3.2 The Integers

Defining and Identifying Integers

Extending the counting numbers to include negative numbers and zero forms the integers. Any other number that cannot be written as $\left\{\dots -3,-2,-1,0,1,2,3,\dots \right\}$ is not an integer.

Graphing Integers on a Number Line

Integers are often imagined as steps along a path. You start at 0, and going to the left is going backward, or in the negative direction, while going to the right is going forward, or in the positive direction. A number line (Figure 3.12) helps envision the integers. This also means that an integer gives magnitude (size) and direction (positive is to the right, negative is to the left). Graphing an integer on the number line means placing a solid dot at the integer on the number line.

Figure 3.12

Comparing Integers

When determining if one quantity or size is larger than another, we know it means there is more of whatever is being discussed. In terms of positive integers, we can envision that larger integers are further to the right on the number line. This idea applies to negative integers also. This means that $\mathit{a}$ is greater than $\mathit{b}$ when $a$ is to the right of $b$ on the number line. We write $a>b$. When $a$ is greater than $b$, we can also say that $\mathit{b}$ is less than $\mathit{a}$. On the number line, $b$ would be to the left of $a$. We write $b<a$.

We need to recognize that $a>b$ means the same thing as $b<a$. This can be seen on the number line in Figure 3.16. On this number line, $a$ is to the right of $b$, so $a>b$. But this means $b$ is to the left of $a$, so $b<a$.

Figure 3.16

Example 3.28

Comparing Integers Using a Number Line

Determine which of −6 and 4 is larger using a number line, and express that using both the greater than and the less than notations.

Solution

To illustrate this, we use a number line (Figure 3.17).

Figure 3.17

Since −6 is to the left of 4, then −6 is less than 4. We can write this as −6 < 4. Another way of expressing this is that 4 is greater than −6. So we can also write $4>-6$.

Example 3.29

Comparing Negative Integers

Determine which of −6 and −2 is larger, and express that using both the greater than and the less than notations.

Solution

To illustrate this, we use a number line (Figure 3.18).

Figure 3.18

Since −6 is to the left of −2, then −6 is less than −2. We can write this as $-6<-2$.

Another way of expressing this is that −2 is greater than −6. So we can also write $-2>-6$.

The Absolute Value of an Integer

When talking about graphing integers on the number line, one interpretation suggests it is like walking along a path. Negative is going to the left of 0, and positive going to the right. If you take 30 steps to the right, you are 30 steps away from 0. On the other hand, when you take 30 steps to the left, you are still 30 steps away from 0. So, in a way, even though one is negative and the other positive, these two numbers, 30 and −30, are equal since both are 30 steps away from 0. The absolute value of an integer $n$ is the distance from $n$ to 0, regardless of the direction. The notation for absolute value of the integer $n$ is $\left|n\right|$.

If we think of an integer as both direction and magnitude (size), absolute value is the magnitude part.

Calculating the absolute value of an integer is very straightforward. If the integer is positive, then the absolute value of the integer is just the integer itself. If the integer is negative, then to compute the absolute value of the integer, simply remove the negative sign. Keeping in mind the number line as a path, when you’ve gone 10 steps to the left of 0, you have still taken 10 steps, and the direction does not matter.

Adding and Subtracting Integers

You may recall having approached adding and subtracting integers using the number line from earlier in your academic life. Adding a positive integer results in moving to the right on the number line. Adding a negative integer results in moving to the left. Subtracting a positive integer results in a move to the left on the number line. But subtracting a negative integer results in a move to the right.

This leads to a few adding and subtracting rules, such as:

Rule 1: Subtracting a negative is the same as adding a positive.

Rule 2: Adding two negative integers always results in a negative integer.

Rule 3: Adding two positive integers always results in a positive integer.

Rule 4: The sign when adding integers with opposite signs is the same as the integer with the larger absolute value.

These rules are good to keep in the back of your mind, as they can serve as a quick error check when you use a calculator.

One use of negative numbers is determining net worth, which is all the weath someone owns less all that someone owes. Sometimes net worth is positive (which is good), and sometimes net worth is negative (which can be stressful).

Multiplying and Dividing Integers

Similar to addition and subtraction, the signs of the integers impact the results when multiplying and dividing integers. The rules are fairly straightforward, but again rely on the direction on the number line. There are only two rules.

Rule 1: When multiplying or dividing two integers with the same sign, the result is positive.

Rule 2: When multiplying or dividing two integers with opposite signs, the result is negative.

Just as before, these rules can serve as a quick error check when using a calculator.

At the end of a season, a team may wish to buy their coach an end-of season gift. It makes sense to share the cost equally among the members. To do so, the team would need to find the average (or mean) cost per member. The average (or mean) of a set of numbers is the sum of the numbers divided by the number values that are being averaged.