# Chapter 2 – Algebra

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

Introduction

So far, you have learned basic arithmetic operations like multiplication, division, etc. You have also learned about two and three-dimensional shapes and calculated their area etc. This branch of mathematics is called geometry. Now, you will learn about the third branch of mathematics called Algebra.

Algebra allows you to use letters and symbols to write formulas, establish rules, and find the unknown values. With the ability to write your formulas and manipulate alphabets, you can use algebra to solve puzzles and apply them in solving day-to-day problems like finding the volume of the container, finding the distance, etc.

Let’s get started and learn about Algebra:

#### Branches in Algebra

You might have come across specific formulas or equations where the problem is solved using alphabets and symbols. These alphabets and symbols are called ‘variables’ in algebra. When you form an equation, variables are used to determine the unknown values. However, since algebra is a vast concept, it is divided into various branches to make it easier to understand and find the correct approach to solve the problem.

1.  Elementary Algebra:

Elementary algebra combines the basic arithmetic operations like addition, subtraction, multiplication, and division with the variables of algebra like a, b, x, and y to form the equations.
For example: a=1, b=5, x=2, y= 4 then the equation can be written as
a+ b= x + y
Elementary algebra is used to evaluate the equations, the equalities, and inequalities of the properties, solving equations with one or more variables, etc.

Advanced algebra helps you understand the higher level of equations to solve the other parts of algebra like:

• Matrix
•  Linear Equations
• Polynomial Equations
• Series and Sequences
• Rational Expression
• Trigonometry
• Graph functions
• Conic Sections
• Probability

3. Abstract Algebra:

Abstract Algebra is the algebra field that deals with the algebraic structure like vectors, lattices, rings, modules, groups, etc. These algebraic systems are not dependent on a specific nature of the algebraic operations, and hence it follows certain concepts to solve the problem.
The concepts of abstract algebra are as follows:

• Sets:
A collection of objects that are determined by a specific property is called a set. For example, a 3×3 matrix or a few two- dimensional vectors for a set.
• Binary Operations:
Binary operations combine two elements to produce a third element.
• Identity Element:
In a set, an identity element is the one that does not change elements when combined with it. For example, 0 and 1 are the identity elements for additions and multiplication as they do not change the value of the element.
• Inverse Elements:
The inverse elements are used as a concept of negation in terms of addition and multiplication. For example, the inverse of ‘a’ will be ‘-a’ while the inverse of ‘a’ in multiplication form will be ‘a-1’.
• Associative Algebra:
The basic premises of the associativity is that when numbers are added, their grouping does not affect the sum of the integers. For example, 1+(2+3) will yield the same result as (1+2)+3.

4. Linear Algebra:

Linear Algebra is a branch of algebra that deals with the spaces between the vectors and their mapping. Linear algebra lets you study lines and planes while also giving you an opportunity to understand the transformation properties of the linear equation. In linear algebra, you will study:
A. Vectors
B. Relations
C.  Linear equations
D. Matrices
E. Computations and relations

5. Commutative Algebra:

Have you ever wondered if there is any branch in mathematics that studies the rings? Well, commutative algebra is the branch that studies the rings, such as polynomial rings and their corresponding ideals.

#### Multiplication of Algebraic Expressions

Two algebraic expressions can be multiplied to find the unknown variable, and the expressions can consist of integer variables and constants. For example, 2ab+5 is an expression, where a & b as variables and 4 & 9 as constants. There are certain rules that must be followed while multiplying the algebraic expression:

• The brackets in the expression must be simplified first.
• If there are no brackets, then the expression can be solved through the BODMAS rule.

#### BODMAS rule in Algebraic Expression

When solving the algebraic expressions with brackets, it is necessary that the bracket is simplified first using the BODMAS rule. BODMAS and simplification of brackets are used for expressions that have multiple arithmetic operations.

By the rule, the expression needs to be solved in the order Bracket, Order, Division, Multiplication, Addition, and Subtraction. If there are multiple bracketed expressions in an expression, then the innermost bracketed expression is solved first and then moved towards the outermost bracket.

#### Laws of Exponents

Exponents give the idea about how many times the base number will have to be multiplied. For example, in the expression 32, 3 must be multiplied 3 times to get the result. Here, 3 is called the base number, and 2 is called the exponent.

The algebraic expressions with exponents follow a particular rule, and the rules are as follows:

• RULE 1: If the numbers being multiplied have the same base, then the exponents must be added. For example, 32 x 33= 3(2+3).
• RULE 2: If the numbers being divided have the same base, then subtract the exponents. For example, 33 x 32= 3(3-2).

RULE 3: When the exponent of the number is raised by another exponent, then multiply the exponents. For example, (32)3=3(2x3).

#### Exponents in Algebra

Exponents in algebra are used when a large number has to be represented in a simple way that can be easily understood. It shows how many times a number must be multiplied by the same number to achieve the result. For example, for the number 100, 10 must be multiplied twice, and it can be denoted as 102.

#### FAQs

1. What are the different types of algebraic equations?
There are five types of algebraic equations: Polynomial, exponential, trigonometric, relational, and logarithmic equations.
2. Are there any branches in algebra?
There are five different branches in algebra: pre-algebra, elementary, abstract, universal, and linear algebra.

Formulas are critical, and one must understand the concept behind them. MSVgo is a learning app that is built on the philosophy that understanding a concept is the core of learning and therefore explains the concepts with examples, animations, or explanatory visualisation.

### 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