# Chapter 1 – Real Numbers

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

Introduction

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's Division Lemma

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

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

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

#### Properties of Real Numbers

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 = 8 both 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 = 40 both equation results in 40

#### Distributive Property

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)

#### Identity Property

• 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.

#### Conclusion

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.

#### 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.

A Step Towards Learning Better

For better insight, you can allude to videos on MSVgo app. MSVgo is an application whose center principle is to convince you to comprehend the conviction with models or informative representations or animations. It is a video library and takes the help of visual medium to enhance the learning process.

### 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