Real numbers

Summary: The real number system is commonplace, but required centuries before it came to be understood in its modern form.

The real number system is often thought of as a number line, with each point on the line corresponding to a number. The set of real numbers includes all the integers and fractions (rational numbers); algebraic irrational numbers, such as the square root of 3 and the cube root of 19; and transcendental numbers such as π, log(2), and e, which do not satisfy any polynomial equation.

In the twenty-first century, students begin to explore whole numbers beginning in the earliest grades, as well as common fractions like 1/2 and 1/4. In the later primary grades they develop knowledge of base-10 decimal places, fractions as portions or divisions of a whole, negative numbers, and equivalent forms for fractions, decimals, and percentages. These notions are further expanded and applied in middle school, including concepts like ratios and proportions, integers, factorization, prime numbers, and exponential and scientific notation for very large numbers. Very large and small numbers, properties of numbers and various number systems, vectors and matrices with real number properties, and number theory may be studied in high school.

The real number system is the principal number system used in calculus, geometry, and measurement. In particular, when one uses coordinate (Cartesian) geometry to describe the plane or space, one labels points by pairs or triples of “real” numbers. In mathematics and the sciences, the word “number” without qualification is generally used to mean “real number.”

Development of the Real Numbers

The ancient theory of length and measurement was very different from current understanding. The ancient Greeks (the civilization about which exists the most complete mathematical history) believed that any set of lengths were commensurable; in modern language, they believed that the ratio of any two lengths (or areas, or volumes) was a rational number. This was not a totally unreasonable belief, since indeed all lengths can be approximated very well by commensurable ones. It is not correct, though; for example, the ratio of the diagonal of a square to its side is the square root of 2.

Greek mathematician and numerologist Pythagoras knew this (it is a simple consequence of what is now called the “Pythagorean Theorem”) and was further able to prove, contrary to the notion of commensurability, that no rational number, when squared, could equal 2. According to some stories, probably apocryphal, this discovery was so contrary to the belief system of Pythagoras and his followers that a discoverer was murdered or committed suicide. Ultimately, geometers were forced to accept the existence of irrational numbers.

The Greek mathematician and astronomer Eudoxus (c. 400–350 b.c.e.) wrote about the theory of proportions in a way that did not assume all lengths were commensurable and is generally credited with laying the groundwork for irrational numbers as legitimate mathematical objects.

Even after mathematicians realized that irrational numbers were required for practical purposes, the understanding of the real number line was somewhat vague and confused. Real numbers were understood, if at all, as things that could be approximated well by rational numbers or by decimal approximations. The major modern contribution to the understanding of real numbers was made by Richard Dedekind (1831–1916), who described the real numbers in terms of so-called “Dedekind cuts.” In addition to its significance for abstract mathematics, Dedekind’s insight also helped to explain some important phenomena in geometry (for example, why a line with points inside and outside a circle must intersect the circle).

This resistance to advancements in the understanding of number, this tendency for even very intelligent people to oppose enlarging the number system, even when doing so enables scientific and technological progress, is not unique to the ancient Greeks. A similar story unfolded much more recently with the development of the complex number system.

Decimal Representations

Every real number has a base-10, or decimal, representation. This consists of three components: a dot (called a “decimal point” in this context), a finite sequence of digits to the left of the decimal point (the integer part), and an infinite sequence of digits to the right (the fractional part). A digit can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Working from the decimal point left, the digits occupy the ones place, the tens place, the hundreds place, the thousands place, and so on; from the decimal point right, the digits occupy the tenths place, the hundredths place, the thousandths place, and so on. In symbols, if the a^k are digits, then the decimal expansion

a_ka_k-1…a₂a₁a₀·a_-1a_-2…a_-k…

represents the real number

a_k10^k +…+a₁10 + a₀ + a_-110^-1+…+a_-k10^-k+….

If from some point rightward, all the digits in the decimal representation of a number are zero, that expression is said to terminate, and the trailing zeroes are typically not written; for example, “7.24” instead of “7.24000.…” Integers have only zeroes to the right of the decimal point, and in such cases even the decimal point is often omitted.

Relying exclusively on decimal expansions as a way to understand real numbers can be problematic. Specifying a real number in this way requires an infinite sequence of digits. Unless there is a pattern, this requires specifying an infinite amount of information. For example, there is no known digit-by-digit description for important numbers like π and e. Dealing with infinite expressions is confusing for many people. For example, some people find it difficult to accept that 0.33333… = 1/3, and even more people find it uncomfortable that 0.99999… = 1.

Almost all real numbers have a unique decimal expansion, but some have two. As 0.99999… = 1 illustrates, every number that can be written so that it ends in an infinite string of 0s also has an expansion that ends in an infinite string of 9s.

Structural Properties of the Real Number System

The real numbers form a field, which means that real numbers can be added, subtracted, multiplied, and divided (except by 0), and that the operations satisfy certain properties (for example, commutative, associative, and distributive laws). The real numbers are actually an ordered field, which means that there is a notion of what it means for one number to be less or greater than another that is compatible with the operations.

There is a natural way to measure distance between two numbers: the distance between numbers a and b is |a-b|, where |•| is the absolute value function. Loosely speaking, this means that one can talk about “closeness” of real numbers to each other; in technical language, the number line has a metric and a topology.

Unlike the set of integers (which is discrete), the real number line is continuous. The discrete/continuous distinction in mathematics is analogous to the digital/analog distinction in science and technology. A digital thermometer has discrete output, moving from 24 degrees to 25. An analog thermometer, on the other hand, can register 24 degrees or 24.65474 degrees or any other number. Unlike both the integers and the rational number system, the real number line is what called “topologically complete.” Because of the ordering on the reals, this can be summarized as: “Any set of real numbers which has an upper bound has a least upper bound.”

In mathematics history, adopting a continuous number system made it possible to develop “limits,” the focal concept of calculus. The development of calculus, in turn, made possible numerous advances in sciences, especially physics and engineering.

Bibliography

Borwein, Jonathan, and Peter Borwein. A Dictionary of Real Numbers. Pacific Grove, CA: Brooks/Cole Publishing, 1990.

Burrill, Claude. Foundations of Real Numbers. New York: McGraw-Hill, 1967.

Drobat, Stefan. Real Numbers. Upper Saddle River, NJ: Prentice Hall, 1964.

Scriba, Christoph, with M. E. Dormer Ellis. The Concept of Number: A Chapter in the History of Mathematics, with Applications of Interest to Teachers. Zurich: Mannheim, 1968.

Stevenson, Frederick. Exploring the Real Numbers. Upper Saddle River, NJ: Prentice Hall, 2000.