Natural Numbers — Definition & Examples

What is a Natural Number?

Natural numbers are the numbers used to count physical things. They start at $1$ and go $1,2,3,4,\cdots$ continuing upward infinitely. We use the symbol $\mathbb{N}$ to denote them. Furthermore, they are a subset of the real numbers. So, every natural number is a real number. Despite this, the natural numbers are still an infinite set. In other words, if we try to count all the natural numbers, it would take forever!

The natural numbers are positive and contain no fractional numbers. There is a debate on whether the number 0 is or isn't a natural number. We will not consider zero a natural number here. However, it is still a whole number. This is what distinguishes natural and whole numbers.

Example 1

We have a list of numbers. $$33,\quad\frac23,\quad-8,\quad0$$

Which numbers are not natural and why?

The first number, $33$, is a natural number. The second number, $\frac23$, isn't because it is a fraction. The third, $-8$, isn't because it's negative. Lastly, $0$ isn't a natural number because it isn't positive. It's the only whole number that is NOT a natural number.

Example 2

You're at the bay and spot a seagull. Curious, you decide to count all the seagulls you can see in the area. When you finish counting, will the number be natural?

The answer is yes! Already, you've seen $1$ seagull (a natural number), so counting anymore will keep it natural. Since you can't count negative seagulls or fractional seagulls, the total will never not be natural. It doesn't even matter if you don't see any more. In that case your total is $1$ seagull.

Solution

• $98$ is a natural number since it's positive with no fractional part.
• $0$ is not natural because it is not positive.
• $-5$ is not natural because it is not positive.
• $2.1$ is not natural because it is a decimal, which contains a fractional part. Precisely, that fractional part is $\frac{1}{10}$.
• $100{,}000{,}000{,}000$ is a natural number since it's positive with no fractional part. And quite a big one at that.

What is a Natural Number?

The set of natural numbers is an infinite set containing the "counting numbers:" $1,2,3,4,...$. The natural numbers start at $1$ and include all positive numbers without a fractional or decimal part.

We use the symbol $\mathbb{N}$ to refer to them. Sometimes you'll also see $\mathbb{N}^+$.

The set of natural numbers is a subset of the set of whole numbers, the set of integers, the set of rational numbers, and the set of real numbers.

Natural numbers have some really cool properties, like the reflexive, symmetric, and transitive properties. Check them out!

(Video) Natural Numbers, Whole Numbers, and Integers

by Learn Math Tutorials

This video by Learn Math Tutorials goes over the first three types, or sets, of numbers: natural numbers, whole numbers, and integers.

$\large{\text{Summary}}$

The natural numbers are commonly referred to as the counting numbers. We use the symbol $\mathbb{N}$ to refer to them. $$\mathbb{N}=1,2,3,4,...$$

If we were to make a number line with tick marks to represent each natural number, it would look like:

The number line starts at $1$ and continues out in the positive direction. We refer to this as infinity, $\infty$. There is also a negative portion, ... but it doesn't have natural numbers on it. So we aren't going to draw that here. However, we can compare natural numbers on the number line. A bigger natural number will be to the right of a smaller one.