1. Leonhard Euler’s Amazing 1735 Proof that
The Basel Problem
old Swiss banknote honoring Euler
Leonhard Euler’s Amazing 1735 Proof that
David Levine
Woodinville High School

2. Beautiful Mathematics
Would you want to play basketball if all you ever saw of it was drills, and never the fun of an actual game?
Today you’ll get to watch one of history’s greatest mathematical artists, Euler (“oiler”), at play
We’ll start with one of math’s snazziest bits of finesse – the Riemann zeta function

3. The Riemann Zeta (ζ) Function
The Greek letter zeta
sum
This simple function is very important in the mathematical fields of analysis and number theory
Bernhard Riemann
( )
One of the most important unsolved problems in mathematics is the Riemann Hypothesis, which states that all the complex roots of the zeta function have a real component equal to ½
Solving the Riemann Hypothesis would lead to a fundamentally greater understanding of how prime numbers are distributed among the integers
Solving the Riemann Hypothesis would probably lead to a fundamentally greater understanding of how prime numbers are distributed

4. The Basel Problem In 1650, Pietro Mengoli asked for the value of
This was the famous Basel Problem
By 1665, ζ(2) was known to be about 1.645
In 1735, the Swiss mathematician Leonhard Euler calculated ζ(2) to 20 decimal places (without a calculator!) and proved, as we will also, that
ζ(2) ≈

5. Leonhard Euler ( )
Did important work in: number theory, artillery, northern lights, sound, the tides, navigation, ship-building, astronomy, hydrodynamics, magnetism, light, telescope design, canal construction, and lotteries
One of the most important mathematicians of all time
It’s said that he had such concentration that he would write his research papers with a child on each knee while the rest of his thirteen children raised uninhibited pandemonium all around him
Discovered that This identity combines the five most basic constants in math in the simplest possible way!
Euler introduced the concept of a function and function notation.

6. Prime Numbers and Zeta
Euler also proved a profound formula that equates a sum of powers of all the natural numbers with a product of powers of all the prime numbers
sum
product
The proof of the formula on this slide is less technical but probably more complex than the proof that zeta(2) = pi^2/6
This formula’s proof isn’t hard to understand, but let’s turn our focus to the main atttraction!

7. Euler’s Really Cool Proof
How did Euler prove that ?
The next eight slides wind through several areas of mathematics to reach Euler’s amazing conclusion
Watch Euler’s brilliance and the proof’s beauty
Euler’s proof begins with an infinite polynomial called a Taylor series, which you’ll see in calculus
First, you need to know what the factorial function is
You’ll need to see the proof several times and reflect on it to understand all the steps

8. The Factorial Function
The factorial function n! is the product of the numbers 1 through n or
For example, n n! en 1
2.7 2
7.4 3 6
20.1 4 24
54.6 5
120
148.4
720
403.4 7
5040
1096.6 8
40320
2981.0 9
362880
8103.1 10 11 12 13
n! grows very quickly as n increases, faster than most other functions
Compare x! to ex
e^x is graphed as a straight line because it’s an exponential function and the y axis is a logarithmic scale
log scale

9. Taylor Series
In 1715, Brook Taylor found a general way to write any smooth function as an infinite degree polynomial
For example, the Taylor series for ex is
Sir Brook Taylor
( )
Each term brings the approximation just a little bit closer to e^x because the terms decrease as the degree increases.
The exponential function is in blue, and the sum of the first n + 1 terms of its Taylor series at 0 is in red. As n increases, the Taylor series gets more accurate.

10. Taylor Series for sin x The sin function’s Taylor series is
largest degree of each
approximation to sin x
The sin function’s Taylor series is 11 7 3
sin x
As the degree of the Taylor polynomial rises, its graph approaches sin x. This image shows sin x (in black) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. 1 13 9 5

11. The Fundamental Theorem of Algebra
Any polynomial of degree n can be written as a product of exactly n (possibly complex) factors
Example:
This degree 4 polynomial has 4 real roots at x = –2, x = –1, x = 2, and x = 3
roots

12. The Roots of sin x The Taylor series for sin x is a polynomial
The Fundamental Theorem of Algebra says that therefore sin x can be written as a product of its roots
The roots of sin x are at x = 0, ±π , ±2π , ±3π, … so we write
Question: Why is there a factor of x after the A? Answer: it corresponds to the root at x = 0
A is some real number
this difference of squares has factors of (x + 3π) and (x – 3π) and roots at ±3π

13. An Exact Expression for sin x
Our expression for sin x has an unknown factor A
Multiply each factor in parentheses by , where n goes up by one each factor
graph of x and sin x
The factors still have the same roots (zeros), but now B is a different real number. What is B?
Where did the x go in the second equation? It was divided by x in the limit of sin(x)/x.
Because the limit of sin x as x approaches 0 is 1, for very small angles of x, sinx = x. This “trick” pops up unexpectedly in more advanced math problems – don’t forget it!
In first year calculus we prove that
, so
the limit as x approaches very closely to 0 without reaching it

14. Multiply all the Factors
(FOIL)
multiply
In the final expression, there are no factors, only terms that are added and subtracted. There is one degree 3 term for each natural number 1, 2, 3,, 4, … There is one degree 5 term for every pair of natural numbers (see the degree five terms in the slide). There is one degree seven term for every triplet of natural numbers, etc. Terms with degree 1, 5, 9, 13, … are added and terms with degrees 3, 7, 11, 15, … are added.
multiply remaining factors
each cubic term comes from one x term and one x2 term, with the rest 1’s

15. …the result has appeared
Euler’s Genius
By multiplying all of its factors, we wrote sin x as
But the Taylor series for sin x is
Euler equated the x3 terms from both expressions
Summing up: we wrote the sin function as the product of its roots and found the cubic terms of the product. We equated those cubic terms with the cubic terms in the Taylor series for sin x. Multiply by –pi^2/x^3 and we have our proof!
multiply by
…the result has appeared
as if from nowhere
-Julian Havil
Voila!

16. Too Good to be True?
Did you think that some parts of this proof were fuzzy?
Euler lived before mathematicians could rigorously complete this proof using modern techniques of real analysis
Does the Fundamental Theorem of Algebra really work for infinite degree polynomials?
Is it really OK to equate the infinite series of cubic terms?
Euler wasn’t wrong, but his proof wasn’t complete

17. Interesting Tidbits
The probability that any two random positive integers have no common factors (are coprime) is also
Euler also proved that and that
Euler found a general formula for ζ(n) for every even value of n
Three hundred years later, nobody has found a formula for ζ(n) for any odd value of n

18. Slide Notes
This presentation was inspired by and based in large part on the book Gamma by Julian Havil, Princeton University Press, 2003
Unless listed below, the photographs are in the public domain because their copyrights have expired or because they are in the Wikipedia commons.
The two Taylor series graphs are in the Wikipedia commons. I annotated the sine graph. I made the other graphs.
Basketball drill photo from downloaded 10/22/10
LeBron James photo from downloaded 10/22/10
Waterfall image from Falls, Cascade Range, Oregon.jpg downloaded 10/24/10 and was reflected horizontally and lightened