Natural number

• 1. Introduction

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  There are seven days in a week. There are three apples on a table. The third day of the month falls where it falls. In each of these small claims lives a natural number, the kind of number people reach for to count and to order without ever thinking about what they are reaching for. These are the numbers 0, 1, 2, 3, and so on, though whether 0 belongs among them has never been fully settled. They feel obvious. A child learns them before learning what a definition is. Yet for most of history they were simply called numbers, with no qualifying word at all. So what made mathematicians decide these particular numbers needed a name, and a rigorous one? Why does a concept this familiar require axioms, set theory, and the careful labor of people like Peano and von Neumann? And how can something used to label a jersey or count a class of students also sit at the foundation of integers, rationals, reals, and complex numbers? The answers run from a 15th-century phrase to a 20th-century standards document, and they begin with two very different jobs that a single number quietly does at once.

• 2. Counting And Ordering

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  How many apples are on the table? A natural number used to answer that question describes the size of a finite collection, a property called cardinality, and the number itself is a cardinal number. The size can be established without counting at all. If every apple can be paired with exactly one orange and every orange with exactly one apple, the two groups have the same cardinality. This one-to-one correspondence is enough to declare the number of apples and the number of oranges equal, in the pictured case both being 3.

  Pairing also settles which collection is larger. When two collections lack the same cardinality, the pairing leaves one group with unpaired objects left over. The collection whose objects are all paired is the smaller, and the one stuck with leftovers is the larger. Size, in other words, can be defined before counting is ever invented.

  A sequence is a list of objects in a specific order, more precisely a function that assigns an object to each position in the list. Here natural numbers do a different job. They label positions rather than measure sizes, making them ordinal numbers, as in the third day of the month. The number 1 marks the first position, 2 the position right after 1, and 3 the position after both 1 and 2 and before 4 and 5.

  The natural numbers are the most common labels for infinite sequences because they form the simplest infinite well-ordered set, with order type omega. They start at either 0 or 1 and run on in fixed order with no end point. Yet they are only the most familiar example. Any well-ordered set would index a sequence just as well, even the letters a, b, c, and so on.

• 3. The Trouble With Zero

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  Starting at 0 or 1 has long been a matter of convention, and the disagreement is old enough to have a paper trail. In 1727, Bernard Le Bovier de Fontenelle argued both sides at once. He allowed that 0 could be a term in a sequence 0, 1, 2, and so on, yet held that 1 was the basic element from which other numbers were built by repeated addition.

  Giuseppe Peano changed his own mind on the matter. In 1889 he used N for the positive integers and began at 1. He later switched to writing N0 and N1, distinguishing the two starting points by notation rather than choosing one outright.

  Most early authors simply excluded 0. But a long roster of mathematicians included it, among them George A. Wentworth, Bertrand Russell, Nicolas Bourbaki, Paul Halmos, Stephen Cole Kleene, and John Horton Conway. Their convention spread, gaining wider adoption in the 1960s.

  ISO 31-11, published in 1978, finally put the inclusion of 0 into a formal standard, a convention the current ISO 80000-2 still keeps. To eliminate the lingering ambiguity, the two readings now travel under clearer names: the sequence starting at 1 is the positive integers, and the sequence starting at 0 is the non-negative integers. The phrase whole numbers points the same way, though it can also mean all integers, positive and negative.

• 4. Symbols On The Page

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  Ten symbols carry the whole system in writing: 0 1 2 3 4 5 6 7 8 9. These are called numerals, and a particular set of symbols with rules for combining them is a numeral system. Each symbol stands for a unique natural number, its value, and can appear alone or strung together with others to form a larger numeral.

  The decimal system, built from Arabic numerals and positional notation, is the universal standard for writing natural numbers in mathematics and everyday life. Because that standard is so dominant, the line between an abstract number and the symbol that names it usually does not matter. Numerals get called numbers without complaint, even in cases where the distinction does matter, as when binary numerals are loosely called binary numbers.

  Numerals can also do work that has nothing to do with quantity. Used as unique identifiers or labels, like the jersey numbers on a sports team, they become nominal numbers. They look like ordinary natural numbers but carry no specific mathematical properties at all. The same symbol that measures a collection can also be a name.

• 5. Building Them From Nothing

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  Formal definitions take the intuitive idea of natural numbers, together with the rules of arithmetic, and rebuild both in the more fundamental terms of mathematical logic. These systems usually treat fixed order as the defining trait and establish it through a primitive notion called the successor. Every natural number has a successor, another unique natural number that follows it.

  The Peano axioms, named for Giuseppe Peano, do not say what the natural numbers are. They list statements that must hold true of natural numbers however those are defined: 0 is a natural number; every natural number has a successor that is also a natural number; 0 is not the successor of any natural number; if two numbers have equal successors then the numbers are equal; and the axiom of induction, which carries a truth from 0 up through every successor to all natural numbers. Some versions put 1 in place of 0.

  Set theory takes the opposite route and defines each natural number as a particular set. The standard construction is due to John von Neumann. It calls 0 the empty set, then defines the successor of any set by joining the set to the set containing it. By the axiom of infinity there exist inductive sets, ones that contain 0 and are closed under this successor step. The intersection of all inductive sets is itself inductive, and that intersection is the set of natural numbers.

  The two methods look nothing alike, yet they agree. The natural number sets built by von Neumann's construction collectively satisfy the Peano axioms. In that construction every natural number n is a set containing n elements, each of which is a natural number less than n, so cardinality and order both fall out of the sets themselves. Order becomes the subset relation, and an older rival construction now survives only as a matter of historical interest.

• 6. A Semiring, Not A Ring

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  Addition on the natural numbers is repeated application of the successor function. The structure it forms is a commutative monoid with identity element 0, and it is a free monoid on one generator. Because this monoid satisfies the cancellation property, it can be embedded in a group, and the smallest group containing the natural numbers is the integers.

  Multiplication, defined once addition is in hand, turns the natural numbers into a free commutative monoid with identity element 1. The generators this time are the prime numbers. Addition and multiplication fit together through the distribution law, the rule that ties the two operations into one compatible system.

  Subtraction is where the structure breaks. The natural numbers are not closed under subtraction, since taking a larger one from a smaller one yields a negative number that lies outside the set. Put another way, they lack additive inverses. That single gap means the natural numbers do not form a ring; they form a semiring, also called a rig.

  Division fails in a similar way, leaving a remainder more often than not, but it has a substitute. Euclidean division, division with remainder, gives a unique quotient and a unique remainder for any pair of numbers. That procedure is the key to divisibility, to the Euclidean algorithm, and to much of number theory built on top of them.

• 7. From Natural Progression To Standard

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  For most of history, what are now called natural numbers were simply numbers, with no special label needed. The label became necessary only as the idea of number widened. Between the late middle ages and the end of the 17th century, the concept expanded to take in negative, rational, and irrational numbers, growing into what is now called the real numbers, and the originals suddenly needed a name to set them apart.

  Nicolas Chuquet used the phrase progression naturelle, natural progression, in 1484. The earliest known appearance of natural number as a complete English phrase comes in 1763. By 1771 the Encyclopaedia Britannica was defining natural numbers inside its article on the logarithm.

  In 19th-century Europe the discussion turned philosophical. Henri Poincaré held that axioms can only be demonstrated in their finite application, and credited the power of the mind with conceiving the indefinite repetition of the same act. Leopold Kronecker put his own view bluntly: God made the integers, all else is the work of man.

  The rigor came from a chain of definitions. In the 1860s Hermann Grassmann offered a recursive definition, implying the numbers were not really natural but a consequence of definitions. Charles Sanders Peirce gave the first axiomatization of natural-number arithmetic in 1881. Richard Dedekind proposed another in 1888, and in 1889 Peano published a simplified version of Dedekind's axioms in his book Arithmetices principia, nova methodo exposita, the work now called Peano arithmetic. Frege's earlier set-theoretic attempt, defining a number as the class of all sets in one-to-one correspondence with a given set, had to be abandoned after it led to paradoxes including Russell's paradox.

• 8. The Numbers Underneath

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  Number theory studies the properties of the operations on natural numbers and their generalizations, while much of combinatorics counts the objects, patterns, and structures defined using them. These are not the only fields resting on the foundation. The most common number systems in all of mathematics are extensions of the natural numbers, each containing a subset with the same arithmetical structure.

  The ladder upward is built by allowing new results to count as numbers. Treat the difference of every two natural numbers as a number and the integers appear, complete with zero and the negatives. Treat the quotient of every two integers as a number and the rationals appear, fractions included. Treat every infinite decimal as a number and the reals appear. Treat every solution of a polynomial equation as a number and the complex numbers appear.

  The natural numbers also mark a limit on what the simplest tools can reach. Peano arithmetic is equiconsistent with several weak systems of set theory, one being ZFC with the axiom of infinity replaced by its negation. And there are statements about the natural numbers, such as Goodstein's theorem, that ZFC can prove but the Peano axioms alone cannot.