Introduction to Integers Lesson

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

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.

Once we start dealing with problems, we will be asked for the opposite of a number using the "-" symbol. The meaning of this symbol changes based on its context:

• 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.

Let's take a look at a few examples in which we will be asked to find the opposite or additive inverse of a number.
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

Note: we will discuss the concept of even and odd numbers at more length in our divisibility rules lesson. Let's try a few examples:
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.

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Skills Check: