11 ## INCOMMENSURABLE MAGNITUDESThe square drawn on the diagonal HERE AGAIN IS THE THEOREM of the previous Lesson: Two straight lines will have a common measure -- they will have the same ratio as two natural numbers -- if and only if the squares on them have the same ratio as two square numbers. What, then, will be the case if two squares are 2 and 1 are not both square numbers. What must we say about the ratio of the sides? The sides do We say that those sides are incommensurable. Problem 1. If one square is three times the size of another square, then a) are their To see the answer, pass your mouse over the colored area. Incommensurable. The squares are in the ratio 3 to 1, and 3 and 1 are not both square numbers. b) if the No. Those sides have no common measure. c) if one side is so many millionths of an inch, could the other side also d) if one side is a multiple of e) Will those sides have the same ratio as two natural numbers? That is, No. f) If one side is a rational number of units, could the other side also be No. They cannot both be rational, because if they were, they would have a common measure. (Lesson 9.) g) Can you express the ratio of those sides? Can you? Problem 2. How will we know when straight lines are incommensurable? The squares on them are not in the same ratio as square numbers. Problem 3. a) If two squares are to one another as 9 is to 20, do their sides have a No. 9 and 20 are not both square numbers. b) If two squares are to one another as 9 is to 25, do their sides have a
The smaller side is Problem 4. Let a square be 10 square meters. a) Does that Yes. 10 and 1 are natural numbers. b) Will its
No. 10 and 1 are not both c) Will its side be a rational number of meters? No. The side has no common measure with 1 meter. Problem 5. a) If one square is four ninths of another square, is its side a fraction of Yes. 4 and 9 are square numbers. b) If one square is four fifths of another square, is its side a fraction of No. 4 and 5 are not both square numbers. Could The square drawn on the diagonal That two magnitudes could be incommensurable was first realized by Pythagoras in the 6th century B.C. To see what Pythagoras saw, consider the square ABCD on the left: On the right, we have joined three equal squares, making a square four times as large. Let us now cut each of those four equal squares in half: Then EDBF is itself as square, and it is composed of four of those equal halves. ABCD is composed of two of them. Therefore EDBF is twice as large as ABCD. Now, DB is called the diagonal of the square whose side is AB. Problem 6. Are the diagonal DB and the side AB commensurable or incommensurable? Incommensurable. The squares on them are in the ratio 2 : 1, and 2 is not a square number. The student should know that this discovery -- Pythagoras realized that since the diagonal and side are not in the same ratio as two natural numbers, he could not Nevertheless, knowledge of a square figure is still rational: The square drawn on the diagonal is Multiples that meet? Consider two magnitudes of the same kind, of them meet, that is, if a multiple of If four On the other hand, if multiples would never meet, no matter how far they might be extended. Multiples of the diagonal and side of a square will never meet. A multiple of one will never be equal to a multiple of the other. Problem 7. Since any two lengths could be measured with a ruler, and we always get a rational number, what sense does it make to say that two lengths are incommensurable? Since lengths are continuous, with no units to count, we always have the problem of measuring exactly. Measurement is limited not only by the fineness of the measuring instrument, but also by the fineness of our eyes to see its readings.
do, or do not, have a common measure, we mean as determined logically, not with rulers. Problem 8. a) Again, what One number is a multiple of the other, a part of it, or parts of it; or a mixture of those. Express the following ratios:
b) Are magnitudes necessarily in the same ratio as natural numbers? No. We cannot always express their relationship in words. c) Therefore, what do we mean by the "ratio" of two magnitudes? ? ? ? The new theory of proportions That magnitudes can be incommensurable completely upsets the theory of proportions. For if the square on AB is twice the square on CD, if they are in the ratio 2 : 1, then the lengths AB, CD are incommensurable; 2 is not a square number. And if the square on EF is also twice the square on GH, then EF, GH are also incommensurable -- yet we expect that whatever ratio AB has to CD, EF will have it to GH. We expect, proportionally, AB is to CD But according to the definition of natural numbers being "in the same ratio," that
will make no sense, because AB is not any Yet we can see that they have the same relationship. Therefore we must create a new definition of "in the same ratio," one that will be applicable to incommensurable magnitudes. We will not present the new definition here. Seeing the need for it -- namely the discovery of incommensurables -- is the climax of our present study. Seeing the need for a new definition of "in the same ratio" was mathematics first logical crisis, and it has always marked the beginning of advanced mathematics. Next Topic: Irrational numbers Please make a donation to keep TheMathPage online. Copyright © 2015 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |