The natural numbers are the ordinary numbers, 1, 2, 3,... with which we count. Sometimes they are called "the counting numbers." They have been called "natural" because much of our experience from infancy deals with discrete objects such as fingers, balls, peanuts, etc. We quickly, if not naturally, learn to count them. The mathematician Kronecker is reported to have said," God created the natural numbers; all the rest is the work of man." The number zero is sometimes considered to be a natural number, but including zero introduces complications which one ordinarily wants to avoid. For instance, counting a group of objects is a process of putting them into one-to-one correspondence with the natural numbers. No one counts starting with zero, "0, 1, 2, 3..." because then the last number used is no longer the number of objects in the group and the answer would be wrong.
Because there is some ambiguity about the meaning of "natural number," the terms "positive integers" (which do not include zero) and "non-negative integers" (which do) are often used instead. Experts in the foundations of mathematics would argue, however, that the natural numbers are different from the positive integers, although the difference is only in how the numbers are defined. For practical purposes, they are the same.
Ultimately all arithmetic is based upon the natural numbers. If one multiplies 1.72 by .047, for example, the multiplication is done with the natural numbers 172 and 47; then the result is converted to a decimal fraction by inserting a decimal point in the proper place-a process that is also done by counting. If one adds the fractions 1/3 and 2/7, the addition is not done directly, but only after converting the fractions to 7/21 and 6/21. Then the numerators are added, using natural-number arithmetic, and the denominators copied. Even computers and calculators reduce their complex and lightening-fast computations to simple steps involving only natural numbers.
Measurements, too, are based on the natural numbers. If one measures an object with a centimeter ruler, he or she relies on the numbers printed near the centimeter marks to count the centimeters, but the eye to count the millimeters. Whether the units are counted mechanically, electronically, or physically, it is still counting, and counting is done with the natural numbers.
The natural numbers can be defined formally by relating them to sets. Then zero is the number of elements in the empty set; 1 is the number of elements in the set containing the empty set; and so on. Another method is to base them on the Peano postulates. Here 2 is defined as the successor of 1; 3 the successor of 2; and so on. Then the postulates specify what properties 1 and its successors will have.
There is one branch of mathematics which concerns itself exclusively with the properties of natural numbers (including natural number-based modular arithmetic). This is the branch known as "number theory." Since the time of the ancient Greeks, mathematicians both amateur and professional have explored these properties for their own sake and for their supposed connections with the supernatural. And in recent times many practical uses, quite apart from counting and computation, have been found for the natural numbers and their special properties. These include check-digit systems, secret codes, and other uses.
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