Rational and irrational numbers

Summary: While the concept of rational numbers is easily understood, mathematicians has struggled with the concept of irrational numbers since antiquity.

Early philosophers and mathematicians explored whether real-life lengths were made up of whole numbers. The discovery of irrational numbers caused great concern and led to the development of number theory and real analysis. A rational number is a real number that can be written as a ratio of two integers. Real numbers that cannot be so written are called “irrational numbers.” So, for example, 17/47 is a rational number, while π or √2 are irrational numbers. Rational and irrational numbers can also be represented using the decimal notation. The rational numbers are precisely those numbers whose decimal representation either terminates after a finite number of digits or is repeating. The decimal representation of irrational numbers does not have a repeating pattern. So, for example, 1/8 corresponds to 0.125 (a terminating decimal), while 1/7 corresponds to the repeating decimal 0.142857142857.… On the other hand, the irrational number π has a decimal expansion that begins with 3.14159265358979323846… and continues indefinitely without any patterns.

Students in twenty-first-century classrooms explore rational numbers in middle school and irrational numbers in high school, and these numbers appear in nature and in daily calculations. For example, e, which is also irrational, is needed to calculate the interest compounded continually on a loan, and π appears in circular or spherical objects. In fact, as a consequence of Georg Cantor’s work, given any real number, there is a higher probability of it being irrational. There are still open problems to explore, such as whether π+e is irrational.

Definition

Irrational numbers are numbers that are not rational; in other words, any number that is not the ratio of two integers is an irrational number. This definition by itself, however, is circular. To be able to use it, one first has to know what a number is.

What is a number? This question is harder to answer than one might expect at first, and, in fact, has been contentious for most of the history of mathematics. The positive integers (or the counting numbers) 1, 2, 3,… directly arise from the daily experience of humans, and it is impossible to trace how long ago humans went from the concrete ideas of three cows, three stones, and three trees and abstracted out the number 3 as a stand-alone concept. The advantage of this abstraction is that people could study operations on numbers and apply them to a large number of settings. If one knows that 3+5=8, this indicates simultaneously that 3 cows together with 5 cows are 8 cows, and that 3 trees and 5 trees are 8 trees. If one knows that 3×5=15, then, while 3 trees and 5 trees cannot be multiplied, this can be used to model many situations. Three boys each having five apples have a total of 15 apples, and a 3-by-5 piece of land has an area of 15. Mathematicians can now concentrate on finding better algorithms and methods for doing number operations. The concept of number was first enlarged to also encompass rational numbers—the ratios of positive integers.

Some 3500 years ago, Egyptians used unit fractions (reciprocals of positive integers) and 2/3 to pose and solve problems. For example, the third problem in the Rhind Mathematical Papyrus is about dividing 6 loaves among 10 men, and the answer given is

Around the same time, Babylonian scribes in Mesopotamia used a base-60 place value system for fractions, but confined themselves to those rational numbers that have a finite sexagesimal representation. In any case, for a very long time, the term “number” meant positive integers and ratios of positive integers—what are now called the “positive rational numbers.”

If the only numbers are rational numbers, then by definition there are no irrational numbers. To enlarge the definition of “number” beyond rational numbers, one has to somehow construct these other numbers, which can be done by proposing that the length of any line segment is a number.

It is believed that in early mathematics it was assumed that given any two line segments, it is possible to find a third line segment—maybe a very small one—that measures both lines a whole number of times. In other words, the length of each of the original line segments is an integer multiple of the third smaller line segment. The third line segment is called a “common unit of measure,” and the original two line segments are called “commensurable.” On the face of it, this assumption may seem reasonable, but, if true, it would mean that the length of any line segment is a rational number. Given an arbitrary line segment of length a, find a common unit of measure for it and a line segment of unit length. If the common unit of measure has length b, then a=mb and 1=nb for some integers m and n. But this means that

from which it can be determined that

is a rational number.

The Pythagorean Theorem, which was known at least 1000 years before Pythagoras, states that given a right triangle whose sides are of unit length, then the length of its hypotenuse will be such that yields 2 if multiplied by itself. Since it has been decided that all lengths are numbers, the length of this hypotenuse must be a number called the “square root of 2” and denoted by √2. One can prove that √2 is not a rational number, thereby proving that not all pairs of line segments are commensurable and that irrational numbers exist.

There are many proofs of the irrationality of √2, but the most common one is as follows: Assume by way of contradiction that √2 is rational and equal to n/m, where n and m are integers. One has many choices for n and m; for example, one could multiply both by 47 and get a new pair of integers with the same ratio—and values of n and m are chosen such that they do not have any common factors. This is, of course, possible. From √2= n/m, one obtains 2m²=n², which means that n² and therefore n is an even number. If ⁿ=2^k, then 2m²=4k² and so m2=2k², which means that m is also an even number. But it had been assumed that n and m have no common factors. A contradiction was reached but, since the logic along the way was impeccable, it must have been that the original assumption that √2 is rational must have been wrong.

Implications

The discovery of irrational numbers led to a crisis in geometry and a need to revisit all the results that depended on the commensurability assumption. Following Eudoxus, Euclid in his very influential book Elements makes a distinction between a number and a magnitude. Roughly, one can think of numbers as the rational numbers and magnitudes as the lengths of line segments—Euclid had an elaborate classification of magnitudes. In an attempt to be rigorous, Euclid treats number and magnitude differently, and hence he does not regard irrational numbers as numbers. For example, he develops the theory of proportions once for magnitudes and once for numbers.

It took the effort of many mathematicians in the middle ages—and most notably mathematicians living in Islamic lands and writing in Arabic—to expand the notion of number to include Euclid’s magnitudes and to have a single treatment of all numbers, rational and irrational. During this period, the decimal number system—first developed in India and crucial in understanding irrational numbers—became widespread. Ninth-century Persian mathematician Al-Mahani gave a definition of irrational numbers (as opposed to Euclidean magnitudes), and ninth-century Egyptian mathematician Abu Kamil used irrational numbers as coefficients in algebraic equations. By the fifteenth century, Persian mathematician Jamshid Kashani (also referred to as al-Kashi) was able to comfortably work with real numbers and their decimal expansions. He treated both rational and irrational numbers as numbers.

In the West, sixteenth-century Flemish mathematician Simon Stevin played an important role in advocating the use of decimal fractions, in eliminating the Euclidean distinction between numbers and magnitudes, and in the understanding of real numbers as numbers. Finally, a modern rigorous construction and definition of real numbers (rational and irrational) was given by nineteenth-century German mathematician Richard Dedekind. He started with rational numbers and defined irrational numbers using the rational numbers.

Bibliography

Gouvea, Fernando. “From Numbers to Number Systems.” In The Princeton Companion to Mathematics. Edited by Timothy Gowers. Princeton, NJ: Princeton University Press, 2008.

Katz, Victor J. A History of Mathematics: An Introduction, 2nd ed. Boston: Addison-Wesley, 1998.