3.1 Signed Number Arithmetic

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 3.1. 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 $b<a$.

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: When adding integers with opposite signs and different absolute values, the result has the sign of the integer with the larger absolute value. If their absolute values are equal, their sum is zero.

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 wealth 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. The two sign rules below apply when both integers are nonzero. Zero times any integer is zero; zero divided by a nonzero integer is zero. Division by zero is undefined.

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 of values that are being averaged.