Mathematical identity

A mathematical identity is a type of mathematical equality, between algebraic expressions that is verified for any value of some variable of all those that intervene in the expression. It is nothing more than the behavior of said expressions, how they will react to any value that we want to study, we could say, figuratively speaking, that it is the “personality” that specific algebraic expressions possess.

A simple example:

ax + bx = x (a + b)

It is an identity since, whatever the values of a, b and x are, the equality that we wrote above will always be fulfilled. So if we put it in numbers:

a = 1, b = 2 and x = 3

ax + bx = x (a + b) -> 1.3 + 2.3 = 3 (1 + 2) -> 3 + 6 = 3.3 -> 9 = 9

Therefore, the equality holds.

Without realizing it you will have seen lots of these identities, when you studied algebraic functions and their properties (distributive, commutative, associative, etc.), here this phenomenon is simply given a concrete name.
I am going to put a few more examples so that you can observe that it is always the same, the only thing that will change are the expressions, but the concept is common to all of them: what both sides of the equality say are constantly being fulfilled. Their primary function is to simplify an expression that would otherwise be complex for us.

>> Examples:

Thus we can know that the sum by difference of the binomials is the difference squared of the terms b and a, without having to multiply one by one.

In this way, although it is correct, we lose valuable time that we would save by knowing that identity, come on, think about the poor mathematicians who broke the coconut so that you would know these identities, do not say that it was in vain.

Jokes aside, we must not forget trigonometric identities (those that use sines and cosines), which is more of the same but refers to angles

These are even more useful, since they take us away from doing cumbersome calculations. I’m not going to say where they come from, because for this we already have a topic related to this.

What I am going to explain to you is the following, we agree that, for an identity to be fulfilled, the equality proposed for any value must be fulfilled, but what would happen if, for a certain value, that equality was not fulfilled ? Would we be facing an identity?

In this case, it is always true, since, whatever the values of a and b are, the difference of their squares will always be equal to the sum by difference.

This is an equation of the second degree, for it, we find two solutions:

x = 1 and x = 2

If we try with any other value, for example 5, the following happens:

With x = 1 however:

** Hint: If you have doubts about whether a solution is correct, substitute it in the equation, to see if it fulfills the equality.

Here I stay with this introduction to mathematical identities, see you in another laguia2000 article.

.