E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 2: Irrational numbersObjectiveWe want to appreciate one of the great moments of mathematics: the insight that there arenumbers which are irrational. It was t he Pythagoreans, who realized this first and - according tolegend - tried even to ”cover the discovery up” and kill Hippasus, one of the earlier discoverers.We have seen in the presentation the following two examples of irrationality proofs. We use thenotation log10for t he logarithm with base 10. For example log10(1000) = 3 because 103= 1000.Two prototype proofs√2 is not rational.Proof: assume the relation√2 = p/q holds. Squaring it gives 2 = p2/q2or2q2= p2.If we make a prime factorization, then on the left hand side contains an odd number of f actors 2,while the right hand side contains an even number of factors 2. This is not possible and so, theoriginal relation was impossible.Here is an other example:log10(3) is irrational.Proof. Exponentiating both sides of log10(3) = p/q gives 3 = 10p/q. Taking the q’th power onboth sides gives3q= 10p.There are no integers p, q doing this (except p = q = 0) since the left hand side is odd and theright hand side is even.Unsolved Math ProblemsWhile one knows from many numbers that they are irrational like π, e,eπ, e−π/2, sin(p/q), cos(p/q) ,log(1 + p/q) for nonzero p/q, or n√mfor non-square m and integer n, there are many unsolvedproblems. One does not know for example, whether π + e, π − e, eeare rational. A general state-ment which would settle these three is called Shanuel’s conjecture.Problemsa) Show that√6 is not rational.b) Assume you know x is an irrational number, can you conclude that√x is irratio nal?c) Is√2 +√3 irra tional?d) Is the golden ratio1+√52irrational?e) log10(7) is irrational.E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 2: rational approximationObjectiveWe rewiew Minkowski’s theorem and apply it to approximate real numbers α by rationa l numbers:Minkowski’s theorem reminderA convex, origin-symmetric region in the plane with area larger than 4 contains a lattice pointdifferent from the origin.A theorem on approximation by fractionsDirichlet’s theorem: Given any real number α < 1. For any n, there exists a fraction p/q with1 ≤ p, q ≤ n such that|α −pq| ≤1nq.To see, why this result is true, we consider the regionG = {−n ≤ x ≤ n, |y − αx| ≤ 1/n } .Hp,qLn-na) Is the region convex and symmetric with respect to reflection at the origin?b) How large is the area of the region?c) Is there a lattice point (p, q) different from zero in that region? If yes, estimate αp − q.d) How does the theorem f ollow?e) Can you deduce from the above theorem that there infinitely many fractions p/q such that|α −pq| ≤1q2E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 2: writing numbersObjectiveIn this worksheet, we look at some historical number systems and write concrete numbers in var-ious systems.1. The Roman systemThe letters I, V, X, L, C, D, M were of Etruscic origin. The subtractive writing principle like in9 = IX, 90 = XC were hardly used by the romans. They would write V IIII, LIIII instead.Problem: How would you write 129 in the roman system?2. The Greek systemThe Greeks used their a lpha bet with 24 letters with additional semitic letters (digamma or stigma,koppa and sampi) to represent numbers. A comma in front of a letter would mean it would bemultiplied by 1000. A dot would mean that the number in front would be multiplied by 10’000.For example,, θτ λη., δζιǫ = 93314715Problem: How would you write the number 6432 in this system?3. Babylonian cuneiformsThis notat ion was used by the Babylonians. The most important document is ”Plimpton 322”, aClay tablet from 1800 BC. Numbers are represented to the base 6 0. Example:Problem: How would Euphrat-Tigris dwellers have written the number 1000 in this system?4. The Egyptian systemThe Egyptians had a similar systemas the Romans but with fewer symbols.Problem: The following stone car vingwas found at Kar nak. What numberdoes it represent?4. Mayan SystemThe Mayans had a base 20 which is na t ur al too in some sense if you count with toes and fingers.Problem: How would the Mayan have written the number