# Natural number

In mathematics, a **"natural" number** is a number from the set {1,2,3,...}.^{[1]} Natural numbers were used initially for counting ("there are three cows in this field"), but they took on the purpose of ordering as well ("She is the 2nd fastest person alive). These are specific instances of the more general notions of cardinality and ordinality which slowly become more complicated as one treats infinite numbers as well. The set of natural numbers is countable- via bijection, this property can be used to prove the countability of the integers and rational numbers.

## Axiomatization

In the late 19th century, Giuseppe Peano (August 27, 1858 – April 20, 1932) elaborated *the* axiomatic system for the Natural Numbers, later named Peano Axioms in his honor.

## Disputes

Some math textbooks, particularly on the college level, include zero. Most elementary school textbooks distinguish between the *whole number*s (i.e., non-negative integers 0,1,2,...) and the *counting number*s (i.e., positive integers 1,2,3,...) - defining natural numbers as identical to counting numbers.

- "Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.
^{[2]}

Students need to check the definition used in their textbook to avoid having test questions marked "wrong".

## Reference

- ↑ 0 is usually included in the list of natural numbers in modern textbooks (Bourbaki 1968, Halmos 1974). Older books sometimes exclude zero, as there is a long history of people thinking that zero is unnatural or not really a number. Bertrand Russell remarked on the trend to include zero in his 1919 book.[1] Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." (Wolfram)
- ↑ Math Is Fun website