Monomials

Monomials are the simplest literal mathematical expressions that could be written. It's possible to build more complex expressions using them as basic building blocks.

Fundamentally, a monomial is composed by the product of a coefficient, i.e. a number, and a set of powers of literal variables where the exponent is a natural number. The set of powers goes under the name of literal part.

This lesson serves as a starting point to introduce the fundamental concepts about monomials necessary to study, later, the operations that can be performed on them. In particular, we will see what the normal form of a monomial is. We will see how to compare two monomials with each other by introducing the concepts of similar monomials, equal monomials and opposite monomials.

We will also define, another important property of monomials: the degree of a monomial.

All these concepts will come in handy in the next lessons where we will see how to work and operate with monomials.

Definition of monomial

A monomial is a mathematical expression consisting of a product of factors which can be either numerical or literal. It represent the simplest literal mathematical expressions that can be written.

The consequence of the above definition is that in order to consider an expression a monomial, it must meet two requirements:

1. The expression must exclusively contain multiplications.
2. The literal factors may only appear as powers with a natural exponent including the zero.

Examples of monomials

Let's look at some monomial examples:

In this case we have the numerical part, , multiplied by three powers of literals with natural exponents.

Again, we have a monomial in which the numerical part is .

Although the fraction can be deceptive, here too we are dealing with a monomial. In fact represents a rational number and it is a valid coefficient.

Similarly, the following expression represents a monomial:

In fact, is still a number, in particular an irrational one, so it is a valid coefficient. It is multiplied by powers of literal variables with natural exponents.

Another interesting example is the following:

At first glance, it may not appear to be a monomial because an addition appears. However, since the addition is between two numbers, it can be reduced to a monomial as it can be replaced by its result:

Below are more examples of expressions that are monomials:

Numbers are monomials

It follows from the definition of a monomial that the following expression is also a monomial:

In fact, the expression above can be rewritten as:

which is always . In fact, the factor is equal to 1 since any number (other than 0) raised to zero always results in 1. Therefore, the above expression can be simplified as . Similarly, can also be rewritten as:

which, similarly, is always .

In short, any number can be seen as a monomial. This is because we can always rewrite it as the product of the number itself by powers of literal variables with an exponent of zero.

Null monomial

Since any number can be considered as a monomial, it follows that zero is also a monomial and, in particular, zero is known as the null monomial.

Examples of expressions that are not monomials.

Let us now look at some examples of expressions which are not monomials:

In this case, there is no multiplication between the variables and , but rather a division. Therefore, the expression violates condition 1 and is not a monomial.

In this case, there is a power of a variable with an exponent of which is not a natural number. Therefore condition 2 is violated and the expression is not a monomial.

Below, let's look at other examples of expressions that are not monomials:

Normal Form of a Monomial

When working with monomials, it is useful in most cases to convert them into their Normal Form:

Let us look at examples of monomials reduced to normal form:

• is in normal form because the literal powers that appear in it have different bases: and . Moreover, only one numerical factor appears.

• is in normal form.

• is in normal form.

Now, instead, let's look at examples of monomials which are not in normal form:

• is not in normal form. In fact two numerical factors appear.

• is not in normal form. The literal variable appears twice.

Reducing a monomial to normal form

To reduce a monomial to normal form, it is sufficient to apply these three properties:

• The commutative property of multiplication.
• The associative property of multiplication.
• The first property of powers: .

To understand better, let's look at an example. Let us take the following monomial:

We observe that it is not in normal form because two numerical factors and multiple literal powers with the same base appear.

First we employ the commutative property of multiplication, in other words we change the order of the factors, grouping the numerical factors and the literal powers with the same base together:

Next we apply the associative property of multiplication, multiplying numerical factors and powers with the same base:

Finally, we use the first property of powers. That is, we substitute, for the product of powers with the same base, the power having as its exponent the sum of the exponents:

Coefficient and literal part of a monomial

When a monomial is in normal form, its coefficient is the numerical part:

In addition, when writing monomials, when the coefficient is equal to or , we avoid writing the numerical part and leave only the sign. For example:

• Instead of writing , we write .
• Instead of writing , we write .

Conversely, the set of powers of the literal variables is called the literal part:

Let's look at some examples:

• : in this case is the coefficient while is the literal part.

• : in this case is the coefficient while is the literal part.

• : in this case is the coefficient while is the literal part.

From now on, when we are dealing with monomials, we always mean monomials already in normal form.

Similar, Equal and Opposite Monomials

Before we can study the operations that can be carried out on monomials, it is very important to know how to compare two or more monomials with each other.

So we begin with the definition of similar monomials:

As can be seen from the definition, the fact that two monomials are similar is independent from the coefficient. Let's look at some examples:

• and are two similar monomials because their literal part is identical:

• and are two similar monomials since their literal part is identical:

• and are not similar monomials: their literal parts differ:

When two similar monomials also have the same coefficient, they are equal monomials:

Let's look at some examples:

• and are two similar monomials because their literal part is identical but they are not two equal monomials because they have different coefficients.

• Conversely, and are two similar monomials because their literal part is identical and they are also two equal monomials because they have the same coefficient.

Similarly, we can define opposite monomials:

For example and are two opposite monomials because they are similar monomials, i.e. with the same literal part, and coefficients equal in absolute value but opposite in sign:

Degree of a monomial

We conclude this introductory lesson on monomials by introducing the concept of the degree of a monomial. In particular, two concepts of degree exist, and both apply only to the case of monomials in normal form:

To make the concept clearer, let us look at a few examples. Take the following monomial in normal form:

This monomial has degree 2 with respect to the letter and degree 1 with respect to the letter . On the other hand, its overall degree is 3, since it is equal to the sum of the degrees with respect to and .

Let's look at some other examples:

Of the above examples, some are particularly interesting:

• has degree 2 with respect to . With respect to , however, it has degree 0. This is because, even if does not appear, the monomial can be rewritten as .

• , as a number, is to all effects a monomial. its degree is, however, 0 with respect to any letter. Therefore, its overall degree is zero.

From this last example, we deduce that any number is a monomial of degree zero both in total and with respect to any letter.

A separate case is the number 0 or null monomial:

In Conclusion

This lesson has been useful in introducing the basic concepts of monomials.

We have seen what a monomial is and what the normal form of a monomial is, and how any monomial can be reduced to normal form using the properties of multiplications and powers. From here, we gave the definition of coefficient of a monomial and literal part of a monomial.

Starting from the normal form, we saw how to compare two monomials with each other by introducing the concepts of similar monomials, equal monomials and opposite monomials.

Finally, in this lesson, we talked about the degree of a monomial. In particular, both the degree of a monomial with respect to a letter and the total degree of a monomial.

All of these concepts are preparatory to the study of operations on monomials, which we will look at in the next lessons.