What is identity? Standard Identities
Consider the following equation:
2(x+3) = 2x+6
What can you say about the above equation? In LHS, if you substitute the value of x =3, you will get 12. Even in RHS if you will substitute the value of the variable x with 3, you will get 12 as the answer.
In mathematics, an equation P = Q is called an identity if the following conditions are satisfied:
- Both side of the equality relation contain some variables.
- Both the sides give the same value, when the variable is substituted with a particular constant.
If an equality relation satisfies above two conditions, it will be an identity. There can be many relations in mathematics which can be called identities, but we need not need to remember all of them. There are few standard identities which are important in algebra, which can make the calculations simple.
The standard identities (algebraic) are as follows:
(a + b)2 = a2 + b2 + 2ab
(a – b)2 = a2 + b2 – 2ab
(a + b)3=a3 + b3 + 3ab(a + b)
(a – b)3=a3 – b3 – 3ab(a – b)
(a + b + c)2=a2 + b2 + c2 + 2ab + 2bc + 2ca
Let us try to prove the first identity and see whether it is valid or not.
Consider the LHS, (a+b)2 = (a+b) x (a+b) = a(a+b)+b(a+b) = a2+ab+ab+b2 = a2 + b2 + 2ab
Let us try to prove the above equality relation in another way.
Consider a square of side (a+b) divided in the following rectangles and squares:
The total area is given as the sum of areas of each figure, i.e. a2 + ab + ab + b2 = a2 + b2 + 2ab, which is equal to RHS. Hence the identity is valid. Therefore, whatever is the method of proving the identity it will always be true.
The standard identities discussed above are the algebraic identities. These standard identities hold true for any value of the variable.