Natural numbers

The natural numbers are the ordinary numbers, 1, 2, 3, etc., with which we count. They are sometimes called the counting numbers. They have been called natural because much of our experience from infancy deals with discrete (separate; individual; easily countable) objects such as fingers, balls, peanuts, etc. German mathematician Leopold Kronecker (1823–1891) is reported to have said, "God created the natural numbers; all the rest is the work of man."

Some disagreement exists as to whether zero should be considered a natural number. One normally does not start counting with zero. Yet zero does represent a counting concept: the absence of any objects in a set. To resolve this issue, some mathematicians define the natural numbers as the positive integers. An integer is a whole number, either positive or negative, or zero.

Operations involving natural numbers

Ultimately all arithmetic is based on the natural numbers. When multiplying 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. The placement of a decimal point is also done by counting natural numbers. When adding the fractions 1/3 and 2/7, the process is also one that involves natural numbers. First, the fractions are converted 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 lightning-fast computations to simple steps involving only natural numbers.

Measurements, too, are based on the natural numbers. In measuring an object with a meter stick, a person relies on the numbers printed near the centimeter marks to count the centimeters but has to physically count the millimeters (because they are not numbered). Whether the units are counted mechanically, electronically, or physically, the process is still one of counting, and counting is done with the natural numbers.

Number theory

One branch of mathematics concerns itself exclusively with the properties of natural numbers. This branch is known as number theory. Since the time of the ancient Greeks, mathematicians have explored these properties for their own sake and for their supposed connections with the supernatural. Most of this early research had little or no practical value. In recent times, however, many practical uses have been found for number theory. These include check-digit systems, secret codes, and other uses.

[ See also Arithmetic ; Fraction, common ; Number theory ]