

For example:

4/2- 1/2 = (4-1)/2 = 3/2

• Multiplication

Closure property pertains to rational numbers under multiplication.

For example:

4/7 x -2/5 = -8/35

• Division

Closure property does not pertain to rational numbers under addition.

For rational number s, s÷0= not defined.

2. Commutative Property

For any two random rational numbers, say ‘a’ and ‘b’, a∗b= b*a. Therefore, the result of the equation should be constant regardless of the order of the operands.

Commutative property pertains to rational numbers under addition because for any two rational numbers a and b-

(a+b) = (b+a)

• Subtraction

The commutative property does not pertain to rational numbers under subtraction because for any two rational numbers a and b-

(a-b) is unequal to (b-a)

• Multiplication

Commutative property pertains to rational numbers under multiplication because for any two rational numbers a and b-

(a x b) = (b x a)

• Division

The commutative property does not pertain to rational numbers under addition because for any two rational numbers a and b-

(a÷b) is unequal to (b÷a)

3. Associative Property

For any three random rational numbers, say ‘a’, ‘b’ and ‘c’, (a∗b)∗c=a∗(b∗c). Therefore, the result of the equation should be constant regardless of the order of the operands.

Associative property pertains to rational numbers under addition because for any three rational numbers a, b and c-

(a+b)+c = a+(b+c)

• Subtraction

The associative property does not pertain to rational numbers under subtraction because for any two rational numbers a, b and c-

(a−b)−c≠a−(b−c) because (a-b)-c = a-b-c whereas a-(b-c) = a-b+c.

• Multiplication

Associative property pertains to rational numbers under multiplication because for any three rational numbers a, b and c-

(a×b)×c=a×(b×c)

• Division

The associative property does not pertain to rational numbers under addition because for any three rational numbers a, b and c-

(a÷b)÷c≠(a÷b)÷c

4. Distributive Property

For any three rational numbers a, b and c-

a(b+c)=ab+aca(b−c)=ab−ac

#### The Negative of Rational Numbers

When zero is added to a random rational number, the result remains constant. Mathematically it is represented as-

For a rational number p/q, p/q+ 0 = p/q

Zero, in this case, is referred to as additive identity.

If (p/q)+(−p/q)=(−p/q)+(p/q)=0 then the additive inverse or the negative of a rational number pq is -pq.

#### Reciprocal of a Rational Number

When one is multiplied to a random rational number, the result remains constant. Mathematically it is represented as-

For a rational number p/q, p/q x 1 = p/q

1, in this case, is referred to as multiplicative identity.

If p/q x r/s = 1 then r/s is the multiplicative inverse of p/q. Also, p/q is the reciprocal of the multiplicative inverse r/s.

#### Representation of rational numbers on the number line

Representation of rational numbers on the number line can be divided into two steps.

Step 1: Equally divide the distance between the two consecutive integers in the ‘n’ number of parts.

Step 2: label the rational numbers on the number line until it contains the number you have to mark.

#### Rational numbers between rational numbers

The number of rational numbers between rational numbers is indefinite.

How to find out rational numbers between rational numbers:

Method 1

Ensure that the two given natural numbers have the same denominator. Once that is settled, you can pick out any rational number that lies between them.

Method 2

We can invariably find a rational number between two rational numbers by mathematically calculating their midpoint or mean.

To sum up

We hope that you understood the nuances of rational numbers and all the properties associated with them.

#### FAQs

1. What is a rational number in maths?

Rational numbers are any numbers that are involved in various mathematical applications, for example, addition, division and etc.

1. What are five examples of rational numbers?

-7, 5/6, -7/9, 3, 9.

1. Is zero a rational number?

Yes, zero is a rational number as it can be written as p/q, where p= 0 and q= non zero.

1. Is 5 a rational number?

Yes. 5 can be written as 5/1.

1. How can you identify a rational number?

Any number that can be written as a fraction, where the denominator is unequal to zero.

1. Is 2/3  a rational number?

Yes. 2/3 is written in the p/q format and 3 unequal to zero.

MSVgo is an excellent educational application that has a video library containing a myriad of videos explaining through animations and visualization on any topic from any subject you desire. Head over to the app to learn the concepts of rational numbers and much more.

### High School Physics

• Alternating Current
• Atoms
• Communication Systems
• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
• Electricity
• Electromagnetic Induction
• Electromagnetic Waves
• Electrons and Photons
• Electrostatic Potential and Capacitance
• Fluid Pressure
• Force and Acceleration
• Force And Laws Of Motion
• Gravitation
• Internal Energy
• Kinetic Theory
• Law of motion
• Light – Reflection And Refraction
• Magnetic Effects Of Electric Current
• Magnetism and Matter
• Management Of Natural Resources
• Mechanical properties of Fluids
• Mechanical properties of Solids
• Motion
• Motion in a plane
• Motion in a straight line
• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
• Oscillations
• Our Environment
• Paths of Heat
• Physical world
• Ray optics and optical instruments
• Semiconductor Devices
• Semiconductor Electronics: Materials, Devices and Simple Circuits
• Simple Machines
• Sound
• Sources Of Energy
• Specific and Latent Heats
• Spherical Mirrors
• Static Electricity
• Systems of Particles and Rotational motion
• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
• Biomolecules
• Carbon And Its Compounds
• Carboxylic acids and Acid Derivatives
• Chemical Bonding and Molecular Structures
• Chemical Energetics
• Chemical Equilibria
• Chemical Kinetics
• Chemical Reactions And Equations
• Chemical Reactions and Their Mechanisms
• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
• Cyanide, Isocyanide, Nitro compounds and Amines
• Electrochemistry
• Electrolysis
• Elements, Compounds and Mixtures
• Environmental Chemistry
• Equilibrium
• Ethers and Carbonyl compounds
• Haloalkanes and Haloarenes
• Hydrocarbons
• Hydrogen
• Ideal solutions
• Introduction to Organic Chemistry
• Ionic equilibria
• Matter
• Matter Around Us
• Matter In Our Surroundings
• Metallurgy
• Metals And Non-Metals
• Mole Concept and Stoichiometry
• Natural Resources
• Organic Chemistry – Basic Principles
• Periodic Classification of Elements
• Physical and Chemical Changes
• Physical and Chemical Properties of Water
• Polymers
• Preparation, Properties and Uses of Compounds
• Principles and Processes of Isolation of Elements
• Redox Reactions
• Relative Molecular Mass and Mole
• States of Matter
• Structure Of The Atom
• Study of Compounds
• Study of Gas Laws
• Study of Representative Elements
• Surface Chemistry
• The d-block and f-block elements
• The Gaseous State
• The p-Block Elements
• The Periodic Table
• The s-Block Elements
• The Solid State
• Thermodynamics

### High School Biology

• Absorption and Movement of Water in Plants
• Anatomy of Flowering Plants
• Animal Kingdom
• Bacteria and Fungi-Friends and Foe
• Biodiversity and Conservation
• Biofertilizers
• Biological Classification
• Biomedical Engineering
• Biomolecules
• Biotechnology and its Applications
• Biotic Community
• Body Fluids and Circulation
• Breathing and Exchange of Gases
• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
• Cellular Respiration
• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
• Environmental Issues
• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
• Human Eye And Colourful World
• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
• Kingdom Protista
• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
• Electric Current And Its Effects
• Electricity And Circuits
• Elements and Compounds
• Fibre To Fabric
• Food production and management
• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
• Getting To Know Plants
• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
• Our Environment
• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
• Structure of Atom
• Synthetic Fibers And Plastics
• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time