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Section 3 The Integers \(\mathbb{Z} \)

Definition 3.1. The Integers.
The natural numbers, their additive inverses, and zero \(0 \text{.}\)

Figure 3.2. Number Line for the integers.

Subsection 3.1 The Integer Operations

For this section we are going to list the operation that we are going to explore. Like in the last section, we start by building some intuition, then we will look at some examples and exercises.

  1. Integer addition
  2. Integer subtraction
  3. Integer multiplication

Feel free to skip ahead to a subsection, but before moving onto the rationals, read over the subsection, Subsection 3.5

Subsection 3.2 Integer Addition

Addition of the integers is similar to addition with the natural numbers. When working with the positive integers the process is the same as with the naturals. With the negative integers, we will need to build some new intuition. Let's start with the number line again.

Figure 3.3. Number Line for the integers.

Now let's add \(-2 \) and \(3 \text{.}\) To do this, start at \(-2 \) and count 3 to the right. The value that we land on is the sum of \(-2 \) and \(3 \text{.}\)

Figure 3.4. Sum of -2 and 3.

To write this up as a mathematical statement we would say, \(-2 + 3 = 1 \text{,}\) and in words we would say, "negative two plus three is one".

Now let's add \(-4 \) and \(7 \text{.}\) To do this, start at \(-4 \) and count 7 to the right. The value that we land on is the sum of \(-4 \) and \(7 \text{.}\)

Figure 3.6. Sum of -4 and 7.

To write this up as a mathematical statement we would say, \(-4 + 7 = 3 \text{,}\) and in words we would say, "negative four plus seven is three".

Now the same question that we explored with the natural numbers (\(\mathbb{N} \)) happens here. If we add them in the other order, will we get the same answer? That is, does \(-4 + 7 = 7 + (-4) \text{?}\) Notice that we needed to include a set of parentheses around the \(-4 \text{,}\) we will address this a bit more in the subsection on subtraction.

To establish the intuition here, we need to think back to why we needed to introduce the negative numbers in the first place. These numbers are the reverse (inverse) of the positive numbers. So, if when we add a positive number we move to the right, when we add an inverse (negative) number we will move to the left.

So, start at positive seven and count left four numbers. We can see that this also ends with a sum of three. Giving us some evidence that \(-4 + 7 = 7 + (-4) \text{.}\) It turns out that this is in fact, true for all integers.

Figure 3.7. Sum of 7 and -4.

One final note before we take a look at some exercises. When adding zero to any number, it represents not moving at all. So the sum of any number with zero gives us that number.

The sum of any \(x \in \mathbb{N} \) with 0 is \(x \text{.}\)

Now let's take a look at a few exercises. Use the following number line as a guide.

Figure 3.9. Number line for exercises.

Next let's take a look at integer subtraction. All the intuition that we have built about adding the negative numbers will help us to understand the operation. And at it's core, adding the negatives is what subtraction is. But it is a good idea to formally introduce the operation.

Subsection 3.3 Integer Subtraction

As noted above, subtraction is the name that we are going to give the operation of adding the negative integers (and rationals in a bit). The one major notational change is that we will drop the addition symbol and use the negative symbol as a replacement of the addition symbol. So,

\begin{equation*} 4 + (-7) = 4 - 7 \end{equation*}

The main thing to note here is that we will preform this operation (\(4 - 7 \)) the same way that we preformed the operation \(4 + (-7) \text{.}\) We do need to take great care here through. Since we have obscured the addition, we won't be able to switch the order when using the subtraction notatio. That is,

\begin{equation*} 4 - 7 \neq -7 + 4 \end{equation*}

So if we are working in a situation that changing the order of operation is needed, we should convert back to the addition form, and then make the change.

We can think about subtraction as removing some from an initial amount. The negative numbers then take on a debt type of feeling. When we talk more from the whole than we have to begin with, we will end with a sum that is less then zero. Financial questions often frame this concept well. Consider the next exercise to explore this context.

Subsubsection 3.3.1 Subtraction as a Difference

Subtraction is sometimes referred to as the difference. This is due to the fact that we can use subtraction as a way to determine how far apart to values are.

Figure 3.18. Number line for exercises.

Consider, \(7 - 3 \text{.}\) Thinking about this as we have above, we would start at seven and count to the left three units. The result is that value of four. We can then say that the difference of seven and three is four. That is to say that 7 is 4 from 3. Using the above number line we can see that this is, in fact, the case.

While there is more to the story of subtraction, we will leave it here for now and move onto our next operation.

Subsection 3.4 Multiplication (and the Table)

Multiplication of integers is a repeated addition. If we are adding 4 to itself 5 times, the total number is five collections, each containing four.

\begin{equation*} 4 + 4 + 4 + 4 + 4 \end{equation*}

Multiplication is a way to denote this repeated process in a short-hand form. The first value in the expression will indicate the amount in each "container". The second value in the expression will indicate how many "containers" we have in all. There are a few different ways that this operation is indicated. Sometimes we use, \(\times \text{.}\) When working with computer we use * . Finally we will often indicate multiplication by putting the numbers adjacent to another like, 4(5). The parentheses are needed so that it doesn't look like 45.

\begin{equation*} 4 + 4 + 4 + 4 + 4 = 4 \times 5 = 4 * 5 = 4(5) \end{equation*}

So in review, the number to the left will be thought of as the amount in each "container" and the number on the right will represent the number of items in each "container". The total count of these will be called the product.

Find the product of 3 and 8. The thinking here is that we have 8 "containers" and there are 3 items in each "container". So the total count would be,

\begin{equation*} 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 3(8) = 24 \end{equation*}

Now one thing that we can do that is useful is to make a table of these products for the values from 1 to 10. This can act as a quick reference, as well as a way to practice these important products. The way to read the table below is to find the first value that in the product in the column to the far left, then find the second value in the product in the top row. Finally, determine where the row and column will intersect. The value in this place is the product.

Figure 3.26. Multiplication Table.

We can use the connection to addition to help us explore the table. For each row, to move along the row we need only add the value on the far left to the one that we are on.

Figure 3.27. Multiplication Table for 7.

So to get from entry to entry we add 7 to each value from left to right. Take some time to explore the other rows to see that this is the case for those values as well.

Use the above multiplication table to answer the following exercises.

Subsection 3.5 What About Division?

Just like the natural numbers and subtraction. The integers won't be able to capture division. The problem is that if we design an operation that "undoes" multiplication, it will require numbers that we do not have in the integers. So, just like in the last jump in collections of numbers, we will now introduce a new collection, the rationals.