MONT 104Q – Mathematical Journeys Solutions for Midterm Exam, November 2, 2015

I.

(A) (10) G. H. Hardy included two theorems and proofs from Euclid’s Elements as prime examples of “real, serious, beautiful mathematics” in his book A Mathematician’s Apology. Give the statements of those two theorems. You do not need to supply the proofs for these.

First theorem: Solution: There are infinitely many prime integers.

Second theorem: Solution: The number \sqrt{2} is irrational (that is, it cannot be written in the form \sqrt{2} = \frac{a}{b} where a, b are integers).

(B) (5) Hardy says “the mathematics which has permanent aesthetic value ... may continue to cause intense emotional satisfaction to thousands of people after thousands of years.” About how much time (to within 100 years) separates the time of Euclid from Hardy’s own time?

Solution: It was about 2200 years, give or take. (Euclid lived about 300 BCE and Hardy was alive 1877 - 1947 CE, so taking 1900 CE as a date for Hardy, that’s about 2200 years.)

II.

(A) (10) State the 5 Common Notions (Axioms) and 5 Postulates at the start of Book I of the Elements of Euclid.

Solution: The Common Notions (This is just one possible way to state all of these. I’ll accept alternates meaning the same, and they don’t need to be in this order)

(1) Things that are equal to the same thing are equal to one another.

(2) If equals are added to equals the wholes are equal.

(3) If equals are subtracted from equals, the remainders are equal.

(4) Things that coincide with one another are equal to one another.

(5) The whole is greater than the part.

The Postulates: (This is just one possible way to state all of these. I’ll accept alternates meaning the same.)

(1) (It is possible) to draw a straight line from any one point to any other point.

(2) (It is possible) to extend a line indefinitely in both directions.

(3) (It is possible) to construct a circle with any given point as center and radius equal to any given line segment.

(4) All right angles are equal.

(5) If a line falling on two other lines makes angles on one side totaling less than two right angles, then the two lines, when extended indefinitely, meet on the side on which the two angles totaling less than two right angles are located.