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Lesson Explainer: Properties of Operations over the Real Numbers Mathematics • 9th Grade

In this explainer, we will learn how to solve problems involving operations and properties of operations on real numbers.

We recall that we can represent any real number as a point on a number line. This allows us to perform operations on real numbers geometrically rather than numerically and this can explain a few useful properties of the operations on the real numbers.

First, let’s consider two real numbers on a number line; let’s consider 2 and 3 as shown.

We know that the point representing 2 on the number line has a displacement of 2 units from the point representing 0. Similarly, we can say that the point representing 3 on the number line is a displacement of 3 units from the point representing 0.

We can then consider the value of 2+3. We can find a point on the number line with a displacement of 2+3 units from the point representing 0 by adding the displacements.

Since there is a point on the number line representing 2+3, this tells us it should be a real number. In fact, we can apply this same process to find a point on the number line representing the sum of any two real numbers. Hence, we should have that the sum of any two real numbers is also a real number. This is known as the closure property of the addition of real numbers, and we can write this formally as follows.

Property: The Closure Property of the Addition of Real Numbers

For any 𝑎, 𝑏 we have 𝑎+𝑏.

It is worth reiterating that this is not a proof of this fact but is an illustration of why we choose this definition. We can find similar reasons for many more properties. For example, let’s go back to our addition of 2 and 3. This time, we will find 3+2.

We follow the same process, and we can note that the order we combine the displacements in will not change the final displacement. Hence, 3+2=2+3. We could apply this to any two or more real numbers to show that we should be able to add real numbers in either order, which explains the following property.

Property: The Commutativity Property of the Addition of Real Numbers

For any 𝑎, 𝑏 we have 𝑎+𝑏=𝑏+𝑎.

Next, we can consider adding 0 to any number. We note that adding a displacement of 0 will not change the overall position, so adding 0 to any real number should not change its value. We call this the additive identity of real numbers since it leaves real numbers unchanged under addition.

Property: Additive Identity of Real Numbers

For any real number 𝑎, we have 𝑎+0=𝑎.

0 is known as the additive identity element (often shortened to additive identity) for real numbers.

We can combine this property with the commutativity of addition of real numbers to show that 0+𝑎=𝑎 for any real number 𝑎.

Next, we can consider what happens when we add a real number to its negative. For example, consider 6+6. The displacement from the origin of the point representing this number should be 6 units in the positive direction and then 6 units in the negative direction.

These displacements cancel to give the point of displacement 0 units from 0. Hence, we should have that 6+6=0.

This can be applied to any real number, and we call 6 the additive inverse of 6 since they sum to give the additive identity.

Property: Additive Inverse of Real Numbers

For any real number 𝑎, we have that 𝑎 and 𝑎+(𝑎)=0.

𝑎 is called the additive inverse of 𝑎.

There are many other properties we can find reasons for geometrically. However, it is beyond the scope of this explainer to find reasons for all of the necessary properties. Instead, we will state the remaining properties.

Property: The Associativity Property of the Addition of Real Numbers

For any real numbers 𝑎, 𝑏, and 𝑐, we have that (𝑎+𝑏)+𝑐=𝑎+(𝑏+𝑐).

There are also a set of equivalent properties for the multiplication operation of real numbers, which we list below.

Property: Properties of the Multiplication of Real numbers

  • For any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏. This is equivalent to saying that the multiplication of real numbers is closed.
  • For any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑏×𝑎. This is known as the commutativity of real number multiplication.
  • For any real numbers 𝑎, 𝑏, and 𝑐, we have that (𝑎×𝑏)×𝑐=𝑎×(𝑏×𝑐). This is known as the associativity of real number multiplication.
  • For any real number 𝑎, we have that 𝑎×1=𝑎. We call 1 the multiplicative identity for the real numbers since it leaves all real numbers unchanged after multiplication.
  • For any real number 𝑎 not equal to zero, we have that 𝑎×1𝑎=1. Where 1𝑎 is called the multiplicative inverse of 𝑎.

Let’s now see some examples of applying these properties to answer questions about expressions involving real numbers.

Example 1: Understanding the Commutative Property of the Multiplication of Real Numbers

Fill in the blank: 21×2=×21.

Answer

We first recall that the multiplication operation of real numbers is commutative. In other words, if 𝑎, 𝑏, then 𝑎×𝑏=𝑏×𝑎. Applying this property with 𝑎=21 and 𝑏=2, we have 21×2=2×21.

Hence, we can fill in the blank with 2.

In our next example, we will determine the additive inverse of a given real number.

Example 2: Finding the Additive Inverse of a Number

What is the additive inverse of 6585?

Answer

We recall that the additive inverse of a real number 𝑎 is the number 𝑎 since 𝑎+(𝑎)=0. Hence, one expression for the additive inverse of 6585 is 6585. However, we can find an exact expression of this inverse by applying some of the other properties of the addition of real numbers.

First, we note that 65+65=0 and 85+85=0. Next, by applying the associativity and commutativity properties of real number addition, we can show that 6585+65+85=65+65+85+85=0+0.

Finally, 0 is the additive identity. Hence, 6585+65+85=0.

We can reorder this using the commutativity of the addition to have the positive term first. This gives us that the additive inverse is 8565.

In our next example, we will explore the idea of the identities of operations on the real numbers.

Example 3: Identifying the Identities of Operations

  1. Which number is the additive identity for the real numbers?
  2. Which number is the multiplicative identity for the real numbers?

Answer

An identity for an operation is an element that leaves all other elements unchanged by the operation.

Part 1

Therefore, the additive identity for the real numbers must leave all real numbers unchanged under addition. We note that, for any real number 𝑎, we have that 0+𝑎=𝑎+0=𝑎.

Hence, 0 is the additive identity of the real numbers.

Part 2

Similarly, the multiplicative identity for the real numbers must leave all real numbers unchanged under multiplication. We note that, for any real number 𝑎, we have that 1×𝑎=𝑎×1=𝑎.

Hence, 1 is the multiplicative identity of the real numbers.

Let’s now determine the multiplicative inverse of a given real number.

Example 4: Finding the Multiplicative Inverse of a Number

Find the multiplicative inverse of 1214.

Answer

We start by recalling that the multiplicative inverse of a nonzero real number 𝑎 is given by 1𝑎 since 𝑎×1𝑎=1.

Therefore, the multiplicative inverse of 1214 is 1. We can simplify this expression by recalling that dividing by a fraction is the same as multiplying by its reciprocal, that is, the fraction obtained by exchanging the positions of the numerator and denominator. Hence, 1=1412.

It is worth noting we do not actually need to use the reciprocal property of fractions to answer this question. Instead, we could have noted that, by using the associativity and commutativity of the multiplication of real numbers, we have 1214×1412=(12)×114×14×112=12×112×114×14.

Then, by using the multiplicative inverse property, we can simplify this expression as follows: 12×112×114×14=1×1.

Finally, since 1 is the multiplicative identity, we have 1×1=1.

Hence, 1214×1412=1.

This can be generalized to justify the property that dividing by a fraction is the same as multiplying by its reciprocal.

Using either method, we have that the multiplicative inverse of the given number is 1412.

In our next example, we will consider if the subtraction operation on real numbers is commutative or associative.

Example 5: Identifying Whether Subtraction is Commutative or Associative

  1. True or False: 26157=(2615)7.
  2. Is subtraction associative or nonassociative in ?
  3. True or False: 715=157.
  4. Is subtraction commutative or noncommutative in ?

Answer

Part 1

We want to compare the equality of the two given values; we can do this by simplifying each expression. Let’s start with 26157. We cannot evaluate the expression inside the parentheses, so, instead, let’s distribute the negative over the parentheses. We have 26157=26+15+7.

We can use the associativity and commutativity of the addition of real numbers to simplify this expression, where we note subtracting 15 is the same as adding 15: 26+15+7=(26+(15))+7=11+7.

We can follow a similar process for (2615)7. We evaluate the expression inside the parentheses to get (2615)7=117.

We know that 7 is nonzero (since 0 is not 7), so we can conclude that 11+7117.

Hence, the statement is false.

Part 2

We recall that saying an operation is associative means that when given an expression involving two operations, we can evaluate the operations in either order. In part 1, we showed that 26157(2615)7.

In other words, if 𝑎=26, 𝑏=15, and 𝑐=7, then 𝑎(𝑏𝑐)(𝑎𝑏)𝑐.

Therefore, we cannot evaluate the subtraction of real numbers in any order, so the operation is nonassociative in .

Part 3

We want to compare 715 and 157. There are a few ways of doing this; one way is to note that 2<7<3, since 2<7<3. Since 7<3, we know that 715<0.

However, 157>0.

Thus, they are not equal, so the statement is false.

Part 4

We begin by recalling that an operation is commutative if the order of the elements in the operation leaves the value unchanged. We can check some values to see if the operation seems commutative. In part 3, we showed that 715157.

In other words, if 𝑎=7 and 𝑏=15, then 𝑎𝑏𝑏𝑎.

Therefore, we cannot reorder the subtraction of real numbers, so the operation is noncommutative in .

It is also worth noting that we can show that division is not commutative or associative in the same way. We see that 1÷2=12,2÷1=2.

Hence, division of real numbers is not commutative. Similarly, (1÷2)÷3=16,1÷(2÷3)=32.

Hence, division of real numbers is not associative.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • The addition operation of real numbers has the following properties:
    • The closure property of the addition of real numbers tells us that the sum of any two real numbers is also a real number.
    • The commutativity property of the addition of real numbers tells us that, for any real numbers 𝑎 and 𝑏, we have 𝑎+𝑏=𝑏+𝑎.
    • For any real numbers 𝑎, 𝑏, and 𝑐, we have that (𝑎+𝑏)+𝑐=𝑎+(𝑏+𝑐).
    • 0 is known as the additive identity for real numbers since, for any real number 𝑎, we have 𝑎+0=𝑎.
    • For any real number 𝑎, we have that 𝑎 and 𝑎+(𝑎)=0. 𝑎 is called the additive inverse of 𝑎.
  • The multiplication operation on real numbers has the following properties:
    • For any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏. This is equivalent to saying that the multiplication of real numbers is closed.
    • For any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑏×𝑎. This is known as the commutativity of real number multiplication.
    • For any real numbers 𝑎, 𝑏, and 𝑐, we have that (𝑎×𝑏)×𝑐=𝑎×(𝑏×𝑐). This is known as the associativity of real number multiplication.
    • For any real number 𝑎, we have that 𝑎×1=𝑎. We call 1 the multiplicative identity for the real numbers since it leaves all real numbers unchanged after multiplication.
    • For any real number 𝑎 not equal to zero, we have that 𝑎×1𝑎=1. Where 1𝑎 is called the multiplicative inverse of 𝑎.
  • The subtraction and division operations of real numbers are not commutative and not associative.

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