Real Number Definition, Properties and Examples

In the notation, real numbers are just the sum of rational and irrational numbers. All the arithmetic operations may be done on these values, and they can also be expressed on the number line. Imaginary numbers, on the other hand, are unreal numbers that cannot be represented on a number line and are often employed to represent complex numbers. The summation of both rational and irrational numbers is considered the set of real integers. They are symbolized by the letter "R" and can either be positive or negative. This category includes all-natural integers, decimals, and fractional numbers.

Types of Real Numbers

Since childhood, we have worked on multiple mathematical operations on real numbers; let us learn about these numbers in detail along with suitable examples.

Natural Numbers: A set of all counting numbers which begins with 1 are known as natural numbers. It is denoted by the capital letter N and includes N = {1,2,3,4,5,6,7……infinite}

Whole Numbers: A set of all countable numbers which begins with 0 are known as whole numbers. It is designated by the capital letter W and contains W = {0,1, 2…., 21…. 5551, …. infinite}

Integers: Integer is represented by capital letter Z and is the summation of whole numbers and negative numbers ranging from -1 to infinity.

Rational Numbers: Fractional numbers of the form a/b where b is never a null value or zero, are known as rational numbers. They are a direct subset of real numbers and contain positive as well as negative numbers.

Irrational Numbers: All the left-out numbers that cannot be expressed as rational numbers and have decimal non-repeating values are termed as irrational numbers. For example, π, sqrt (2).

Properties of Real Numbers

Commutative property, associative property, distributive property, and identity property are the four major properties related to real numbers. Consider the three real numbers "k, l, and m." Let us use these numbers to derive all the above-mentioned properties.

Commutative Property

For two numbers, k and l, the commutative property for addition is k + l = l + k. The commutative property for the multiplication of real numbers is k × l = l × k.

Associative Property

Now for three numbers, we define the associative property for addition and multiplication. Let the numbers be i, j, and k. The standard form for addition is given as: i + (j + k) = (i + j) + k and the general form for the multiplication is i * (j * k) = (i * j) * k.

Distributive Property

For the same three numbers k, l, and m, the distributive property is defined as: k (l + m) = kl + km and (k + m) l = kl + ml.

Identity Property

The identity property of real numbers is the numbers which when added or multiplied to the given value, results in the formation of the same value. The additive identity of a real number is 0, because x + 0 = x, and the multiplicative identity of a real number is 1, because 1 * x = x.

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Accuracy Precision and Error in Measurement Ampere Amplitude Modulation

Anemometer : Measurement of Wind Speed

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118 Elements and Their Symbols and Atomic Numbers Acetone Acid And Base Difference Acids Bases and Salts Actinides Addition Reaction Adsorption Chromatography Aldol Condensation Alkali Metals Ammonium Chloride Applications of Redox Reactions Argon Arrhenius Equation Rate Constant And Temperature Atomic Number Mass Number Aufbau Principle Aufbau Principle, Paulis Exclusion Principle and Hands Rule Bakelite: Structure and Uses Benzene Reactions Benzoic Acid - Structure and Properties Biodegradable and Non Biodegradable Calcium Oxide Cathode Ray Experiment Charles Law Chemical And Its Composition Chemical Formula of Common Compounds Classification Of Carbohydrates and Its Structure Classification of Organic Compounds Coefficient of Viscosity - Definition, Formula and Example Colligative Properties beRelative Lowering of Vapour Pressure Colloidal Solution Components of Air Condensation Conformation Cracking Meaning Crystallization Decantation Decomposition Reaction Dehydration of Alcohols Difference Between Alkali and Base Difference Between Atom And Ion Difference Between Atom and Molecule Difference Between CNG and LPG Difference between Compound and Mixture Difference Between Element And Compound Difference Between Exothermic and Endothermic Reactions Difference Between Mixture and Solution Difference Between Molecule and Compounds Difference Between Organic And Inorganic Compounds Difference between Physical and Chemical Change Difference between Vapor and Gas

Disaccharides

Discovery of Protons and Neutrons Distillation Effects of Burning Fossil Fuels Electromeric Effect Electron Configuration Electronic Configuration of First 30 Elements Electronic Configuration of Iron Electroplating Process Endothermic Reaction Enthalpy Change Enzyme Catalysis Ester Ester Hydrolysis Ethanoic Acid Exothermic Reaction Filtration Fuel Types Functions of Nucleic Acids Glycerin Glycine Structure Glycogen Gravimetric Analysis Group 17 Trends Properties Heat Capacity Cp Cv Relation Heavy Water Homogeneous Mixture and Heterogeneous Mixture Hunds Rule Hybridization Hydrochloric Acid Hydrogen Peroxide Introduction: Importance of Forest IR Spectroscopy Iron Krypton Lewis Dot Structures Metallic Bonds Metals and Nonmetals Methanol Mineral Resources Mixtures Modern Periodic Table and Its Significance Mole Concept Monosaccharides NaHCO3 Natural Polymers Neutralization Reaction Non Metals Oxalic Acid Oxygen

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Physical and Chemical Properties of Carbon

Physical And Chemical Properties of Water Physical Change and Chemical Change Polarity Polymers Uses Polyvinyl Alcohol Positron Potassium Potassium Dichromate - Formula, Properties and Uses Potassium Permanganate Precipitation Reaction Preparation of Benzene Preparation Properties and uses of Sodium Chloride Properties of Boron Family Properties of Solution Redox Titration Relation Between Normality And Molarity Rusting Iron Prevention

Rutherford Atomic Model

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Synthetic Polymers

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Gattermann Reaction

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Mannich Reaction

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Reactivity Series

Difference between isotropic and anisotropic

Introduction to p-Block elements

Plant fibres

Inert Gases: Uses

Benzoin Condensation

Difference between primary cell and secondary cell

Partition Chromatography

Applications of colloids

Displacement Reactions

Handpicking

Biogas: Uses of Biogas

Acid Rain

Types of Chemical Reactions

SN2 Reaction Mechanism

Complexometric Titration

markovnikov rule

Kohlrausch Law

Electrode

Valency Chart

Order of Reaction

Band Theory

Difference Between Rusting and Corrosion

Potentiometric Titration

Frenkel Defect

Deforestation

Stephen Reaction Mechanism

Gattermann – Koch Reaction Mechanism

Electronegativity

Industrial Waste: Types

Ethylene

Charge to Mass Ratio of an Electron

Fehling Solution

Photochemical Reactions

Difference Between Endpoint and Equivalence Point

Second Order Reaction

Etard Reaction

Forms of water

Preparation of Acetanilide

Metallic Minerals and Non-Metallic Minerals

Ferric Chloride Henry's Law Preparation of Mohr's salt

Schottky Defect

Claisen rearrangement

Calcium sulphate

Sodium carbonate

Rubber

Ethyl acetate

Conductometric Titration

Resorcinol

Difference Between Evaporation and Condensation

States of Matter

Boric Acid

Types of Minerals

Finkelstein Reaction

Electronegativity Chart

Difference between Solid, Liquid and Gas in tabular form

Aromaticity

Drawbacks of Rutherford's Atomic Model

Thermosetting Polymers

Preparation of Alkanes

Examples of Bases

Suzuki Coupling Reaction

Standard Electrode Potential

Disadvantages of Plastics

Reducing Agent

Van't Hoff Factor

Hard Water and Soft Water

Organometallic compounds

Gay Lussac's Law

Slaked Lime

Borax

Tetravalency of Carbon

Boyle's Law

Ellingham Diagram

Crystal Defects : Point Defects

HVZ Reaction (Hell-Volhard-Zelinsky Reaction)

Classification of Drugs

Preparation of Sulfuric Acid

Lucas test

Xenon Difluoride

Bravais Lattice

Classification of Oxides

Elimination Reaction

preparation of polythene, Teflon and polyacrylonitrile

Difference between cations and anions

Uses of Limestone

Fructose

Azeotropes

Lewis Acid and Base

Emulsification

Dielectric properties of solids

Zwitterion

Birch Reduction Mechanism

Oxidation and Reduction

Electrophilic Substitution Reaction

Suspensions

Aromatic Compounds

Ores and Minerals

Adsorption theory of heterogeneous catalysis

Wittig Reaction

Difference between evaporation and boiling

Intensive And Extensive Properties Of Matter

Electronic Configuration of Group 16 Elements

Difference Between Elements and Atoms

Sodium Oxide

Gypsum

Difference Between Alloy and Composite

Chemical Equations

Thorium

Lead Acid Battery

Isotopes of Hydrogen

Figure Rules

Electrophilic Addition Reactions Of Alkenes

Isothermal Expansion of an Ideal Gas

Potassium Chloride - KCl

Aluminum Oxide

Difference Between Acetic Acid and Glacial Acetic Acid

Difference Between Polar and Nonpolar

Difference Between Baking Powder and Baking Soda

Test for Phenolic group

To prepare colloidal solution of starch

Uses of Mica

Pi Bonds

Potassium Chlorate

Schmidt Reaction

High density Polyethylene

Huckel's Rule

Discovery of Proton

Aluminium Ore: Extraction of Aluminium

Benzene Hexachloride

Caustic Potash or Potasium Hydroxide

Electromagnetic Radiation - Wave Nature

Atomic Number and Mass Number, Isotopes and Isobars

Preparation of Potash Alum

Beckmann Rearrangement

Heterogeneous Equilibrium

Butane

Factors Affecting Electrolytic Conductance

Scandium

Homogeneous Equilibrium

Phenols Nomenclcature

Galvanic Corrosion

Chlorine Trifluoride

Robinson Annulation

Copper

Pyridine Silver Carbonate Harmful Effects of Radiation

Water

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Cm to inch converter

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Surface Area of a Sphere

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Logarithm Table

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General Equation of a Line

Factors of 100 Factors of a Number

Multiplicative Inverse

Linear Differential Equations

Inverse Cosine

Comparing Quantities using Percentage

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Food Web

Fragmentation

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Invertebrates

Kingdom Monera Protista Fungi Law of Segregation and Law of Dominance

Life process

Light-dependent Reactions

Lipids

Living Things

Macromolecule Meiosis 1 Stages and Process Mendelian Disorders Microbodies Modern Synthetic Theory of Evolution Monera Nephron Function Renal Tubules Nuclear Membrane Nucleolus Nucleus Our Environment Parasitism Parts of Plants Passive Transport Plant Cell Plant Kingdom Plantae Plant Respiration Plants Plastids Protista Red Data Book Reproductive System of Earthworm Ribosomes Scientific Names of Animals and Plants Seed germination Sensory Perception Snail Skeletal System Spermatogenesis Structure Of Eye Taxonomic Hierarchy Taxonomy Algae Thallophyte

Types of Soil

Vacuoles What Is Adaptation Agricultural Practices Animal Kingdom Alimentary Canal Coelenterata Conservation of Forest and Wildlife Difference between Endosmosis and Exosmosis Difference between Afforestation and Deforestation Difference between Algae and Fungi Difference between Cereals and Pulses Difference Between Fragmentation and Regeneration Difference Between Neurosis and Psychosis Difference Between Nucleotide and Nucleoside

Difference Between Sea and Ocean

Difference Between Turtles and Tortoises

Difference Between Wildlife Sanctuary and National Park

Modes of Plant Reproduction

Ecological Pyramid and Its Types

Double Fertilization in Angiosperms

Ecosystem Enzymes

Fertilisation in Plants

Five Kingdoms Classification

Ganongs Potometer

Health and Hygiene

Herbivores and Carnivores

Heterotrophic Nutrition Human Digestive System

Vertebrates and Invertebrates

Morphology of Leaves

Modifications of Root

Phylum Aschelminthes

Diffusion

Osmosis

Prokaryotic Cells

Reproduction

Flora And Fauna

Overview of Food Chain

Difference Between Rabi And Kharif Crops

Vegetative Propagation Angiosperms Characteristics Soil Profile

Types of Pollution

Human Excretory System Sense Organs Rainwater Harvesting

Prokaryotic And Eukaryotic Cells

Living And Non Living Things

The Nucleus

Irrigation

Endoplasmic Reticulum

Between Data And Information

Mitochondria

Nitrogen Cycle

Diagram of Animal Cell

Meristematic Tissue

Law of Segregation Law of Dominance

Eukaryotic Cells

Sources Animal Plant Products

Photosynthesis Respiration Ozone Layer Depletion Kingdom Fungi Aerobic Anaerobic Respiration Cell Organelles Peristalsis Bacteria Oogenesis

Leaves Morphology Types Modification

Mitosis and Meiosis

How Do Organisms Reproduce

Diagram of Stomata

Monohybrid Cross Inheritance One Gene Mechanism of Breathing

Soil Erosion

Plasmolysis

Urine Formation Osmoregulation

Porifera

Flagella

Amoeba

Aerobic Respiration

Pteridophyta

Arteries And Veins Difference

Reflex Action

Stomata

Cell Wall

Diagram of Neuron

A Guide To Composition And Function Of Lymph

Difference Between Ligaments And Tendons

Insectivorous Plants

Nutrition Modes Living Organisms

Mendel Laws Of Inheritance

Nutrition In Amoeba

Calorific Value

Air Pollution Control

Fertilization In Plants

Binary Fission

Omnivores

Saprophytes

Asexual Reproduction

Deficiency Diseases

Light Reaction Vs Dark Reaction

Global Warming

Wildlife Sanctuary

Viviparous Oviparous Embryo Development

Food Preservation Methods Food Poisoning

Crops

Biofertilizers

Biodiversity Conservation

Difference Between Endocrine And Exocrine Glands

Greenhouse Effect Gases

Diagram Of Digestive System

Crop Production And Management

Types Of Pollination

Introduction and Characteristics of Living Beings Meristematic Cells Taxonomical Aids _Flora, Manual, Monograph And Catalogues Types of leaves structure of skeletal muscle spirometry human respiratory system herbarium dinoflagellates general characters and structure of leaf halophiles and their features Exchange of Gases Simple Permanent tissue Morphology of root and its types Complex permanent tissues Regions of Root History of classification and its need Breathing And Exchange Of Gases Transport of Gases Regulation of Respiration Disorders of Respiratory System Interaction of Haemoglobin with Other Gases Amniocentesis Anatomy of flowering plants ASSISTED REPRODUCTIVE TECHNOLOGIES (ARTs) Meristematic Cells Phloems Population explosion Reproductive Health RCH (Reproductive and Child Health Care) Medical termination of Pregnancy (MTP) Plant kingdom Biological Systems of Classification & Branches of Taxonomy Pteridophytes Gymnosperms Angiosperms Life cycle Patterns Anatomy of Roots Contraceptives Anatomy of stem Algae Bryophytes Tissues and its types STDs (Sexually Transmitted Diseases) Cell - The Basic Unit of Life & its Historical background Cell theory and its modification An Overview of Cell Secondary growth in dicot stem Tissue system and types Prokaryotic (Bacterial) cell Ribosomes Eukaryotic cell Cell: The basic unit of life Cytoskeleton Centrosome and Centriole Cilia and flagella Nucleus Chromosomes Evolution Movement and Locomotion Human Reproduction Absorption and Translocation of Solutes Soil as a reservoir of Essential Elements LOCOMOTION AND MOVEMENT Mechanism of Muscle Contraction Morphology in Flowering Plants Contractile proteins Muscular System Thyroid Puberty Hormones Origin of Universe Hypothalamus Cell wall Mitochondria Special Chromosomes Microbodies Introduction, Hydroponics and the study of mineral requirements of the plants Criteria of essentiality and Categorisation of Essential elements Deficiency symptoms of essential elements and toxicity The Stem The Root Cytoplasm and cell membrane Mineral Nutrition Disorders Related To Reproductive System Phylum Ctenophora Role of Macronutrients Role of Micronutrients Phylum - Coelenterata / Cnidaria Reptilia Aves Mammalia Anatomy of the leaf Animal kingdom Phylum Annelida Phylum Mollusca Phylum Echinodermata Vertebrata Chordates Disorders of circulatory system Abiotic Factors Male Reproductive System Kingdom Animalia Responses to Abiotic Factors Female Reproductive System Mammary Glands Pregnancy and Embryonic Development Cell Cancer Disorders of the Digestive System Infertility Morphology of Cockroach Blood and Plasma Alternation of Generations and Classification Predation Phaeophyceae and Rhodophyceae Phylum Coelenterata Phylum Porifera Phylum Platyhelminthes Chromosomal Disorders Competition Taxonomic Aids Alternation of generation in Pteridophytes Disorders of muscular and skeletal system General Characteristics of Living Digestive Glands Stomach Testis Buccal cavity Seed Common Diseases in Humans Permanent Tissue Permanent Tissues Xylem Pteridophytes Characteristics Parasitism and Commensalism Water Pollution and Its Control Differentiation, Dedifferentiation and Redifferentiation Organism and its Environment Carbohydrates Greenhouse Effect Gases Amino Acids Dynamic State of Body Constituents Photosynthesis Chloroplast Features of Living Beings Blood Group Lipids Kingdom Protista dense connective tissue connective tissue cartilage connective tissue blood plant cell viruses the stem Connective Tissue Bone Viroids Prions Lichens Animal cell Prokaryotic Cell Taxonomic Aids Herbarium and Museum Sphygmomanometer and ECG Population Attributes Noise pollution Diaphragm Parts of a Plant Photoperiodism Poultry Farming Nucleic Acids Cell Cycle Principles of Biotechnology Antibiotics IVF Glands Kranz Anatomy Sexual Reproduction in Plants Cytokinin Plant Development Biofertilizers Photorespiration Auxin Nutrient Cycling Types of Flowers Taxonomic Hierarchy and Categories Epidermal Tissue System Calyx and Corolla Ethylene Fertilization and Implantation Types of Fruits Inflorescence Types of Growth Macronutrients Family Liliaceae ETC and Oxidative Phosphorylation Classification of Wood Ground Tissue System Secondary Growth in Stem Plant Vascular System Electron Transport System Transpiration Difference Between Types of Cells Ovary Structure Birth Control Epithelial Tissue Endomembrane System Cell Envelopes and Cell Membrane Muscle Types Axial Skeleton Acquired Immunity Lung Volumes and Capacities Suspension and Migration Prokaryotic Cells

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