Mathematical Identities

A Mathematical Identity is an equality that relates one literal algebraic expression to another and has the characteristic of being always verified regardless of the values assigned to the variables contained in the expressions themselves.

The concept of mathematical identity is fundamental in the study of algebra and other mathematical disciplines. For this reason, in this lesson, we will study it in detail.

Mathematical Identity

Given two algebraic expressions, which we indicate with and , we can relate them through the equality relation, in this way:

This notation means that is always equal to .

However, this relation is not necessarily verified. In other words, it's not a given that is always equal to . To confirm this relation, it must be proved.

When expressions are simple, verifying the equality is almost immediate. Consider the following examples:

All of these equalities are always verified. In particular, the second and third ones are always verified regardless of the value we assign to the variable or the variable .

Other relations are a bit more complex to verify. Consider the following example:

In this case, we simply need to expand the expression that appears before the equality symbol using the square of a binomial:

By doing this, we have shown that the two expressions are equal. Specifically, they are equal regardless of the values we assign to variables or .

However, other equality relations are not always verified.

In some cases, they are never verified. Some examples include:

The first example is straightforward since will never equal .

The second doesn't represent an equality because no value will ever be equal to itself plus one.

In other cases, equality is verified only for specific values assigned to the variables.

For instance, consider:

This equality holds only if the variable assumes certain values. Specifically, this equality holds true if is either or , as they are the only values whose square is .

When an equality between two expressions is always verified, it's called a Mathematical Identity or simply Identity.

Components of a Mathematical Identity

An identity consists of two expressions. Each one of these expressions is called a Member of the identity. Specifically, the expression to the left of the equals sign is termed the First Member, while the expression on the right is called the Second Member:

Examples

Now, let's try to determine whether the following equality relations are identities.

Let's begin with a first example:

Now for a second example:

Existence Conditions of an Identity

In discussing mathematical identities, we have always implied one detail: the numeric set in which we consider the identity to be true is the set of real numbers.

For instance, revisiting the following identity:

By stating that this relation is an identity for all possible values that we can assign to and , we are considering all values belonging to the set of real numbers. Hence, a more accurate representation of the identity would be:

Going forward, when we talk about identities, we will always be considering the set of real numbers, , unless specified otherwise.

In some cases, however, the identity is always true for all real values except for specific ones. In particular, there might be some real values that render one of the members meaningless.

To clarify, consider the following equality relation:

It's easy to see that this is an identity. By expanding the first member, we get:

However, in eliminating the variable , we have overlooked a crucial detail. Although we can assign any real value to the variable and the identity holds, the same is not true for . In fact, if we assign the value 0 to , we can no longer simplify the first member. We would have:

This means, the first member would result in the expression , which is undefined.

Therefore, the above relation is an identity only if .

The condition is called the Existence Condition of the identity.

In Summary

In this lesson, we've seen that an equality between two algebraic expressions is a mathematical identity when it holds true regardless of the values we can assign to the literal variables.

Generally, unless specified otherwise, we'll always assume that the variables can take any value from the set of real numbers.

However, we can't always assign any value to the variables of an identity. There might be forbidden values that render the expressions meaningless. In such cases, we must adhere to existence conditions.

In other situations, an equality might only hold true for specific values. In this case, we're dealing with equations, which we'll begin to study in the next lesson.