Statistics Definitions > Natural Numbers and Whole Numbers

**Contents** (Click to skip to that section)

- Natural Number
- Whole Numbers
- Why Is a Natural Number a Whole Number?
- Whole Numbers Example
- Closed Sets and Wholes
- Properties of Whole numbers

## What is a Natural Number?

**Natural numbers** are numbers that we use to count. They are whole, **non-negative** numbers. We often see them represented on a **number line**.

The line in the above image starts at 1 and increases in value to 5 The numbers could, however, increase in value forever (denoted by the dotted line in the image). Natural numbers, therefore, can continue to infinity.

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. ** Integers are like whole numbers, except they can be negative or zero as well. For example: -10, -3, 0, 1 5.

Related article: Integer Sequences (CalculusHowTo.com).

## 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, 1.7^{2}.**Fractions**: ½, 1/27, 2 ½, 99/100.**Negative numbers: -10, -99, -521.**

## Closed Sets and Wholes

In set theory, the whole numbers follow a few rules. The set of whole numbers is:

**Closed under addition and multiplication.** Take two whole numbers a and b. If you add then ( a + b = c), then “c” will also be a whole number. The same is true for multiplication: a · b = d.

Let’s take a look at a couple of specific example with numbers instead of variables:

- 6 + 7 = 13
- 6 · 7 = 42

**The set of whole numbers is not closed for division and subtraction.** If a is a whole number, then there’s another whole number b that results in a non-whole number solution. In notation, that’s:

- a – b = c
- a / b = d

Where “b”, “c” and “d” are not whole numbers.

**Examples**:

**Subtraction:**

6 and 10 are whole numbers,

but 7 – 9 = -2, which is not a whole number.**Division**:

4 and 5 are whole numbers, but 4/5 is not a whole number.

## Properties of Whole numbers

- Whole numbers are
**commutative**for addition and multiplication. You can’t subtract two whole numbers in any order and get the same result.

In notation: For every a, b in the set of whole numbers, a + b = b + a and a · b = b a.

**Example**: 10 – 1 is not the same as 1 – 10. - Whole numbers are
**associative**for addition and multiplication. The order of addition isn’t important (they can be grouped in different orders).

For every a, b, and c in the set of whole numbers, a(b·c) = (a·b) · c and (a + b) + c = a + (b + c). - The set of whole numbers includes an additive identity (0). Zero is the additive identity of whole numbers. In notation, a + 0 = a for every whole number a.
- The
**multiplicative identity**is 1. Multiply any whole number by 1 and you get the same result. In notation, 1 · a = a.

## Natural Number: References

Extending the Natural Numbers to the Wholes