Basic concepts of Algebra in Mathematics

Important Definitions and Basic Concepts of Algebra in Mathematics.
A Representation of Algebra.
Image by Gerd Altmann from Pixabay

What is Algebra?

Algebra is a branch of mathematics that deals with arithmetic operations on variables. These variables are represented by various symbols. Arithmetic operations include variables as well as the numerals as constants.

Variables are special parts of algebra. When we don’t know the exact value of any quantity but we still have to do the calculations, then we use this variable as a symbol or a letter. By using symbols or letters, it becomes easier to perform calculations. Apart from these, we also use mathematical signs and symbols such as <, >, ≤, ≥, =, or ≠.

There can be different types of numeric constants. (Click here to know more about the numbers)

Combinations of variables, numeric constants, and symbols can be:

  • Expressions (Polynomials)
    Example, 5x + 8y + 9
  • Equations
    Example, 5x + 8y – 9 = 10
  • Inequalities
    Example, 5x + 8y – 9 < 10

Basic arithmetic operations

Basic arithmetic operations are performed by following the principles of BODMAS, BIDMAS, or PEMDAS. These principles decide the “order of operations” when we have multiple operations to be done all together in the same calculation.

BODMAS means

  • B – Bracket
  • O – Order
  • D – Division
  • M – Multiplication
  • A – Addition
  • S – Subtraction

BIDMAS means

  • B – Bracket
  • I – Indices
  • D – Division
  • M – Multiplication
  • A – Addition
  • S – Subtraction

PEMDAS means

  • P – Parenthesis
  • E – Exponents
  • M – Multiplication
  • D – Division
  • A – Addition
  • S – Subtraction

Brackets or Parenthesis

Brackets are symbols used to group objects things together. There are 3 types of brackets used to group objects together in any arithmetic operation.

( ) – Parentheses / Small brackets / Round brackets
{ }
– Curly brackets / Braces
[ ] – Square brackets / box brackets

Exponent or Order

An exponent is a representation where one number is multiplied n-times (raised to power n) by the number itself.
For example, ( 4 is an exponent in 5⁴ )
Here, the power (n ) is 4 which means that 5 is multiplied 4-times by itself. (5⁴ = 5 x 5 x 5 x 5 = 625)

Addition, Subtraction, Multiplication, and Division are other arithmetic operations.

Order of Operations

In any expression, inequality or equation, BODMAS/PEMDAS/BIDMAS is followed to perform the arithmetic operations.
>>> Group of entities in the brackets are solved first.
>>> Exponents are solved next.
>>> Division and Multiplication are done next.
>>> Addition and Subtraction are next performed.

Some Important Algebraic Formulae

  1. ⟨w + x⟩⟨y + z⟩ = wx + xy + yz + zw
  2. ⟨x + y⟩⟨x + z⟩ = x² + xy + yz + zx
  3. ⟨x + y⟩⟨x – y⟩ = x² – y²
  4. ⟨x + y⟩² = ⟨x + y⟩⟨x + y⟩ = x² + 2xy + y²
  5. ⟨x – y⟩² = ⟨x – y⟩⟨x – y⟩ = x² – 2xy + y²
  6. ⟨x – y⟩² = ⟨x + y⟩² – 4xy
  7. ⟨x + y⟩² = ⟨x – y⟩² + 4xy
  8. ⟨x + y + z⟩² = x² + y² + z² + 2xy + 2yz + 2zx
  9. ⟨x + y – z⟩² = x² + y² + z² + 2xy – 2yz – 2zx
  10. ⟨x – y – z⟩² = x² + y² + z² – 2xy + 2yz – 2zx
  11. ⟨x + y⟩³ = x³ + 6x²y + 6xy² + y³
  12. ⟨x – y⟩³ = x³ – 6x²y + 6xy² – y³

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