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What Are the Properties of Real Numbers?

properties of real numbers: Colorful diagram showing the different types of numbers

In math, there are many different sets of numbers to work with. One set of numbers is known as real numbers. When it comes to the properties of real numbers, there are several important properties to know:

  • Additive identity
  • Multiplicative identity
  • Commutative property of addition
  • Commutative property of multiplication
  • Associative property of addition
  • Associative property of multiplication
  • Distributive property of multiplication

A property is just a way of expressing a characteristic or something that is true about real numbers. Knowing these properties will help you reduce or simplify algebraic expressions. Let's look at each property in detail to understand them better.

Identity Properties of Real Numbers

Keep in mind that all these properties apply to any real number, including positive numbers, negative numbers, numbers with fractions, and numbers with decimals. The first two properties are called identity properties. There is an additive identity and a multiplicative identity.

0 is called the "additive identity" to all real numbers. This simply means that for any real number a:

properties of real numbers: Equation that shows additive identity

In other words, whenever you add 0 to a number, it equals itself. For example, 5 + 0 always equals 0.

1 is called the "multiplicative identity" to all real numbers. This identity states that for any real number a:

properties of real numbers: Equation that shows multiplicative identity

In other words, multiplying a number by 1 always results in itself. For example, 8 x 1 still equals 8.

Commutative and Associative Properties of Real Numbers

Next, the commutative property states that changing the order of adding or multiplying real numbers does not change the result. There are both additive and multiplicative commutative properties to all real numbers.

Here’s an example of the additive commutative property:

properties of real numbers: Example of an equation with additive commutative property

properties of real numbers: Example of an equation with additive commutative property

It does not matter in what order you add the numbers together — the result is still 11.

Here’s an example of the multiplicative commutative property:

properties of real numbers: Example of an equation with multiplicative commutative property

properties of real numbers: Example of an equation with multiplicative commutative property

It does not matter in what order you multiply the numbers together, the result is still 24.

Associative Property

The associative property states that changing the grouping of adding or multiplying real numbers does not change the result.

Here’s an example of the additive associative property:

properties of real numbers: Example of an equation with additive associative property

Example of an equation with additive associative property

Here’s an example of the multiplicative associative property:

Example of an equation with multiplicative associative property

Example of an equation with multiplicative associative property

Distributive Properties of Real Numbers

The distributive property applies when you are simplifying expressions with parenthesis. It is also known as the distributive property of multiplication, and it states that for any real numbers a, b, and c:

Example of an equation with distributive property of multiplication

This typically comes up when you are doing an order of operations problem. Remember the P in PEMDAS stands for parenthesis. In the simple example 2(3 + 5), if you were to calculate without the distributive property, you must first calculate the term inside the parenthesis:

Equation showing the order of operations PEMDAS

If you were to calculate by utilizing the distributive property, you would simplify first to:

Equation showing the order of operations PEMDAS

Then, you do the M (multiplication) in PEMDAS before you do the A (addition) to get the same result.

Equation showing the order of operations PEMDAS

Working With the Properties of Real Numbers

To summarize, these are well-known properties that apply to all real numbers:

  • Additive identity
  • Multiplicative identity
  • Commutative property of addition
  • Commutative property of multiplication
  • Associative property of addition
  • Associative property of multiplication
  • Distributive property of multiplication

The commutative and associative properties are properties that apply to both addition and multiplication when it comes to the order and grouping of adding and multiplying real numbers. The distributive property holds true for all real numbers when you are simplifying an expression with a parenthesis in it.

And don't forget that the identity property of addition states that adding 0 to any real number equals itself, and the identity property of multiplication states that multiplying any real number by 1 also equals itself. These properties all hold true for any real number and are useful properties to understand and learn!

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