Whole numbers, counting numbers, natural numbers and integer numbers are all closely related to one another. Counting numbers, as their name implies, compose the set {1, 2, 3,...} (in this symbolism the brackets denote the elements of a set and the end dots denote a continuation of the sequence). Natural numbers are usually defined to be the set of counting numbers with zero added, that is the set {0, 1, 2, 3,...}. Integers numbers constitute the set {... -3, -2, -1, 0, 1, 2, 3,...}. Various authors have defined whole numbers differently: defining it as equivalent to the set of counting numbers, equivalent to the natural numbers, or to the set of integers. One reason for the various definitions of something as fundamental as whole numbers is most likely due to the fact that mathematics as a science has evolved over time. New number systems were created and previously defined systems and terms were changed or modified. In this article, whole numbers will be identified with the set of counting numbers, that is as the set {1, 2, 3,...}. Since integers can be partitioned into negative integers, zero (0), and positive integers, whole numbers can also be defined as the set of positive integers.
Before the modern concept of whole or counting number, people had not yet differentiated the "number" of a group from the objects of which it is composed. For example, people used different words to describe a pair of oxen and a pair of men. Whole numbers provided a way to compare groups on the basis of the quantity of elements they possess. Hence a group of 5 things has more elements than a group of 2 things. When used to describe quantity, whole numbers are called cardinal numbers. Now it is quite natural to arrange the elements of a set or group in a particular order. This is done all the time: the tallest to the shortest, the lightest to the heaviest, etc. Arranging the whole numbers according to their cardinality (the quantity of elements each number denotes), and proceeding from smaller to larger, results in the familiar sequence {1, 2, 3,...}. When whole numbers are used to place an ordering on some group, they are referred to as ordinal numbers. For example, customers arriving at a shop can be given a "number" to acknowledge who is number 1 in line, number 2 in line, and so forth. The arrival of a customer is, then, in the same order as the ordinal number they possess. Calling out the whole numbers in sequence yields the order in which the customers are served.
Historically, in 1430 the first known citation for the term whole number came from the Art of Nombryng and appears as the Middle English word "hoole", where its (now obsolete) definition is "a number composed of three prime factors". The term is found in its modern sense in a book anonymously published in 1537 titled An Introduction for to leerne to reken with the Pen and with the Counters, after the true cast of arismetyke or awgrym in hole numbers, and also in broken. In 1839 author J.R. Young refers to "a whole number or 0" in his book Elements of the Integral Calculus, and later referred to "a positive whole number".
The set of whole numbers is closed under the arithmetic operations of addition and multiplication, which means that the sum or product of any two whole numbers is another whole number. However, under subtraction, the inverse operation of addition, the whole numbers do not form a closed set. For instance, the difference of 5 - 2 is a whole number, but the difference of 2 - 5 is not a whole number. The stipulation that subtraction be defined for all values of the set of numbers it operates on leads directly from the whole numbers to the set of integers. In the set of integers the arithmetic expression "a - b" is closed regardless of whether a b or b a. A similar situation holds for division, the inverse operation of multiplication, where the quotient 4 / 2 is a whole number while the quotient of 5 / 2 is not. Once again, as for the case of subtraction leading to integers, the stipulation that division is defined for all values of the set of numbers it operates on leads from the whole numbers to the set of positive rational numbers. The synthesis of the positive rational numbers with the integers leads inevitably to the set of (positive and negative) rational numbers. All four arithmetic operations (addition, subtraction, multiplication and division, are valid and closed for all rational number values (except for division by zero).
The previous paragraph shows how the interaction between the whole numbers and the arithmetic operations leads to successive extensions of the number system (and the operations themselves) within the framework of arithmetic. The expansion and redefinition of the number system continued within the context of algebraic investigations so that ultimately mathematicians devised the real and complex number systems. Further developments within mathematics lead to the construction of mathematical objects such as vectors, matrices, quaternions, etc. The nineteenth century witnessed a concerted effort on the part of many scientists to reexamine the foundations of mathematics. Inasmuch as the more advanced number systems can trace their development back to the whole numbers, one consequence of this effort was a reexamination of the whole numbers. Italian mathematician Giuseppe Peano (1858-1932) made a noteworthy accomplishment in that effort concerning the properties of the set of whole numbers. Peano's axioms (sometimes referred to as Peano's postulates) characterize the whole numbers by the use of five axioms and the principle of mathematical induction. (The axioms are sometimes stated with reference to the natural numbers.) Along with men like German mathematicians Julius Wihelm Richard Dedekind (1831-1916) and Georg Ferdinand Ludwig Philipp Cantor (1845-1918), Peano's axiomatic system helped show how the real number system, and hence most of mathematics, can be derived from a postulate set for the whole numbers.
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