Property 1: add equal numbers or formulas on both sides of the equation. The two sides are still equal, just as the two ends of the balance are balanced, adding or subtracting the same weight items at both ends of the balance. The two ends of the balance are balanced.

Property 2: both sides of the equation multiply or divide equal and non-zero numbers or formulas at the same time. The two sides are still equal, just like an object whose two ends shrink or enlarge at the same time, the two ends of the balance keep balance.

An identity has multiple variables and one variable. If there is only one variable on both sides of the identity, then the identity is the relationship between two analytical expressions. It comes from e ix = cosx+isinx (triangular representation of complex numbers), if x=π, e π I+ 1 = 0.

Mathematically, an identity is an equation that holds no matter what value a variable takes in a given range. An identity has multiple variables and one variable. If there is only one variable on both sides of the identity, then the identity is the relationship between two analytical expressions.

Given two analytic expressions, if any number of common parts (or subsets of common parts) of their domains or the values of arrays are equal (see function), the two analytic expressions are said to be the same.