# The Concept of Identity Transformation Expression. Identity

Let′s consider two expressions: `f(x)=x^2-2x` and `g(x)=4x-5`. For `x=2` we have `f(2)=2^2-2*2=0`; `g(x)=4*2-5=3`. The numbers 0 and 3 are called** corresponding values of expressions** `x^2-2x` and `4x-5` for `x=2`.

Let′s find the corresponding values the same expressions for `x=1` :

`f(1)=1^2-2*1=-1` ; `g(1)=4*1-5=-1` ;

for `x=0` :

`f(0)=0^2-2*0=0` ; `g(0)=4*0-5=-5`.

The corresponding values of two expressions can be equal to each other (So, in above example equality holds `f(1)=g(1)` and can differ from each other ( in the considered example `f(2)!=g(2)` ; `f(0)!=g(0)` ).

If corresponding values of two expressions, that contain same variables, are same for all allowable values of variables, then expressions are called **identically equal**.

The **identity** is the equality, that is correct for all allowable values of variables, that it contains.

So, identically equal expressions `x^5` and `x^2*x^3` , `a+b+c` and `c+b+a`, `(2ab)^2` and `4a^2b^2` .

The examples of identities: `a+b=b+a`, `a+0=a`, `(a+b)*c=ac+bc`, `a*1=a`, `x^5=x^2*x^3` .

The proportion `(2a)/(a-1)=(10a)/(5(a-1))` is identity for all values of `a`, besides `a=1`, because for `a=1` the denominators of fractions convert into zero, i.e. the fractions dont′t make sense. Replacement the expression `(ac)/(bc)` by expression `a/b` (we have reduced by `c` ) makes identically transformation of expression `(ac)/(bc)` with restriction `b!=0, c!=0` . So, `(ac)/(bc)=a/b` - identity for all values of variables, besides `b=0, c=0` . The correct numerical expressions is also called identity.

The replacement one expression by another, that is identically equal to it, is called **identically transformation of expression. **