__maths__>__Foundation of Algebra with Numerical Arithmetics__>__Numerical expressions, equations, identities, and in-equations for Algebra__### Identities Explained with Numerical Arithmetics

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 in-equalities in algebra.

*click on the content to continue..*

What does 'identity' mean?

- It is just a name.
- equality of two expressions; left and right hand side are identical
- equality of two expressions; left and right hand side are identical

Answer is '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

• `5xx(2+3) = 5xx2+5xx3` *(left hand side equals to right hand side )*

• `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+2a^2+b = 3a^2+b`

• `(2+x)x = x^2+2x`

Consider the expression `(2 + 3) xx (1 + 5)`. Which of the following equals the given expression?

- `= 2xx1 + 3xx1 + 2xx5 + 3xx5`
- `= 2xx1 + 3xx1 + 2xx5 + 3xx5`
- `=2 + 3xx1 + 5`

The answer is "`= 2xx1 + 3xx1 + 2xx5 + 3xx5`"

`color(coral)((2 + 3))xx color(deepskyblue)((1 + 5))`*`2+3` is a number by closure property and by distributive property it is distributed over addition *

`=color(coral)((2 + 3))xx color(deepskyblue)((1))` + `color(coral)((2 + 3))xx color(deepskyblue)((5))`*by distributive property * `=color(coral)((2))xx color(deepskyblue)((1))` + `color(coral)((3))xx color(deepskyblue)((1))`

`=color(coral)((2))xx color(deepskyblue)((5))` + `color(coral)((3))xx color(deepskyblue)((5))`

This proves that for any number `p, q, b, c` the following is true. `color(coral)((p+q))xx color(deepskyblue)((b+c))`

`= color(coral)(p) color(deepskyblue)(b) + color(coral)(q)color(deepskyblue)(b) + color(coral)(p) color(deepskyblue)(c) + color(coral)(q)color(deepskyblue)(c)`

The left hand side might be complex to compute, wherein the right hand side is easier to compute. For example `98 xx 98` involves large values to multiply and add. The same in `100xx100 -100xx2-100xx2+2xx2` is simpler.

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.

*slide-show version coming soon*