Lesson Objectives

- Demonstrate an understanding of the whole numbers
- Learn about a new set of numbers known as the integers
- Learn how to create a number line for the integers
- Learn the difference between a positive number, negative number, and zero
- Learn how to find the additive inverse or opposite of a number

## What are Integers

Up to this point, we have worked exclusively with the whole numbers: {0,1,2,3,4,…}. Every number larger than zero is known as a positive number. Although it is rarely used, we can place the "+" symbol in front of a number to indicate that the number is positive.

3 could be written as: +3

11 could be written as: +11

We typically don’t use the "+" symbol in front of a number, as we assume if no sign is present, the number is positive. Every number that is less than zero is known as a negative number. We place a "-" symbol in front of a number to indicate that it is a negative number. You have already seen negative numbers in your everyday life. Some common scenarios would be a negative balance in a checking account (a balance that is less than zero), or a negative temperature (a temperature that is below zero).

-3 » represents the number read as: negative three. This number represents 3 units less than 0.

-11 » represents the number read as: negative eleven. This number represents 11 units less than 0.

What about Zero?

Zero is not positive or negative, we can just think about it as a center point. In most cases, people will assume that zero is positive since there is no sign present. This is a misconception as zero is not a positive number.

Integers: {…,-3,-2,-1,0,1,2,3,…}

Notice the three dots or ellipsis in each direction now. The one on the left shows a pattern where we decrease by 1 forever. After -3, comes: -4, -5, -6, -7, so on and so forth. We already have seen that the ellipsis on the right shows that we increase by 1 forever. There is no smallest or largest integer. Let’s look at a number line for the integers and think about the relationship between two numbers: What if we asked the following question?

Is -6 > -3?

Many of you may answer yes since 6 is normally a larger number than 3. In this case, the answer is no. The correct relationship is:

-6 < -3 : -6 is less than -3.

Recall that numbers decrease as we move left and increase as we move right. -6 lies to the left of -3 on the number line and is, therefore, a smaller number: This may be a bit difficult to remember at first, but you can simply remember that a bigger negative will result in a smaller number. Let’s look at a few examples:

Example 1: Replace each ? with the correct inequality symbol.

-4 ? -8

-4 > -8

-4 lies to the right of -8 on the number line and is a larger number. We can also think of this as -4 is a smaller negative and therefore a larger number. -7 ? -6

-7 < -6

-7 lies to the left of -6 on the number line and is a smaller number. We can also think of this as -7 is a larger negative and therefore a smaller number.

Opposites are numbers that have the same distance from zero on the number line but lie on opposite sides. Every integer other than 0 has an opposite. As an example, 2 and -2 are opposites, they are each 2 units away from zero on the number line: To find the opposite or additive inverse of a number, we simply change the sign of the number. Let's take a look at a few examples:

Example 2: Find the opposite of each number.

-6, 7, -19, and 43

Example 3: Simplify each.

-(14)

-(14) = -14

For this problem, we are asked to find the opposite of 14. We change the sign and end up with -14 as our answer.

Example 4: Simplify each.

-(-(-5)).

-(-(-5)) = -5

For this problem, we start at the innermost set of parentheses and work outward:

-(-(-5))

From the highlighted section, we are asked, what is the opposite of -5, the answer is 5. Let's replace -(-5) with 5 and move on:

-(-(-5)) = -(5)

Now we have one more operation, let's find the opposite of 5 which is -5.

-(5) = -5

-(-(-5)) = -5

In some cases, you will run into many "-" symbols. Instead of working through each iteration, we can use a shortcut and count the total number of "-" symbols:

Example 5: Simplify each.

-(-(-(-(-14)))).

-(-(-(-(-14)))) = -14

We can count the total number of "-" signs and see the result is 5. Since 5 is an odd number, we know the result is negative.

Example 6: Simplify each.

-(-(-(-(-(-(-(-23)))))))

-(-(-(-(-(-(-(-23))))))) = 23

We can count the total number of "-" signs and see the result is 8. Since 8 is an even number, we know the result is positive.

3 could be written as: +3

11 could be written as: +11

We typically don’t use the "+" symbol in front of a number, as we assume if no sign is present, the number is positive. Every number that is less than zero is known as a negative number. We place a "-" symbol in front of a number to indicate that it is a negative number. You have already seen negative numbers in your everyday life. Some common scenarios would be a negative balance in a checking account (a balance that is less than zero), or a negative temperature (a temperature that is below zero).

-3 » represents the number read as: negative three. This number represents 3 units less than 0.

-11 » represents the number read as: negative eleven. This number represents 11 units less than 0.

What about Zero?

Zero is not positive or negative, we can just think about it as a center point. In most cases, people will assume that zero is positive since there is no sign present. This is a misconception as zero is not a positive number.

### Integers

The integers are a set of numbers that include all whole numbers and their negative counterparts. In other words, you can take each positive whole number and slap a negative on it to get its negative counterpart.Integers: {…,-3,-2,-1,0,1,2,3,…}

Notice the three dots or ellipsis in each direction now. The one on the left shows a pattern where we decrease by 1 forever. After -3, comes: -4, -5, -6, -7, so on and so forth. We already have seen that the ellipsis on the right shows that we increase by 1 forever. There is no smallest or largest integer. Let’s look at a number line for the integers and think about the relationship between two numbers: What if we asked the following question?

Is -6 > -3?

Many of you may answer yes since 6 is normally a larger number than 3. In this case, the answer is no. The correct relationship is:

-6 < -3 : -6 is less than -3.

Recall that numbers decrease as we move left and increase as we move right. -6 lies to the left of -3 on the number line and is, therefore, a smaller number: This may be a bit difficult to remember at first, but you can simply remember that a bigger negative will result in a smaller number. Let’s look at a few examples:

Example 1: Replace each ? with the correct inequality symbol.

-4 ? -8

-4 > -8

-4 lies to the right of -8 on the number line and is a larger number. We can also think of this as -4 is a smaller negative and therefore a larger number. -7 ? -6

-7 < -6

-7 lies to the left of -6 on the number line and is a smaller number. We can also think of this as -7 is a larger negative and therefore a smaller number.

### How to find the Additive Inverse or Opposite of a Number

Now that we understand the concept of the integers, let’s think about how to find the additive inverse or opposite of a number. In your class, you may hear these two terms used interchangeably. They mean the exact same thing. If someone wants the opposite of a number or the additive inverse of a number, they are asking for the same value.Opposites are numbers that have the same distance from zero on the number line but lie on opposite sides. Every integer other than 0 has an opposite. As an example, 2 and -2 are opposites, they are each 2 units away from zero on the number line: To find the opposite or additive inverse of a number, we simply change the sign of the number. Let's take a look at a few examples:

Example 2: Find the opposite of each number.

-6, 7, -19, and 43

- The opposite of -6 is 6. We just change the sign from "-" to "+". Note that each number is 6 units away from zero on the number line.
- The opposite of 7 is -7. We just change the sign from "+" to "-". Note that each number is 7 units away from zero on the number line.
- The opposite of -19 is 19. We just change the sign from "-" to "+". Note that each number is 19 units away from zero on the number line.
- The opposite of 43 is -43. We just change the sign from "+" to "-". Note that each number is 43 units away from zero on the number line.

- placed directly next to a number to imply a negative number: -3 » negative 3
- placed outside of grouping symbols to ask for the opposite of what’s inside: -(3) » the opposite of 3, which is negative 3
- although we haven’t gotten to multiplication with integers, you will remember that placing a number outside of parentheses implies multiplication. We can think of a negative outside of parentheses as multiplication by -1. When we multiply by -1, we simply change the sign or find the opposite of the number inside of the parentheses.

Example 3: Simplify each.

-(14)

-(14) = -14

For this problem, we are asked to find the opposite of 14. We change the sign and end up with -14 as our answer.

Example 4: Simplify each.

-(-(-5)).

-(-(-5)) = -5

For this problem, we start at the innermost set of parentheses and work outward:

-(-(-5))

From the highlighted section, we are asked, what is the opposite of -5, the answer is 5. Let's replace -(-5) with 5 and move on:

-(-(-5)) = -(5)

Now we have one more operation, let's find the opposite of 5 which is -5.

-(5) = -5

-(-(-5)) = -5

In some cases, you will run into many "-" symbols. Instead of working through each iteration, we can use a shortcut and count the total number of "-" symbols:

- an even number (number such as: 2, 4, 6, 8, 10, 12,…) will give you a positive result
- an odd number (number such as: 1, 3, 5, 7, 9, 11,…) will give you a negative result

Example 5: Simplify each.

-(-(-(-(-14)))).

-(-(-(-(-14)))) = -14

We can count the total number of "-" signs and see the result is 5. Since 5 is an odd number, we know the result is negative.

Example 6: Simplify each.

-(-(-(-(-(-(-(-23)))))))

-(-(-(-(-(-(-(-23))))))) = 23

We can count the total number of "-" signs and see the result is 8. Since 8 is an even number, we know the result is positive.

#### Skills Check:

Example #1

Replace the "?" with "<" or ">".

-4 ? -3

Please choose the best answer.

A

-4 < -3

B

-4 > -3

Example #2

Simplify each.

-(-15)

Please choose the best answer.

A

15

B

-15

Example #3

Simplify each.

-(-(-(-(-29))))

Please choose the best answer.

A

29

B

-29

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