12 ## IRRATIONAL NUMBERSThe relationship of arithmetic to geometry The invention of irrational numbers THE JOB OF ARITHMETIC when confronted with geometry, that is, with things that are continuous -- length, area, time -- is to come up with the name of a number to be its measure. For if we say that a length is 3½ meters,
then the number "3½" names the ratio of that length to 1 meter. (Lesson 7.) With a different unit of measure -- 1 inch, 1 foot, 1 mile -- the "length" of that line would be a completely different number. But is there a number to name every ratio to a unit? If the square on AB is to the square on CD in the ratio 2 : 1, then AB, CD themselves are incommensurable, and no rational number can name their ratio. For, any two rational numbers always have a common measure. Incommensurability has been and is the logical sticking point in the relationship of arithmetic to geometry. It is not only an intellectual problem, but it has been an extremely practical one in the history of mathematics. For if we want to involve AB, CD in some calculation, what number would we write? AB = ? × CD. The invention of irrational numbers We have seen that when two squares are in the same ratio as two If we insist, however, that there be a × = 2. is that number which when multiplied by itself -- when squared -- is 2. "Square root of 2" is its name. It is not any whole number, any fraction, any mixed number or any decimal -- it is not any rational number. It is a pure creation.
-- which is
But that is impossible. Since is in lowest terms, then There is no number of arithmetic -- no rational number -- whose square is 2. Naturally we wonder, "How much is ?" We can only answer approximately. As a fraction, we saw it is almost . As a decimal, it is approximately 1 1 We could come closer to by approximating it with more decimal digits. But no decimal squared will ever be exactly 2. is irrational. It was argued for many centuries whether "really" is a number. Mathematicians thought of as a convenient symbol with which to calculate, but as for its being a In the next Lesson, we will investigate in what sense irrational numbers exist. And we will return to our original inquiry: If AB, CD are lengths, will there always be a number AB is to CD as 1 is to In any event, is not the only irrational number. Let us illustrate that by asking: The square roots of which natural numbers are rational?
= 1 is rational. , are irrational. = 2 is rational. , , , are irrational. = 3 is rational. And so on.
Problem 1. Say the
Problem 2. Which of the following are rational numbers and which are irrational?
A common measure with 1 We have seen that every rational number has the same ratio to 1 as two natural numbers. We can always and 1 are incommensurable. Problem 3. What number is a common measure of each pair? a) 1 and 5 1
c) 1 and 2 d) 1 and None. Problem 4. Why were irrational numbers invented?
To have a name for the relationship of incommensurable magnitudes. In algebra, it was so that the solution of any quadratic equation—e.g. Problem 5. The squares on the sides of triangle ABC are in the ratio 1 : 4 : 3. Express the ratio of each pair of sides as a ratio of numbers, whether rational or irrational. a) AB : BC = 1 : 2 b) BC : CA = 2 : c) CA : AB = : 1
Problem 6. One square is four fifths of another. a) Are the sides commensurable? No. 4 and 5 are not both square numbers. b) Express the ratio of the sides as a ratio of numbers. 2 : c) Are the Yes. They are in the same ratio as natural numbers. Next Topic: The existence of irrational numbers Copyright © 2020 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com |