A **natural number **is a counting number, a member of the set {1,2,3,4,….}. The set of natural numbers is designated by ℕ.

## Formal Description of Natural Numbers

More formally, we can define the natural numbers as the set ℕ= {x| x=1, or there is some y in ℕ where y = {x + 1}. In plain English, that means that ℕ is the set where x either is one, or x is 1 more than some other number y in ℕ. The easiest way to come up with examples of natural numbers is simply to start at one and begin counting; each of the numbers that result when you add 1 to a natural number is another natural number.

All natural numbers are integers. In set theory, we would say that ℕ is a subset of ℤ.

## Examples of Natural Numbers

Examples of natural numbers include** 4, 79, and 56,793. **

-5, -7, and 9.8 are *not *natural numbers.

## Alternate Definition of Natural Numbers

The set of natural numbers is sometimes defined as the set {0,1,2,3,4,….}. In this case it is equivalent to the set of non negative integers. The essential properties of natural numbers remain the same whether or not we include 0.

## Properties of a Natural Number

The set of natural numbers is *closed *under addition and multiplication. If you add or multiply natural numbers, you’ll get a natural number as a result. That is, for any a and b in ℕ, a + b = c and a * b = g will also be in ℕ.

The set of natural numbers isn’t closed under subtraction or division. For every natural number *a*, there exist natural numbers *b *and* c* such that a – b = e and a / c = f, where e and f are *not *natural numbers.

Natural numbers make an infinite set. For any natural number, there is another natural number that is one greater.

Another important property of natural numbers is that they can be **ordered**. Formally, we’d say that for any a, b in ℕ a > b if and only if a = b + k for some k in ℕ. This means that natural numbers include the set of ordinal numbers and the set of cardinal numbers.

## Sources

Constructing the Natural Numbers

Natural Numbers and Whole Numbers

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