### Unit 1: Number Properties

In this unit, we will discuss some of the basic algebraic properties you may already know, that many math instructors call "common sense" properties. We see uses of these properties every day.

For example, the commutative property tells us we can rearrange the order of the numbers and still get the same result: 3 + 2 = 5, and 2 + 3 = 5. The same is true for multiplication: 2 × 3 = 6, and 3 × 2 = 6.

Other algebraic properties are less intuitive. For example, what happens when we multiply or divide a number by zero? Understanding these number properties will give you the foundation you need for this course.

**Completing this unit should take you approximately 4 hours.**

Upon successful completion of this unit, you will be able to:

- apply the commutative law of addition and multiplication;
- apply the associative law of addition and multiplication;
- apply the identity property of addition and multiplication;
- apply the inverse property of addition and multiplication;
- apply the zero property of multiplication and division; and
- apply the distributive property.

- apply the commutative law of addition and multiplication;

### 1.1: Commutative Law of Addition and Multiplication

It is important to note that the commutative property only holds for addition and multiplication. It does not work for subtraction or division. We can make a simple example to show this: 10 − 2 = 8, but 2 − 10 = −8.

The results are not the same when we change the order of the numbers in subtraction. Likewise, 20/2 = 10, but 2/20 = 0.10. Again, the results are not the same when we change the order of the numbers in division.

Read the first half of this section. Do example 7.5 and Try It exercises 7.9 and 7.10 to practice using the commutative property.

Watch this video for additional explanation and examples of the commutative law of addition.

Watch this video for additional explanation and examples of the commutative law of multiplication.

### 1.2: Associative Law of Addition and Multiplication

The associative law of addition and multiplication tells us that the grouping of numbers in addition and multiplication does not change the result. In math, when we put parentheses around a set of numbers, we do the calculation in the parentheses first. For example, 5 + (2 + 1) = 8. When we calculate this, we first calculate 2 + 1 = 3. Then, we add 5 to get 8. We can also write (5 + 2) + 1 = 8. Here, we first calculate 5 + 2 = 7, and then add one to get a sum of eight. We get the same result regardless of how we group the numbers.

The same law works for multiplication. We can compute (2 × 2) × 3 = 12. We can change the grouping and write 2 × (2 × 3) = 12. The placement of the grouping does not change the answer.

Read this section up to example 7.6. Then, do example 7.6 and Try It exercises 7.11 and 7.12.

Watch these two videos for more examples of the associative law as it relates to addition and multiplication.

Take this quiz to practice your understanding of the associative and commutative laws.

### 1.3: Identity Property of Addition

In math, the identity is a number that can be added or multiplied to another number to give the same number. The identity property of addition states that if you add zero to a number, the sum will be that number. That is, 0 + 5 = 5.

Watch this video. You should be able to recognize an equation that shows the identity property of addition.

### 1.4: Inverse Property of Addition

The inverse property of addition states that a number plus its negative (or inverse) will equal zero. That is, 5 + −5 = 0.

Read the first paragraph of this section to see examples of how the inverse property of addition is applied. Then, do example 7.34 and Try It exercises 7.67 and 7.68.

Watch this video for additional examples.

### 1.5: Identity Property of Multiplication

Much like the identity property of addition (section 1.3), the identity property of multiplication states the identity for multiplication. For multiplication, the identity property states that if you multiply any number by 1, the answer is that number. For example, 3 × 1 = 3.

Watch this video for more examples of the identity property of multiplication.

### 1.6: Inverse Property of Multiplication

The inverse property of multiplication tells us that any number times its inverse (1 divided by the number) will equal one. For example, 2 × ½ = 1.

Read this section to see examples of how the inverse property of addition is applied. In the first multiplication example, we can use the inverse property for fractions as well as whole numbers. Do example 7.35 and Try It exercises 7.69 and 7.70.

Watch this video for more examples.

### 1.7: Multiplication by Zero

When we multiply any number by zero, the result is always zero. For example, 15 × 0 = 0. Likewise, because of the commutative property, 0 × 15 = 0.

Read this section to see this definition written formally.

### 1.8: Dividing by Zero Is Undefined

Because division does not follow the commutative property, we must consider two cases when doing division with zero. First, we can divide zero by a number. Zero divided by any number is zero. For example, 0/5 = 0.

Now we must consider the case when a number is divided by zero. When we divide a number by zero, the answer is undefined. For example, 5/0 = undefined. In math, undefined means that there is no possible answer.

Watch this video. Try to sum up the argument being made in the video in your own words as to why it is impossible to divide a number by zero. What problem happens as we divide one by smaller and smaller numbers, approaching zero? What happens when we divide −1 by smaller and smaller numbers, approaching zero? What inconsistency occurs?

### 1.9: Distributive Property

The distributive property allows us to distribute a multiplier to a sum in a parentheses. For example, if we have 2 (3 + 5), we can rewrite this using the distributive property as: (2 × 3) + (2 × 5). This often makes it easier for us to do mental math, or to at least simplify a large calculation.

Read this article up to Practice Set A. The beginning explains why the distributive property works. Pay close attention to the examples in Sample Set A, which show how to use the distributive property. Do the examples in Practice Set A.

Watch these videos for more examples.

### Unit 1 Assessment

Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment