Identities - Revision Notes

In these revision notes for Identities, we cover the following key points:

• What is an identity in math?
• Where does a conditional identity differ from an identity?
• What is another name used for conditional identities?
• What are inconsistent equations?
• How to make the distinction between identities, conditional identities and inconsistent equations?
• How to solve a question in math by proof?
• How to disprove a claim in by counter-example in math?

Identities Revision Notes

Identities are special equations, which give a true result for whatever value of their variable(s).

On the other hand, equations that have a fixed number of roots are known as 'conditional identities', as they are true only for specific values of the variable(s).

There is also a third category of equations known as 'inconsistent equations', which are always false for all values of their variable(s).

The general form of identities (after having made all possible operations) is

where n is the order (highest power) of the original equation. In first - order identities, we have n = 1, in second - order ones n = 2 and so on. In other words, all variables, coefficients and constants become zero after all operations and simplification are made.

On the other hand, if we have an equation that after having made all possible operations and simplifications takes the form

where a and b are numbers (a is a coefficient and b is a constant) and x is a variable, then we are dealing with a conditional identity, which is nothing more but a normal equation.

Last, if we have an equation that after having made all possible operations and simplifications gives a general form

where b is a constant, then we are dealing with an inconsistent equation, i.e. with an equation that has no solutions.

The eight well - known algebraic identities are called so because they are true for every value of their variables.

Solving by proof is a very powerful method used in mathematics to solve questions of a high level of difficulty. Proof as a method is closely related to identities as in many cases, we have to confirm that a relationship is always true.

To solve an exercise that requires proving that something is always true, we start from a known fact; then, step - by - step we move towards the final solution (proof). Solving by proof is a widely applied method, especially in geometry.

Disproving by a counter-example is another powerful tool applied in advanced mathematics to reject a given claim. Thus, often it is not worth trying to prove a supposition but it is enough to reject (disproof) it through a counter - example. A supposition must be always true to be accepted. If it is 1000 times true and only 1 time false, it is definitely regarded as false.