# Chapter 2 – Algebra

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

Introduction

Algebra, which deals with number theory, geometry, and analysis, is one of the oldest branches in the history of mathematics. The definition of algebra sometimes notes that the study of symbols and laws in mathematics requires the manipulation of these symbols in mathematics. If you observe carefully, Algebra covers nearly everything from solving elementary equations to the analysis of abstractions.

In several chapters in Maths, algebra equations are included that students can study in their scholarships. In algebra, there are also some formulas and algebraic identities present.

#### What is Algebra?

In order to rewrite the equations, the algebra variables should be used to describe the unknown quantities that are coupled in such a way.

In our everyday lives, algebraic formulas are used to find the distance, the number of containers, and to work out the market rates when and when necessary. By making use of letters or other symbols describing entities, Algebra is very helpful in expressing a mathematical equation and relationship. It is possible to solve problems on algebraic identities.

#### Elementary Algebra

In a contemporary elementary algebra tutorial, Elementary Algebra covers the typical topics covered. Along with logical operations like +, -, x, ÷, multiplication requires numbers. Yet numbers are also represented by symbols in the field of algebra, which are referred to as variables such as x, a, n, y. It also facilitates the traditional formulation of arithmetic laws such as a + b = b + a and is the first step that illustrates the systematic discovery of all the properties of a real number system.

As opposed to pre-algebra, this algebra has a high level of equations to solve. This section has algebraic identities for two variables. Advanced algebra allows you to go through other algebra elements, such as:

• Equations of differences
• Matrices
• System Solving of Differential Equations
• Function graphing and linear equations
• Conic Sections
• Equation with Polynomial
• Polynomials of extremists and expressions
• Series and sequences
• Expressions of rationality
• Trigonometry
• Discrete and probability mathematics

#### Abstract Algebra

Abstract algebra is one of the algebra divisions that discovers the truths of algebraic structures regardless of the particular existence of certain operations. Such operations have certain properties, in particular situations. We should also assume that these properties have certain ramifications.

Abstract algebra covers algebraic structures such as fields, sets, modules, rings, lattices, spaces for vectors, etc.

The abstract algebra principles are below—

• Sets- Sets are characterized as the selection of objects that are identified by a particular set property. For eg, a set of all 2-2 matrices, the set of two-dimensional vectors present in the plane, and the various finite groups form.
• Binary Events- When the definition of addition is conceptualized, binary operations are given. Without a package, the definition of all binary operations will be pointless.
• Identity Element- To give the idea of an identity element for a particular operation, the numbers 0 and 1 are conceptualized. Here, 0 for the addition operation is called the identity element, while 1 for the multiplication operation is called the identity element.
• Inverse Elements- A negative number comes up with the definition of Inverse components. We write “-a” as the inverse of “a” for addition, and the inverse form is written as “a-1” for multiplication.
• Associativity- There is a property known as associativity when integer numbers are applied, in which the aggregation of added numbers does not affect the sum. Consider (3 + 2) + 4 = 3 + (2 + 4) as an example.

#### Linear Algebra

A subset of algebra that refers to both applied and pure mathematics is linear algebra. The linear mappings between the vector spaces are dealt with. It also deals with the analysis of aircraft and lines. That is the study of linear sets of equations with the properties of transformation. It is basically seen in all fields of mathematics. This refers to linear equations with their expression in vector spaces and via the matrices for linear functions. The following are the important topics discussed in linear algebra:

• Linear Equations
• Spaces of Vectors
• Relations
• Matrices and decomposition of matrices
• Relationships and Computations

#### Commutative Algebra

One of the divisions of algebra that explores commutative rings and their ideals in commutative algebra. The theory of algebraic numbers, as well as algebraic geometry, relies on the algebra of commutations. It covers algebraic integers rings, polynomial rings, and so on. There are several other fields of mathematics, such as differential topology, invariant theory, order theory, and general topology, that depend on commutative algebra in various ways. In modern pure mathematics, it has occupied a remarkable position.

#### Conclusion

This chapter helped us in learning the basics of algebra. We learned the proof of algebraic identities, and how to use algebraic identities for three variables in solving questions.

#### FAQs

1. What are the basics of algebra?

Algebra is a mathematics division designed to make it faster and easier to solve certain types of problems. Unlike arithmetic, which is entirely based on known number values, algebra is based on the concept of unknown values called variables.

2. What are the branches of algebra?

• Pre-algebra.
• Elementary algebra.
• Abstract algebra.
• Linear algebra.
• Universal algebra.

3. Who invented algebra?

Al-Khwarizmi is algebra’s father.

4. What is the main purpose of algebra?

By using letters of the alphabet or other symbols to represent entities as a form of shorthand, the purpose of Algebra is to make it easy to state a mathematical relation and its equation. Algebra then allows you to substitute values for unknown quantities in order to solve the equations.

5. How is algebra used in real life?

Algebra is used every day in our morning routine. When you wake up, by the end of the day, at least you have some objectives to achieve. The alarm is another good instance. People set up the alarm to wake up in the morning, but they do not understand that algebraic addition has just been executed.

### 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