1 Activities for this section:

Where have the integers been?

1.1 Motivation

From the point of view of arithmetic, the whole numbers have had two serious deficiencies.

• We can add any two whole numbers and get another whole number, but the same isn’t true for subtraction.
• We can multiply any two whole numbers and get another whole number, but the same isn’t true for division.

We fix the second deficiency by introducing the set of fractions – we can now divide any whole number by any other whole number which is not zero and get a meaningful answer. In fact, the same is true for the entire (now expanded) set of integers and fractions together. To solve the first deficiency, we want to do something similar. The expansion of the whole number system to the system of integers, that is, introducing negative numbers to go along with positive numbers, is designed to resolve this problem.

From an application point of view, we need integers to be able to quantifiably describe situations in which we have what we will call an artificial zero.

The terminology for absolute and artificial zeros helps us to distinguish these two types of situations, but we will not in general focus on identifying which situation is which or require you to use this terminology. After all, we want to work here with negatives as well as positives, so we will generally be in situations with an artificial zero!

In situations with an artificial zero, or a zero arbitrarily set, it makes sense to be talking about amounts that are less than zero. Here are some examples.

• Temperature. In the Fahrenheit and Celsius scales, zero is set at a chosen level, but it is possible for temperatures to be colder than that set zero. Notice that the Kelvin scale is different: zero is set at the point there is completely no heat, so a temperature less than zero Kelvin is not possible.
• Finances. In finances, a person is said to have zero financial worth when they don’t have any money. Also, it makes sense for a person to have a negative financial worth in the sense of owing someone money. So, having a worth of $-5$ dollars means you need to somehow gain $5 before you can say you are “even”, with no debt and no profit. Notice that this can get complicated! If oranges are the currency, one could say one has $-5$ oranges instead of $-5$ dollars even though it is impossible to physically have $-5$ oranges or $-5$ dollar bills. The idea of owing someone oranges or dollars makes this idea work.
• Sports. In football, we can take zero to be the line of scrimmage (or starting line for the play), or where the play begins, and we report the result of the play based on this zero. For instance, if a player who is running gets tackled behind the starting line, we say that the play has resulted in negative yardage.
• More sports. In golf, “par” is defined as the number of shots experts think a golfer should need in order to be able to get the golf ball into the hole. We can take par as our zero in this situation. However, if the golfer takes fewer shots to get into the hole, we say the golfer is so many shots “under par”, and if the golfer takes more shots to get into the hole, we say the golfer is so many shots “over par”.

1.2 Representations of Integers

There are several main ways we represent integers, or several important tools we will discuss.

Stories. We use situations similar to those with an artificial zero described above. These stories are useful because we can start to picture what we would like to do with integers, why we would like to do these things, and most importantly we can use the stories to check whether or not our answers make sense.

Chips. We use one black chip to represent one positive unit and one red chip to represent one negative unit. If I have one black chip together with one red chip, I have the same amount in value as if I have no chips at all. We might say that one red chip cancels one black chip.

One of the most important things to keep in mind while working with chips is that we can easily change the representation of our number without changing the value of our number. Let’s investigate this phenomenon.

Number lines. Previously, we marked zero and one on our number lines, and used the spacing between zero and one to mark all of the other positive numbers to the right of zero. With integers, we extend the number line to the left of zero as well, using the same spacing between zero and one.

1.3 Comparison of Integers

Like with the other number systems we’ve studied, when we have two integer amounts, we would like to be able to tell which one is greater, lesser, or if the two amounts are equal. While we could just state a rule for this, as with all mathematics, if it is possible to come up with a rule from a sense-making point of view, it will be more understandable and useful.

Your first guess might be to try a one-to-one correspondence. However, with negative numbers we are sometimes dealing with sets of objects that don’t physically exist, and so making a one-to-one correspondence is very difficult in this case. Also, when we work with models like the chips, our physical objects have different meanings, and so they cannot be placed in a one-to-one correspondence without extra care!

However, like in other number systems, we can use our story problems to look at real-life situations and ask, “Which situation is better?” or “Which number has more value in this situation?” For instance, we have things like the following.

• Which financial worth is better? (“Better” here usually means more money.)
• Which team’s total running yards is better? (“Better” here usually means more yards, with positive yards being better than negative yards.)
• Which temperature is better? (Here, “better” depends on the situation! We should probably ask instead, “Which temperature is hotter?”)
• Which golf score is better? (Here, “better”, with respect to par, would actually be the more negative number!)

Notice that in each case, one needs to define which attribute means greater and which means lesser.

We can also use a number line and take advantage of what we did with positive numbers, but extend our ideas to negative numbers as well.

1.4 Checks and Bills

We will end this section with two sets of conventions that we plan to use to help make sense of operations with integers. The first is a model that often helps us to write meaningful story problems, called the “checks and bills” model.

If we are using chips to represent the checks and bills in our stories, a black chip will mean $$1$, and a red chip will mean $$-1$.

1.5 Operations and Number Lines

When we model our operations using number lines, we will use the following conventions.

• An addition sign means to walk or move forward.
• A subtraction sign means to walk or move backward.
• A positive number means to face right (towards the positive numbers).
• A negative number means to face left (towards the negative numbers).

The result that you see after following these procedures should answer the question, “Where on the number line are we now?” We can also often use our story situation, whether it is checks and bills or something else, to understand how to move on the number line. Some people think of this as asking the question, “Did we get good news or bad news?”