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Chapter 1 – Real Numbers

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


Real numbers is considered one of the most pivotal chapters in mathematics from the point of view of exams. Practically speaking, any number that comes in your mind can be termed as a real number. But believe it or not, real numbers can be rational numbers as well. Real numbers are easy to comprehend and understand. When the length of any number needs to get measured, the real number gets called for action. So let us continue our discussion and explore the world of real numbers together. 

Real Numbers

The arrangement of rational and irrational numbers is called real numbers. It encompasses all the numbers of the number lines. It tends to be positive, negative, and even zero. Yet, do you realize why real numbers are called real numbers? Simple, since they are not a piece of imaginary numbers.

Examples of Real Number- 1, 0, -45, 1/5, 178

Euclid was proclaimed as one of the renowned mathematicians who facilitated the new methodology to study geometry. One of his most profound contributions to number theory was the introduction of Euclid’s Division Lemma. 

According to Euclid’s Division Lemma theory, on the off chance that you have any two positive integers like a and b, in such case, there surfaced two novel integers that fulfil the following condition-

a = bq + r = 0 ≤ r < b

The Euclidean division algorithm gets hinged on Euclid’s division lemma. Whenever you have to calculate HCF (Highest Common Factor), Euclid’s division algorithm gets employed. HCF gets called the largest number that gets divided by two positive integers that mean when divided; it results in zero as a reminder.

The Fundamental Theorem of Arithmetic gets profoundly known as the unique factorization theorem in number theory. It expresses that each number greater than 1 can either be a prime number itself or can be part of the product of prime numbers. For instance: 

1200 = 24 . 3 . 52

Revisiting Rational Numbers and Their Decimal Expansions means the decimal expansion of any rational number can either be terminating or non-terminating repeating. 

  • Terminating Decimal Expansion- When in the process of the division after the finite number of steps, any number that gets ceased is called Terminating Decimal Expansion. For example – 6.25
  • Non- terminating Decimal Expansion- When no number gets ceased in the course of division, it is called as Non- terminating Decimal Expansion. For example – 0.333333

The basic properties of Real Numbers get recouped to make complex mathematics simple and easy to solve. Here is the rundown of the real number’s properties:

  • Associative Property – This property states that any real number in multiplication or addition, when added or multiplied in various ways, will not change the results. 

Illustration of the associative property of addition 

(2 + 5) + 1=1 + (6 + 1) 
7 + 1=1 + 7 
8=8both equation results in 8

Illustration of the associative property of multiplication 

(2 x 4) x 5=2 x (4 x 5) 
8 x 5=2 x 20 
40=40both equation results in 40

It states that the upshot of the real number (x) that results in the sum of two real numbers ( y+z) claims equal to the sum of each part of real numbers. Here, numbers inside the bracket get added first, and then the expression gets multiplied by the given number. 

Example of Distributive Property:

2(1+3) = 1(2) = 8
2(1)+ 2(3) = 2+ 6 = 8
2(1+3) = 2(1)+ 2(3)

  • This property is composed of two subpart- Additive and Multiplicative Identity. 
  • The additive identity property: – a+0 = a, i.e. 4+0 = 4. Here, the number 0 is called an additive identity element.
  • The multiplicative identity property:- a.1 = 1.a = a, i.e. 3.35 x1 = 3.35. Here, the number 1 is called a multiplicative identity element.

In a nutshell, real numbers can have integers, whole numbers, and natural numbers but not imaginary numbers because those numbers are hard to express in the number line and complex in nature. You can say various properties of a real number as discussed above can be used to orchestrate any algebraic expression.

Frequently Asked Questions (FAQs)

  • Is 0 a real number?
    Zero can be both a real number as well as an imaginary number as the square root of non-positive real numbers is called the imaginary number and zero is a non-positive number.
  • What is the set of all real numbers?
    The amalgamation of a set of rational numbers and the set of irrational numbers get collectively called a set of real numbers that include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
  • What is considered a real number?
    All the combinations of rational and irrational numbers get called real numbers. Real numbers can also be whole numbers like 0, 1, 2, 3, 4, 8, 9, and so on. Any numbers that are not imaginary in nature get considered a real number.
  • What are not real numbers?
    Those numbers that incorporate the square root of any negative number like the square root of – 1 are not real numbers. – 1 can get indicated as i. For example – 6 + 2i and 4i are non real numbers.
  • How do you identify real numbers?
    The simple and the best way to identify real numbers is when numbers can get plotted easily on the number line,they are real numbers. For instance– 0.5, -2, and π.

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High School Physics

  • Alternating Current
  • Atoms
  • Communication Systems
  • Current Electricity
  • Dual nature of Radiation and Matter
  • Electric Charges and Fields
  • Electricity
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  • Units and Measurement
  • Vectors, Scalar Quantities and Elementary Calculus
  • Wave Optics
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  • 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
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  • Basic concepts of Chemistry
  • Biomolecules
  • Carbon And Its Compounds
  • Carboxylic acids and Acid Derivatives
  • Chemical Bonding and Molecular Structures
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  • 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
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  • The d-block and f-block elements
  • The Gaseous State
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  • The Solid State
  • Thermodynamics

High School Biology

  • Absorption and Movement of Water in Plants
  • Adolescent Issues
  • Anatomy of Flowering Plants
  • Animal Kingdom
  • Bacteria and Fungi-Friends and Foe
  • Biodiversity and Conservation
  • Biofertilizers
  • Biological Classification
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  • Biomolecules
  • Biotechnology and its Applications
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  • Body Fluids and Circulation
  • Breathing and Exchange of Gases
  • Cell – Unit of Life
  • Cell Cycle and Cell Division
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  • 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
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  • Health and Its Significance
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  • Kingdom Fungi
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  • Neural Control And Coordination
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  • Organism and Population
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  • Plant Growth and Development
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  • Principles of Inheritance and Variation
  • Reproduction and Development in Angiosperms
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  • The Endocrine System
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  • The Nervous System and Sense Organs
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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
  • Algebra – Quadratic Equations
  • Binomial Theorem
  • Calculus – Applications of Derivatives
  • Calculus – Applications of the Integrals
  • Calculus – Continuity and Differentiability
  • Calculus – Differential Equations
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  • Geometry – Area
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  • 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
  • Chemistry in Your Life
  • 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
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  • Forests: Our Lifeline
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  • Fun With Magnets
  • Garbage In, Garbage Out
  • Getting To Know Plants
  • Health and Hygiene
  • Heat
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  • Life Processes: Nutrition in Animals and Plants
  • Light, Shadows And Reflections
  • Materials: Metals And Non-Metals
  • Matter and Its States
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  • 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
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  • Reaching The Age Of Adolescence
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  • Reproduction In Plants
  • Respiration In Organisms
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  • Separation Of Substances
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  • Some Natural Phenomena
  • Sorting Materials Into Groups
  • Sound
  • Stars And The Solar System
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  • 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

  • Addition
  • 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
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