maths > algebra

Identities Explained with Numerical Arithmetics

what you'll learn...

overview

In this lesson, identities are explained in general.

•  identity is a statement of one expression modified as per PEMA / CADI properties to arrive at a different expression.

•  These two expressions are identical.

•  And, identities are studied because one form of identity can be modified into its equivalent form for some purpose.

It is very important to go through this once to understand identities in algebra.

'identical' means 'exactly equal'

'Identity' means 'equality of two expressions; left and right hand side are identical'

Identities are statements that specify two expressions as equal.

In numerical terms, some examples of the identities are
•  $5×\left(2+3\right)=5×2+5×3$$5 \times \left(2 + 3\right) = 5 \times 2 + 5 \times 3$
(left hand side equals to right hand side )
•  $4-0=4$$4 - 0 = 4$
(left hand side and right hand side are equal.)

Similar to identities of numerical expressions, algebraic identities are statements that specify two algebraic expressions as equal.
(left hand side equals right hand side)
•  ${a}^{2}+2{a}^{2}+b=3{a}^{2}+b$${a}^{2} + 2 {a}^{2} + b = 3 {a}^{2} + b$
•  $\left(2+x\right)x={x}^{2}+2x$$\left(2 + x\right) x = {x}^{2} + 2 x$

example

Consider the expression $\left(2+3\right)×\left(1+5\right)$$\left(2 + 3\right) \times \left(1 + 5\right)$.

That does not equal $=2+3×1+5$$= 2 + 3 \times 1 + 5$

That equals $=2×1+3×1+2×5+3×5$$= 2 \times 1 + 3 \times 1 + 2 \times 5 + 3 \times 5$

$\left(2+3\right)×\left(1+5\right)$$\textcolor{c \mathmr{and} a l}{\left(2 + 3\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(1 + 5\right)}$

$2+3$$2 + 3$ is a number by closure property and by distributive property it is distributed over addition
$=\left(2+3\right)×\left(1\right)$$= \textcolor{c \mathmr{and} a l}{\left(2 + 3\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(1\right)}$ + $\left(2+3\right)×\left(5\right)$$\textcolor{c \mathmr{and} a l}{\left(2 + 3\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(5\right)}$

by distributive property $=\left(2\right)×\left(1\right)$$= \textcolor{c \mathmr{and} a l}{\left(2\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(1\right)}$ + $\left(3\right)×\left(1\right)$$\textcolor{c \mathmr{and} a l}{\left(3\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(1\right)}$

$=\left(2\right)×\left(5\right)$$= \textcolor{c \mathmr{and} a l}{\left(2\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(5\right)}$ + $\left(3\right)×\left(5\right)$$\textcolor{c \mathmr{and} a l}{\left(3\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(5\right)}$

This proves that for any number $p,q,b,c$$p , q , b , c$ the following is true. $\left(p+q\right)×\left(b+c\right)$$\textcolor{c \mathmr{and} a l}{\left(p + q\right)} \times \textcolor{\mathrm{de} e p s k y b l u e}{\left(b + c\right)}$
$=pb+qb+pc+qc$$= \textcolor{c \mathmr{and} a l}{p} \textcolor{\mathrm{de} e p s k y b l u e}{b} + \textcolor{c \mathmr{and} a l}{q} \textcolor{\mathrm{de} e p s k y b l u e}{b} + \textcolor{c \mathmr{and} a l}{p} \textcolor{\mathrm{de} e p s k y b l u e}{c} + \textcolor{c \mathmr{and} a l}{q} \textcolor{\mathrm{de} e p s k y b l u e}{c}$

The left hand side might be complex to compute, wherein the right hand side is easier to compute. For example $98×98$$98 \times 98$ involves large values to multiply and add. The same in $100×100-100×2-100×2+2×2$$100 \times 100 - 100 \times 2 - 100 \times 2 + 2 \times 2$ is simpler.

summary

Algebraic identities are equations of two expressions wherein one expression is modified per PEMA precedence / CADI Laws and Properties of Arithmetics to derive the other equivalent expression.

Outline

The outline of material to learn "Algebra Foundation" is as follows.

→   Numerical Arithmetics

→   Arithmetic Operations and Precedence

→   Properties of Comparison

→   Properties of Multiplication

→   Properties of Exponents

→   Algebraic Expressions

→   Algebraic Equations

→   Algebraic Identities

→   Algebraic Inequations