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Section 2 Natural Numbers \(\mathbb{N} \)

Definition 2.1. Natural numbers.
Often called the counting numbers. These numbers start with 1 and count up by 1. So, 1,2,3,4... . Note: These numbers sometimes include 0, but not in all cases. The symbol that is often used for the natural numbers is \(\mathbb{N} \text{.}\)

Subsection 2.1 Adding Natural Numbers

Adding natural numbers is a binary operation, counting up from one number by the amount of the second. Thinking about a number line can be a useful way to picture this operation. Let's start with the number line with the natural numbers on it. Throughout the module we will add more context to this line as we explore more collections of numbers.

Figure 2.2. Number Line (Nautral Numbers).

The above number line has a very light dotted line to indicated that there is more to come, but the phrase "number line" is a bit of a misnomer since it is really just the tack-marks that we are considering here.

With this number line in mind, let's add two natural numbers together.

Find \(2 + 5 \)

To add these numbers together, start at the \(2 \) on the number line, and count \(5 \) to the right. The number that we land on it called the sum, and we would say, "two plus five equals seven".

Figure 2.4. Number Line Adding 2 and 5.

Now find, \(5 + 2 \)

To add these numbers together, start at the \(5 \) on the number line, and count \(2 \) to the right. The number that we land on it called the sum, and we would say, "five plus two equals seven".

Figure 2.5. Number Line Adding 5 and 2.

The above example seems to imply that the order of the addition does not matter and this is, in fact, true. That is \(m + n = n + m \) for \(m,n \in \mathbb{N} \text{.}\) This property is called the commutative property, and we say that addition of the natural numbers is commutative.

Subsection 2.2 "What about subtraction"

One question that comes next is, "what about subtraction?". We can use the same idea that we built above using the number line to subtract two natural numbers. Let's start with an example, and then we will consider an issue that will arise with subtracting natural numbers.

Find \(5 - 2 \)

To subtract these two numbers we start at the \(5 \) on the number line, and count \(2 \) to the left. The number that we land on it called the difference, and we would say, "five minus two equals three".

Figure 2.8. Number Line subtracting 2 from 5.

With that example in mind, what's issue? What would happen if we considered \(2 - 5 \) ? When we switch the order, we see that the result will not be a natural number.

Figure 2.9. Number Line subtracting 5 from 2.

Infact, it isn't even on our number line. This is the central idea behind all the different collections of numbers. As we work with each collection, something mathematical will require new numbers and these new numbers will help us move back up the tree (Section 1).

So to be able to find \(2 - 5 \) we need additional numbers. These additional numbers, with the inclusion of the natural numbers and the number zero, \(0 \text{,}\) make up the integers \(\mathbb{Z} \text{.}\) We will call these new numbers, the additive inverses.