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.