## Natural Numbers

When we count using our fingers, the numbers that we use are \(1,2,3,…\) etc.

In our day-to-day life, we typically refer to these types of numbers as *counting* numbers (for obvious reasons!).

In maths, on the other hand, we instead typically refer to them as **natural numbers**. The symbol \(\mathbb{N}\) is used denote *all* of the natural numbers.

### Key Point

A number is said to be a* natural number* if it is contained within the following list:

\begin{align}\mathbb{N}=1,2,3,…\end{align}

The numbers \(3\) and \(1{,}234{,}567\) are both examples of natural numbers.

In contrast, the numbers \(7.3\) and \(-5\) are both examples of numbers that are *not* natural numbers. (More on these types of numbers later!)

### Cause of Confusion

\(0\) is *not* considered a natural number as we do not use it when counting with our fingers.

Note that while e.g. \(5\) and \(11\) are both natural numbers, there are also other natural numbers in between them.

The natural numbers \(5\) and \(6\), on the other hand, have no other natural numbers in between them. These numbers are therefore said to be **consecutive**.

### Key Point

Two natural numbers are said to be *consecutive* if there are no other natural numbers between them.

\(15\) and \(20\) are not consecutive numbers and \(15\) and \(200\) are also not consecutive numbers. However, there is obviously a *lot more* natural numbers between the first pair of numbers compared to the second pair.

Rather than demonstrating this by listing out all of these numbers, it would be better if we could instead see this *visually*. This is achieved by using what is known as a **number line**.Â

### Key Point

The *number line* for natural numbers, shown below, is a visual way of depicting all of the natural numbers.

###### Figure 1

Number lines allow us to see the “distance” between one number and another.

The arrow to the right of the above number line indicates that this number line “keeps going forever” as the natural numbers get larger and larger.

In contrast, there is no arrow to the left as there are no natural numbers less than \(1\).

(Later, we shall look at number lines for different types of numbers in which there will be arrows pointing *both* to the left and right!)

Rather than comparing just two natural numbers, we can instead write down a *list* of numbers.

Three such lists are shown below:

\(458, 2, 31, 600, 11\)

\(4, 10, 23, 58, 59\)

\(100, 67, 42\)

Note that the *ordering* in each of these lists are different: the first list does not appear to have any order, the numbers are getting bigger in the second list and the numbers are getting smaller in the third list.

### Key Point

A list of numbers is said to be in *ascending* order if the next number in the list is always larger than the previous number, e.g. \(4, 10, 23, 58, 59\).

A list of numbers is instead said to be in *descending* order if the next number in the list is always smaller than the previous number, \(100, 67, 42\).

## Guided Example 1

Consider the following list of natural numbers:

\begin{align}17,205,48, 49,3,612, 42\end{align}

Rearrange this list in:

**(a)** ascending order

**(b)** descending order

**(a)** To rearrange this list in ascending order, we must start with the smallest number in the list (\(3\) in this case) and increase the numbers in the list form there.

Upon doing so, we obtain:

\begin{align}3,17,42,48,49,205,612\end{align}

**(b)** To rearrange this list instead in descending order, we must start with the larger number in the list (\(612\) in this case) and decrease the numbers in the list form there.

As we have already stated the list in ascending order in part **(a)**, we can do this quite quickly in this case by simply reversing that list:

\begin{align}612,205,49,48,42,17,3\end{align}

Finally, we can also split all of the natural numbers into two smaller groups, depending on what happens when we divide each of the numbers by two.

### Key Point

A natural number is said to be *even* if there is no remainder when it is divided by two.

\(2,4,6…\) are therefore the even natural numbers.

A natural number is said to be *odd *if there is a remainder of one when it is divided by two.

\(1,3,5…\) are therefore the odd natural numbers.

All natural numbers are either odd or even.