Download Lecture 2: Irrational numbers

E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010
Problems
Lecture 2: Irrational numbers
a) Show that
√
6 is not rational.
Objective
We want to appreciate one of the great moments of mathematics: the insight that there are
numbers which are irrational. It was the Pythagoreans, who realized this first and - according to
legend - 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 the
notation log10 for the logarithm with base 10. For example log10 (1000) = 3 because 103 = 1000.
b) Assume you know x is an irrational number, can you conclude that
Two prototype proofs
√
Proof: assume the relation
2 is not rational.
√
2 = p/q holds. Squaring it gives 2 = p2 /q 2 or
c) Is
√
2+
√
3 irrational?
2q 2 = p2 .
If we make a prime factorization, then on the left hand side contains an odd number of factors 2,
while the right hand side contains an even number of factors 2. This is not possible and so, the
original relation was impossible.
Here is an other example:
log10 (3) is irrational.
d) Is the golden ratio
√
1+ 5
2
Proof. Exponentiating both sides of log10 (3) = p/q gives 3 = 10p/q . Taking the q’th power on
both sides gives
3q = 10p .
There are no integers p, q doing this (except p = q = 0) since the left hand side is odd and the
right hand side is even.
e) log10 (7) is irrational.
Unsolved Math Problems
While one knows from many numbers
that they are irrational like π, e,eπ , e−π/2 , sin(p/q), cos(p/q),
√
m
for non-square m and integer n, there are many unsolved
log(1 + p/q) for nonzero p/q, or n
problems. One does not know for example, whether π + e, π − e, ee are rational. A general statement which would settle these three is called Shanuel’s conjecture.
irrational?
√
x is irrational?
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010
a) Is the region convex and symmetric with respect to reflection at the origin?
Lecture 2: rational approximation
Objective
We rewiew Minkowski’s theorem and apply it to approximate real numbers α by rational numbers:
b) How large is the area of the region?
Minkowski’s theorem reminder
A convex, origin-symmetric region in the plane with area larger than 4 contains a lattice point
different from the origin.
c) Is there a lattice point (p, q) different from zero in that region? If yes, estimate αp − q.
A theorem on approximation by fractions
Dirichlet’s theorem: Given any real number α < 1. For any n, there exists a fraction p/q with
1 ≤ p, q ≤ n such that
p
1
.
|α − | ≤
q
nq
To see, why this result is true, we consider the region
G = {−n ≤ x ≤ n, |y − αx| ≤ 1/n } .
d) How does the theorem follow?
Hp,qL
-n
n
e) Can you deduce from the above theorem that there infinitely many fractions p/q such that
p
1
|α − | ≤ 2
q
q
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010
3. Babylonian cuneiforms
Lecture 2: writing numbers
This notation was used by the Babylonians. The most important document is ”Plimpton 322”, a
Clay tablet from 1800 BC. Numbers are represented to the base 60. Example:
Objective
In this worksheet, we look at some historical number systems and write concrete numbers in various systems.
1. The Roman system
Problem: How would Euphrat-Tigris dwellers have written the number 1000 in this system?
4. The Egyptian system
The letters I, V, X, L, C, D, M were of Etruscic origin. The subtractive writing principle like in
9 = IX, 90 = XC were hardly used by the romans. They would write V IIII, LIIII instead.
The Egyptians had a similar system
as the Romans but with fewer symbols.
Problem: How would you write 129 in the roman system?
Problem: The following stone carving
was found at Karnak. What number
does it represent?
2. The Greek system
The Greeks used their alphabet 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 be
multiplied by 1000. A dot would mean that the number in front would be multiplied by 10’000.
4. Mayan System
The Mayans had a base 20 which is natural too in some sense if you count with toes and fingers.
For example,
, θτ λη., δζιǫ = 93314715
Problem: How would you write the number 6432 in this system?
Problem: How would the Mayan have written the number 441?