## Natural numbers

**Natural numbers** are numbers that we use to count. They are whole, **non-negative numbers.**

A set of natural numbers is typically denoted by the symbol *ℕ*. For example:

*ℕ* = {1, 2, 3, 4, 5, 6, 7…}

**What are Whole Numbers?**

The set of natural numbers that includes zero is known as the whole numbers. A set of whole numbers are typically denoted by W. For example, the following is a set of whole numbers:

W = {0, 1, 2, 3, 4, 5, 6, 7…}

Perhaps confusingly, some authors don’t include zero in the set of whole numbers. In that case, it is the same as the set of natural numbers.

### Why is a Natural Number a Whole Number?

As mentioned above, natural numbers must be whole and positive. This makes sense for a number of reasons, including the fact that they are counting numbers. Let’s say a teacher wants to count the number of students in her class: She can only count the whole children.

We often see in statistics published across the internet numbers that seem to contradict the “wholeness” of people. For example, “the average family size is 3.1”. It should be fairly clear that it is impossible to have .1 of a person, but this number is just an average. The average number of cars per household is calculated by adding up the total number of cars and dividing by the number of households. Once we divide we are no longer working with natural numbers. Rather, we are left with a real number, in this case a fraction.

The **sum** or **product** of natural numbers are also natural numbers. For example, 5 + 5 = 10 (all three of which are natural), or 10 · 15 = 150.

Likewise, it makes no sense in the physical world of “natural” numbers to say that we have “negative something”. Rather, we say that we have zero of something where there are none. Using our teacher example from above, if the teacher currently has no students in her class she has zero students; It makes zero sense in the real world to have negative students.

The complete set of whole numbers is equal to the set of non negative integers.

## Whole Numbers Example

A few examples of whole numbers: 3, 15, 998, 2, 232, 589.

All of the following are not whole numbers:

- Decimals: 0.1, 5.23, 15.999, 1.72.
- Fractions: ½, 1/27, 2 ½, 99/100.
- Negative numbers: -10, -99, -521.