Appendix 1 ## RATIONAL AND IRRATIONAL NUMBERSA proof that square root of 2 is not rational Which natural numbers have rational square roots? The decimal representation of irrationals CALCULUS IS A THEORY OF MEASUREMENT But let us start at the beginning. The following numbers of arithmetic are the counting-numbers or, as they are called, the natural numbers: 1, 2, 3, 4, and so on. (At any rate, those are their symbols, the Arabic numerals.) If we include 0, we have the whole numbers: 0, 1, 2, 3, and so on. And upon including their algebraic negatives, we have the integers: 0, ±1, ±2, ±3, and so on. ± ("plus or minus") is called the double sign. The following are the square numbers, or the perfect squares: 1 4 9 16 25 36 49 64, and so on. And these are their roots: 1 2 3 4 5 6 7 8, and so on. The student is no doubt familiar with the radical sign: . = 5. "The square root of 25 is 5." Since it is with those symbols—1, 2, 3, 4—that we do written calculations, it has become common to call the symbols themselves "numbers." But a number is more than a symbol. (@ is a symbol; it is not a number.) The number is that to which the symbol refers. Rational and irrational numbers 1. What is a rational number? A rational number is simply a number of arithmetic: a whole number, a fraction, a mixed number, or a decimal. They are the numbers with whose names and symbols we count and measure. Their negative images nowadays are also called rational. 2. Which of the following numbers are rational?
To see the answer, pass your mouse over the colored area. All of them. All decimals are rational. That long one is an approximation to π, which, as we shall see, is not equal to any decimal. For if it were, it would be rational. 3. Why the name The natural numbers of arithmetic have a ratio to one another. Every number of arithmetic can be written as a fraction, which has the same ratio to 1 as the numerator has to the denominator.
A number that has the same ratio to 1 as two natural numbers we say is rational. Every number of arithmetic proceeds from 1, which is the source. The natural numbers are its multiples, the fractions are its parts. Every number thus has a relationship to 1, which we can always name. ("The number we write as 5 is five times 1." "The number we write as ¾ is three fourths of 1.) Language, we will see, cannot express the relationship of an irrational number to 1. Upon extending these to the negative numbers of algebra: 4. A rational number can always be written in what form?
That is the Finally, we can in principle (by Euclid VI, 9) place any rational number exactly on the number line. We can say that we truly know a rational number. * We will now see that Consider √2 ("Square root of 2"). is close because
—which is To see that there is no rational number whose square is 2, suppose there were. Obviously, it is not a whole number. It will be in the form of a fraction in lowest terms. But the square of a fraction in lowest terms is also in lowest terms. No new factors are introduced and the denominator will never divide into the numerator to give 2—or any whole number. There is no rational number whose square is 2 or any number that is not a perfect square. We say therefore that is an irrational number. 5. Which natural numbers have Only the square roots of the square numbers; that is, the square roots of the perfect squares. = 1 Rational Irrational Irrational = 2 Rational , , , Irrational = 3 Rational And so on. This follows from the same proof that is irrational. * An irrational number and 1 are incommensurable. What does that mean? Every fraction -- every number of arithmetic -- is a sum of unit fractions. And 1 can be written as a multiple of any unit fraction. Every number of arithmetic and 1 will therefore be a multiple of the same unit fraction. For example,
We say in geometry that is a common measure of and 1. It measures three times and 1 five times. When two numbers or two lengths have a common measure, then each will be a
Now an irrational number is not a multiple of any unit fraction. It is not a number of fifths, or tenths, or thousandths, or trillionths. For if it were, it would be rational. An irrational number thus has no common measure with any rational number. Specifically, it has no common measure with 1. We say that they are incommensurable. Every rational number has a common measure with 1. Incommensurability with 1 is what characterizes an irrational number. We cannot Pythagoras effectively knew that in the 6th century B.C. He realized that the relationship of the diagonal of a square to the side was not as two natural numbers. Their relationship, he said, was To say that the diagonal and side of a square are not as two natural numbers, is to say that they are incommensurable. (Knowledge of a square figure, however, is still rational: The square drawn on the diagonal is twice the square on the side.) Irrational numbers have been called surds, after the Latin For the earliest use of the terms rational and irrational, see here. The decimal representation of irrationals A number is not inherently a decimal. However, when we express a Moreover, there will not be a predictable pattern of digits. For example, ≅ 1.4142135623730950488016887242097 (This symbol ≅ means "is approximately equal to.") Now, with rational numbers, you sometimes see = By writing the equal sign = and three dots (ellipsis) rather than ≅, we mean: "It is not possible to express exactly as a decimal. However, we can approximate it with as many decimal digits as we please according to the indicated pattern; and the more decimal digits we write, the closer we will be to ." We say that any decimal for is inexact. But the decimal for , which is The decimal for an irrational number is always inexact. If we write ellipsis— = 1.41421356237. . . —we mean: "No decimal for will be exact. We could continue its rational approximation for as many decimal digits as we please by means of the algorithm, or method, for calculating each next digit (not the subject of these Topics); and again, the more digits we calculate, the closer we will be to ." We often hear that the decimal goes on forever. Obviously, we cannot produce such a sequence of digits nor could we be aware of one. It is nothing but a thought. What would it even mean to say that a decimal that never ends "exists"? It certainly does not exist in this world. In quantum mechanics, it is not logical to say that even an electron exists until it is observed. We are taught of course that certain decimals go on forever, and so we think that's mathematics—that's the way things are. We do not realize that it is a man made conceit. This writer asserts that what we can actually bring into this world—1.41421356237—has more As for the phrase "goes on forever," we may take it to mean what "infinite" has historically meant in mathematics: There is no limit to the number of digits that we could calculate. To sum up, a rational number is a number we can name and know exactly, either as a whole number, a fraction, or a mixed number, but not always exactly as a decimal. An irrational number we can never know exactly in any form. No decimal that you or anyone will ever see is equal to , or π, or any irrational number. We know an irrational number only as a rational approximation. That approximation, moreover, is the only relationship of an irrational number to 1. 6. What is a A The term real number was coined by René Descartes in 1637. It was to distinguish it from an imaginary or complex number (An actual measurement can result only in a rational number. Problem 1. We have categorized numbers as real, rational, irrational, and integer. Name all the categories to which each of the following belongs.
8. What is a real variable? A variable is a symbol that takes on values. A Calculus is the study of functions of a real variable.
Problem 2. Let
* See The Evolution of the Real Numbers starting with the natural numbers.
Appendix 2: Are real numbers really numbers? Copyright © 2022 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |