M425 Exam 1 Spring 2002 -- Solutions
1. Prove that the decimal expansion of a rational number is either terminating or repeating.
Note that the decimal expansion of a rational number can be found using long division. When that is done, either the remainder is 0 at some stage (in which case the expansion is terminating) or the remainder must repeat. In fact it must repeat after at most (denominator - 1) steps, because every remainder is necessarily less than the denominator. Once the remainder repeats, the string of remainders will begin repeating, and the decimal expansion will begin repeating.
2. a) What was the first great crisis of mathematics described in the text?
The discovery of the existence of irrational numbers
b) Why was it a crisis? Give at least one philosophical and one mathematical reason.
The Pythagoreans made the discovery. Since they based their entire philosophy on the integers, and the rationals (made from integers), the existence of other numbers shook their world view. Also, many of their proofs assumed that any two intervals are commensurable, and thus those proofs were no longer valid.
c) How was the mathematical part of the crisis at least partly solved?
Eudoxus came up with a new definition of proportions that allowed new proofs to replace those lost because of incommensurability.
3. Briefly describe the phases in the development of geometry mentioned in our text.
Various names are given to these stages in the text. The first may be called subconscious geometry. In that stage, people become aware of shapes and similarities of shapes between various objects. The second stage may be called scientific. In that stage relationships are arrived at experimentally, by measuring and estimating. So formulas are found, but not proved. The third stage is demonstrative, in which the rules discovered experimentally are verified in some logical manner, or proved.
4. Find the greatest common divisor of the numbers 34,671 and 5467. Show your work and name the method you use.
By repeated divisions, it is found that the GCD is 7. It is a good idea to always check a GCD computation by verifying that the GCD found does, indeed, divide both of the original numbers given!
5. a) Why did we study how to do the various operations of arithmetic geometrically?
b) Draw pictures and explain how to divide 5 by 3, geometrically.
c) Draw pictures and explain how to find the square root of 3, geometrically.
We study the various operations geometrically because that is what the Greeks did. They had very primitive systems of arithmetic and algebra, and so they used their facility in geometry instead. The division (and multiplication) of numbers involves introduction of a unit segment and the observation of proportional sides of similar triangles. The square root operation involves drawing a semicircle with diameter 1 + 3 and erecting a perpendicular at the point between 1 and 3. That vertical line is the square root of 3, because it is the mean proportional between 1 and 3. .
6. State and prove Pappus' theorem, and explain how it implies the Pythagorean theorem.
The statement and proof are in the text. It is important to note how the various parallelograms produced are seen to have equal bases and altitudes, and therefor equal areas. This implies Pythagoras' Theorem because of course squares are special cases of parallelograms. The fact that the square on the hypotenuse is in fact produced by the rule given in Pappus' Theorem should be verified geometrically.