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What are the Properties of Math?

Math is a universal language. From algebra to calculus, the principles and properties of math are always the same.
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What are the Properties of Math?
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Math is a universal language. Every mathematician knows this, and there have been studies that back up the idea that math is a useful tool that can be used across cultures.

You might not be able to speak every language, but you can do math in any language, validating that math is the same everywhere. Furthermore, from algebra to calculus, the principles and rules of math are always the same. No matter where you are, 2 plus 2 will always equal 4.

Thankfully, some math properties make the subject a little easier to understand. In this article, we'll look at some of the most important basic properties of math as they relate to real numbers.

What are the Properties of Math?

In mathematics, a property is a characteristic that always remains true. Number properties are at the core of mathematics and help us to understand and work with complex equations and problems. Certain properties will always hold true whether you're dealing with addition, multiplication, division or anything else.

And when you know a property exists, you can use it to your advantage. For instance, if you're trying to solve a complex equation, you can use properties to simplify it and make it easier to solve.

Why Knowing the Properties of Math is Important

Unlike some other areas of mathematics, the properties of math do not just apply to equations and problems but are fundamentally related to real numbers.

Real numbers refer to all natural, rational, whole, integers and negative numbers.

Understanding the properties of math can also help you develop a better understanding of mathematics as a whole.

Understanding Number Properties

Number properties are the set of rules that dictate how numbers behave when they are added, subtracted, multiplied or divided.

Numbers have four main properties: distributive, associative, commutative and identity property, and each governs how a mathematical operation is carried out.

Without further ado, let's look at some of the most important properties of mathematics.

1. The Commutative Property

The commutative property is one of the most fundamental properties of mathematics. The commutative property states that the order in which you add or multiply numbers doesn't matter.

In other words, a + b = b + a and a × b = b × a.

From the example, with commutative property, you can change the position of the real numbers, but this rearrangement will not affect the sum. Even if you change the position of two numbers, the sum will remain the same.

Hence, just like the additive commutative property expressed here:

5 + 2 = 7 and 2 + 5 = 7

Notice that the sum remained the same despite the rearrangement.

We can consider another example with the equation 3×4. If we use the commutative property, we can rearrange the equation to 4×3. If you follow the rule, multiplying the two numbers in a different grouping/order will give the same answer.

2. The Associative Property

The associative property states that the order in which you group numbers doesn't matter. The associative property derives its name from the word ‘associate’.

We can define associative property as the grouping of three or more numbers in different orders and still arriving at the same sum after the addition or multiplication of the numbers.

In other words, no matter how the numbers are grouped, you can get the same answer.

Here are some examples

1 + (2+3) = (1+3) + 2

5 × (3 × 4) = (4 × 5) × 3

Here is another example, consider the equation (5+3)+2. Using the associative property, we can rearrange the equation to 5+(3+2).

With the associative property, we can group the numbers in any order we like and still arrive at the same sum or product.

The Associative Property of Addition

The associative property of addition means that when adding three or more numbers, regardless of the way you order the numbers, the sum or outcome will remain the same.


2 + (3 + 6) = (2 + 3) + 6

a (b + c) = (a+ c) = b

The Associative Property of Multiplication

The associative property of multiplication states that regardless of how numbers are grouped or rearranged, the final answer or product will be the same after multiplying the numbers.


4 (2 × 3) = ( 3 × 4) = 2

y(x × z) = (y × x) = z

Hence, Associative Property of Addition: 3 + (4 + 5) = (3 + 4) + 5

Associative Property of Multiplication: 2 (3 ×  4) = (2 ×  3) 4

Though associative and commutative properties may seem similar, there is a difference between them. While the associative property deals with three or more numbers, the commutative property only has two.


Associative property: 4 (2 × 3) = ( 3 × 4) = 2

Commutative property: 5 + 2 = 7 and 2 + 5 = 7

3. The Distributive Property

The distributive property is another important property of mathematics, and it states that you can distribute a number across an addition or subtraction equation.

For example, consider the equation 3 (4 + 5). 

Using the distributive property, we can distribute the number 3 across the addition equation to get 3 × 4 + 3 × 5.

The distributive property is also called the distributive property of multiplication over addition and subtraction because the operands outside the bracket can be distributed between the operands or numbers in the parentheses.

For instance, in equation 2(3-4), the number 2 will be distributed or used in multiplying both numbers in the bracket before subtraction is done.

Thus, 2 (3 - 4) = 2×3 - 2×4

The same principle also goes for the distributive property of multiplication over division. If you can understand and remember the distributive property, you'll be well on your way to mastering algebra.

4. The Identity Property

Identity property states that when you introduce identity elements to numbers, it does not change the value of the number. Rather the number remains the same.

Identity property elements are 0 for addition and subtraction and 1 for multiplication and division.

For instance, 5 + 0 = 5

3 × 1 = 3

Additive identity property

Additive identity property is when the number 0 is added to any number; the value of the number does not change.

For example, 2 + 0 = 2

0 + 4 = 4

Multiplicative identity property

The multiplicative identity element is 1. The multiplicative identity property states that when the number 1 is multiplied by any other number, it does not change the value of that number.

For example, 3 × 1 = 3

1 × 7 = 7

When to Consider Finding a Math Tutor

Learning concepts like the properties of math may open up a whole new world for you. However, it may be time to seek a math tutor if you need more help understanding these concepts. You may seriously need to consider finding a tutor when:

  • You're consistently getting low grades in math class
  • You're having difficulty understanding new concepts
  • You're feeling overwhelmed or stressed about math class or math lesson
  • You're falling behind in math class

If you're struggling with math, don't be afraid to seek out help. There's dignity in admitting that you need help. In fact, getting a tutor may be the best decision you ever make.

A good math tutor can help you understand concepts, catch up in class, and improve your grades. If you're struggling with math, consider finding a tutor today. 

At Learner, we can connect you with a tutor to help you reach your math goals. Contact us today to learn more.

What to expect when working with an online math tutor at Learner

We understand the difficulties that can come with learning math online, and that's why we've developed a system that makes it easy for you to get the help you need when you need it.

Whether you are looking to brush up on your math skills and ace your exams, or you are starting from scratch, we can help. We offer a wide range of services, from online math tutoring to test prep.

When you work with an online math tutor at Learner, you can expect the following:

A personalized learning experience

Our tutors will develop a custom math tutoring plan based on your individual needs. Your success is the most important thing to us, and we will do everything we can to help you reach your goals.

Flexible scheduling

We know you have a busy schedule, and we understand that you must juggle school and other obligations. That's why we offer flexible scheduling to fit your busy schedule. You can choose when and where to learn, and we will support you.


Struggling with math is challenging, and we get it. Our tutors are empathetic and will never make you feel bad for struggling. They understand the difficulties of trying to learn math, and they will help you overcome these difficulties.

We also understand the definition of success and what it means to you and your educational pursuits. We can help you understand the concepts, catch up in class, and even improve your grades.


We can relate to feeling frustrated when trying to learn math. We understand that the concepts can be confusing and difficult to grasp.

We know you must have spent hours studying in the library, studying your math textbook, writing down questions you attempted to answer, and even watching tons of videos on YouTube.

Our tutors are patient and understanding. We will take the time to listen to your concerns and help you overcome any challenges you may be facing, track your progress in your math course, and ensure you pass.

Math Expertise

Whether it's calculus, algebra, or anything in between, our online math teachers are experts in their field. They will be able to answer any question and help you understand even the most difficult concepts.

If you're looking for help with math, look no further than Learner. Our tutors are passionate about helping students succeed.

Affordable rates

We offer competitive rates and flexible payment options to make sure that our services are affordable for everyone. With Learner, it doesn't cost an arm and a leg to get the help you need.


No matter what level you are working at, the properties of math are still an important part of math. To succeed in mathematics, you must possess a strong understanding of the properties of math, including the rules that govern them, just as we have discussed.

From commutative to associative, there are a variety of properties that govern how every mathematical operation is carried out. And we have explained what each is and how they are used in mathematics. Be sure to go through the examples again to grasp the topic better.

If you are struggling with the concept of math properties or other topics like algebra, calculus, or geometry, it is important to seek professional help. A math tutor can work with you one-on-one to help you understand the concept and improve your grades.

At Learner, we offer affordable, flexible, and personalized math tutoring services to help you succeed. Contact us today to learn more.

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About the author:

Mike developed his passion for education as a math instructor at Penn State University. He expanded his educational experience launching and running an Executive Education business - training over 100,000 students per year. As the CEO of Learner, Mike focuses on accelerating learning and unleashing the potential of students. 

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