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Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this is the 3rd largest city in the country"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics.
History of natural numbers and the status of zeroThe natural numbers presumably had their origins in the words used to count things, beginning with the number one. The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. For example, the Babylonians developed a powerful place-value system based essentially on the numerals for 1 and 10. The ancient Egyptians had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. A much later advance in abstraction was the development of the idea of zero as a number with its own numeral. A zero digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element.1 The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but they did not pass it along to anyone outside of Mesoamerica. The modern concept dates to the Indian mathematician Brahmagupta in 628 AD. Nevertheless, zero was used as a number by all medieval computists (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it. Instead, the Latin word for nothing, nullae, was used. The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. However, independent studies also occurred at around the same time in India, China, and Mesoamerica. In the nineteenth century, a set-theoretical definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number. Wikipedia follows this convention, as do set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers. The term whole number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers. NotationMathematicians use N or <math> \mathbb{N}<math> (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition. To be unambiguous about whether zero is included the following are sometimes used to indicate the positive integers:
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