When you learn mathematics, you may be taught about the commutative property (or commutative law) and the associative property (or associative law). The commutative property and the associative property are taught in elementary or junior high school.

The commutative property and the associative property are formed by addition and multiplication. The commutative property and the associative property cannot be established by subtraction and division. However, subtraction can be converted to addition and division can be converted to multiplication. This means that we can use these rules in all calculations.

It is important to understand these rules in mathematics. Understanding the commutative property and the associative property will allow us to perform all kinds of calculations. Although we don’t use division anymore in math in junior high school and above, the commutative property and the associative property can help us understand why this is the case.

We will explain how to think about and apply the commutative law and the associative law.

## An Overview of the Commutative Law and the Associative Law

What is the content of the commutative property and the associative property that is taught in elementary or secondary school? The meanings of these are as follows.

• Commutative property: the law that gives the same answer even if the numbers are interchanged
• Associative property: the law that gives the same answer even if you change the place of parentheses.

Regarding the commutative property and the associative property, both of which are used in so many situations, they are essential knowledge when solving math problems. These laws are used in addition and multiplication. But in practice, they can also be applied to subtraction and division.

In other words, we can use the commutative property and the associative property in all calculations. This makes it convenient, and these laws are useful for everyone, including those who do mathematical calculations in high school, college, and even in the business world.

## Under the Commutative Property, Replacing the Numbers Gives the Same Answer

In the commutative property, if you swap numbers, the answer will be the same. The commutative law is satisfied by addition.

For example, we have the following.

It doesn’t matter how you rearrange the order; this is called the commutative property of addition.

The same is true for multiplication. The commutative property is satisfied not only in addition but also in multiplication. As shown below, the answer is the same in multiplication, even if the numbers are interchanged.

It doesn’t matter how you multiply, and this is called the commutative property of multiplication.

### Isn’t the Commutative Property Established by Subtraction and Division?

However, it is said that the commutative law is not valid for subtraction and division. Let’s try to calculate by actually swapping numbers.

For example, if we interchange the numbers for subtraction, we get the following.

Thus, the answers are different. In one equation, the answer is 2, and in the other equation, the answer is -2. Since we subtract different numbers, the answers will naturally be different. This is the reason why it is said that the commutative property is not valid in subtraction.

The same is true for division. In the case of division, it is as follows.

Using the commutative property with division changes the answer to the question. Therefore, the commutative law is not valid for division.

### Correcting the Signs and Making Them Negative Numbers, the Commutative Property Is Established

Since the commutative property is only valid for addition and multiplication, its usage is limited and seems to be meaningless. However, the commutative law that we learn in elementary school or junior high school mathematics is very important.

Why is the commutative property so important even though it only applies to addition and multiplication? It’s because by changing subtraction to addition, the commutative property becomes valid. For division, the commutative property is also available by changing it to multiplication.

Addition and subtraction are the same. Subtraction can be converted to addition. For example, the following are all the same equations and the same answer.

• $4-2=2$
• $4+(-2)=2$

Once you learn positive and negative numbers, you can understand that addition and subtraction are the same.

As mentioned above, the commutative property is valid for addition. So, by changing subtraction to addition of negative numbers, the commutative law is established.

What is important is the fact that the commutative property is valid even for negative numbers. We have explained above that the commutative property is not established for subtraction. In fact, however, we can use the commutative property of addition even for subtraction, because the commutative law is established by changing subtraction to addition.

-Use Fractions in Division and Use the Commutative Property

The same is true for division. By converting division to multiplication, we can use the commutative property of multiplication.

How do we convert division into multiplication? This is done by using fractions. By using reciprocal numbers, we can convert division into multiplication of fractions. Divisions are also multiplication of fractions, as shown below.

As mentioned above, the commutative property is valid in multiplication. As shown below, the answer is the same in multiplication, even if it is a fraction.

If you want to correct subtraction to addition, you just need to change the sign of the plus and minus. But in the case of division, you have to convert it into fractions. This requires a bit of work. In any case, you can use the commutative property even in division.

## The Associative Property That Can Change the Place of Parentheses

When we study the commutative property, there is the associative property that we learn at the same time. What kind of law is the associative law?

The associative property allows us to change the place of the parentheses freely. The same as the commutative property, the associative property is only valid for addition and multiplication. As shown below, the answer is the same even if we change the position of the parentheses in addition.

In math, the rule is to calculate the inside of the parentheses first. It doesn’t matter where the parentheses are placed in addition.

In the same way, the associative property is valid for multiplication.

Why is the associative law valid? It is because the meaning is the same with or without the parentheses. The answer is the same even if you remove the parentheses as follows.

The answer is the same no matter where you put the parenthesis, because it doesn’t matter if there are parentheses or not. The associative property is just a law of the obvious.

### The Associative Law That Can Be Calculated From Anywhere

The associative property is, in essence, a law that you can calculate from anywhere. For example, we have all done multiplication calculations by changing the order of the numbers. For example, how would you calculate the following?

• $2×6×13$

From left to right, the calculations are as follows.

• $\textcolor{red}{2×6}×13=\textcolor{red}{12}×13=156$

You need to calculate $12×13$, which makes the calculation more difficult. So let’s use the associative property and change the order of the numbers. For example, instead of $2×6×13$, we change the equation to $6×13×2$. In this case, we get the following.

• $\textcolor{red}{6×13}×2=\textcolor{red}{78}×2=156$

Thus, the calculation is easier; multiplying a number with fewer digits is less likely to result in calculation errors than multiplying two numbers by two digits.

### The Formula of Subtraction and Division Is Formed by Changing the Sign

The associative property is valid for addition and multiplication formulas. It is the same as the commutative property that cannot be applied to subtraction and division. For example, in subtraction, changing the parentheses will change the answer as follows.

If there are parentheses in the equation, we must first calculate the inside of the parentheses. So when there is subtraction, the answer changes by using the associative property.

The same thing happens in division. The answer changes when the parentheses change, as shown below.

For subtraction and division, the associative property is not valid.

-Calculate by Addition or Multiplication

However, in practice, it doesn’t matter if there is subtraction or division. Just like the commutative property, subtraction can be corrected to addition and division can be corrected to multiplication. It is as follows.

It can be modified into an equation for addition or multiplication, so that the associative property is established. The answer is the same no matter where you put the parenthesis. In other words, you can do the calculation from any place first.

In general, the associative property is not available for subtraction and division. However, by correcting it to addition or multiplication equations, the associative law becomes valid. Therefore, the associative property is a rule that can be used in all calculations.

## Use Multiplication of Fractions by Reciprocal Numbers Instead of Using Division in Math

Why do we need to understand the commutative property and the associative property, including elementary school and junior high school math? This is because they are essential for easier calculations. Understanding the commutative property and the associative property is especially important for division calculations.

When you convert subtraction to addition, the method is simple. Just make the addition and then make it a negative number. It is as follows.

• $2\textcolor{red}{-3}=2\textcolor{red}{+(-3)}$

For division, on the other hand, you can’t calculate it as it is. In order to change division to multiplication, you have to modify it to multiply fractions. By using reciprocal, we must change the shape of the numbers.

• $2\textcolor{red}{÷3}=2\textcolor{red}{×\displaystyle\frac{1}{3}}$

You can calculate from anywhere by using only multiplication equations.

We don’t use division in math above secondary school because the commutative property and the associative property is not valid. We don’t use division in math, including junior high school and high school math. Instead, we calculate after converting division into fractions.

And in the case of division, there are many cases where the numbers are indivisible. With fractions, on the other hand, you can get an answer even if you can’t divide. It is important to understand that division is less useful and has almost no use in mathematics.

The reason why we all convert division to multiplication in secondary school and above in math is so that the commutative property and associative property can be used. This makes it much easier to calculate and less likely to make calculation errors.

## Exercises: the Commutative Property and the Associative Property for Positive and Negative Numbers

Q1: Do the following calculation.

1. $(6 ÷ 15) × 5$
2. $-4 ÷ 3 ÷ 6 × 15$

A1: Answers.

For addition and subtraction, you can solve problems without trouble when you use the commutative law and associative law. On the other hand, if division is involved, you need to correct it to multiply by a fraction. So use the reciprocal number and make it a fraction.

(a)

$(6\textcolor{red}{÷15})×5$

$＝(6\textcolor{red}{×\displaystyle\frac{1}{15}})×5$

$=6×\displaystyle\frac{1}{3}=2$

In the division, we always correct it to multiply of fractions. Then, in this calculation, we calculate $\displaystyle\frac{1}{15}×5=\displaystyle\frac{1}{3}$ first. It’s easier to calculate than $6×\displaystyle\frac{1}{15}$.

(b)

$-4÷3÷6×15$

$=-4×\displaystyle\frac{1}{3}×\displaystyle\frac{1}{6}×15$

$=\textcolor{red}{-4×\displaystyle\frac{1}{6}}×\textcolor{blue}{\displaystyle\frac{1}{3}×15}$

$=\textcolor{red}{-\displaystyle\frac{2}{3}}×\textcolor{blue}{5}$

$=-\displaystyle\frac{10}{3}$

First, we convert division into multiplication of fractions. Then, by using the commutative property and the associative property, we replace the numbers to make them easier to calculate.

Calculating with $-4×\displaystyle\frac{1}{3}$ and $\displaystyle\frac{1}{6}×15$ is difficult. It is also difficult to multiply fractions by each other.

On the other hand, $-4 × \displaystyle\frac{1}{6}$ and $\displaystyle\frac{1}{3} × 15$ allow you to reduce the number, making the calculation easier. In a multiplication-only equation, you can move the numbers freely. In this way, we use the commutative law and the associative law.

## Learn the Definition of the Law and Use it in Formulas

There are several laws in mathematics. One of those laws is the commutative property and the associative property. Why do we need to understand the rule? It is because the laws make it easier to calculate the equations and reduce calculation errors.

Two of the most important laws in mathematics that we all use are the commutative property and the commutative property. Even in elementary school math, we all use the commutative property and the commutative property.

However, you have to understand the rules and definitions of the laws in order to use them in mathematical formulas. The commutative property and the commutative property are only valid for equations with addition or multiplication. So if there is subtraction or division, correct it to addition or multiplication. Note that it is easy to correct subtraction, but with division, you must change it to a fraction.

Also, if you learn the commutative law and the associative law, you can understand why no one uses division in math above junior high school. Division is not useful. Be sure to correct the multiplication of fractions by reciprocal numbers before you calculate equations.