Math E-320: Spring 2013

Teaching Math with a Historical Perspective

Mathematics E-320:

Instructor: Oliver Knill

Office: SciCtr 432

Email: knill@math.harvard.edu

- Chaos in the three body problem can be seen in this page written more than 13 years ago.
- Janice mentioned a youtube video lyrics.

Li Zhou (a visitor of the math in movie page) wrote about the movie Silk, in which the Menger sponge appears.

This construction might need a bit time to be understood because it is ergodic theoretic, but it is true that any real number defines so a probability space and a sequence of random variables. It is known from ergodic theory that almost all real numbers in the interval [0,1] have the property that they generate independent random variables like that. Of course, not all numbers do. If we take the number 5/7 for example then the hull consists of a probability space with 7 elements on which the probability measure P is the normalized counting measure |A|/7. Now the random variables X

- One slide joked that we had the "New Math" revolution in 1965, the "Math wars" in 1989 so that if we arithmetically continue, we should have a Math revolution every 34 years and so one in 2013. Cale mentioned that this is confirmed in that we move in this time away from individual state standards and have one set of national standards in math.
- The Venn Diagram at the Gonville and Caius College.

- Jeff knew a limerick:
"Integral zee-squared dee-zee from one to the cube root of three times the cosine
of three pie over nine equals log of the cube root of ee".
which is from Betsy Devine and Joel E. Cohen's book "Absolute Zero Gravity" page 37.
Integrate[z^2,{z,1,3^(1/3)}] Cos[3Pi/9]==Log[E^(1/3)]

- Here is the link to the Pecha-Kucha performance.

"An infinite number of mathematicians walk into a bar. The first one orders one beer. The second one orders half of a beer. The third, a quarter of a beer. The bartender looks at them all and says 'You really need to know your limits,' and pours two beers."

- As in previous years, we have proven Wilson's theorem and the little Theorem of Fermat. Both are great theorems also to practice algebra. Wilsons theorem shows that factorials can be fun, Fermats theorem illustrates the principle of induction as well as a property of Binomial numbers.
- We mentioned the 48th Mersenne prime found by the GIMPS project. The largest known prime has more than 17 Million digits.
- We watched part of Calculus of Love, where the Goldbach conjecture appears. The entire movie is quite short and can be found on the above website.
- By the way, here are my own encounters with Goldbach.
- Jeff mentioned the number 867-5309
which is a prime twin, and 617-867-5309 is also prime but not a prime twin.
The Wikipedia page on 867-5309 mentions that the number is the fourth most common
7 digit password chosen. In class, we have built the Mathematica procedure
PrimeTwinQ[x_] := And[PrimeQ[x], PrimeQ[x + 2]];

And here is the verificationPrimeTwinQ[8675309]

which gives the answer True.

Pi^2(7^(E/1-1/E)-9) = 867.5309...

T[x_] := If[x == 1,1,DivisorSigma[1,x]-x];

Here is the orbit of a number like 123456
ListPlot[NestList[T, 123456, 100],Joined->True] |

- Update June 5, 2013: A
A new Scientist article on the quest to close the prime pair gap: from the article:
*The work relates to a longstanding problem called the twin prime conjecture. A prime number can only be divided by 1 and itself, and twin primes are those just two numbers apart, like 3 and 5, or 29 and 31. The conjecture, put forward in 1849, says there are an infinite number of these pairs, but no one has managed to prove or disprove it. Last month Yitang Zhang of the University of New Hampshire in Durham took an important step towards this goal by showing there are infinite number of primes that are separated by 70 million or less. It was the first time someone had put an upper limit on the gap between pairs of primes. Since then, mathematicians have been competing online to shrink the limit. Zhang proved that a set of 3.5 million numbers with a certain mathematical property produces an infinite number of similar sets and that these contain at least two primes. He then showed for a particular series of sets that these primes are at most 70 million apart. But with a fairly quick calculation it is possible to choose other sets with a smaller gap.*

An other theorem which does not appear in textbooks and which I had learned in high school is on the Worksheet. In some way, it provides an other possibility to compute the product of two numbers.

We were debating whether Hypocrates can be proven without Pythagoras. Here is an entirely visual proof in a picture book edited by Jackson:

Having had no clay handy, we wrote onto Chewing gum. And it seemed have
worked pretty well. We discussed the dilemma how the Babylonian mathematicians would distinguish between numbers like 60 ^{3} and 60^{2} because there was
no zero available. Mirjanda mentioned place holders which in some sense also
introduce zero. If this is counted as zero, then also the Babylonians have dealt
with zero.
Here is an article
with a picture of
places, where zero was invented. And
this picture shows the
placeholder for zero. Here is an article about the history of "0". Also this Scientific american article only talks about the zero found in India. I believe the realization that the Mayans had introduced "0" independently and earlier came only later. Added March 11, 2013: An interesting article in Slate about an arithmetic question. |

- As we noticed when working on the worksheet, the parabola was a bit too small (we used two parabolas on
one sheet of papers). Here is a PDF of the parabola alone, if you want to
print it out and experiment on your own.
- Here is the video of George Hart about the opening of the MoMath Museum. This is where I learned about the parabola.
- We have not yet found out, who was the inventor of the "parabola computation". I have not seen anywhere
yet the fact that one has really implemented the full real multiplication group (R,*) and not cleared yet
the question whether there is a relations with elliptic curve multiplication on y
^{2}= a x^{3}- x which for a=0 degenerates to a parabola. Naively, there is no connection because in the limit a to zero, there is no intersection any more with the curve. But also the parabola multiplication does not deal with the parabola alone but with the**"fork curve"**y^{3}= - x y, as you can see here in Wolfram alpha. There could be deeper connections although. - As mentioned in the worksheet, this construction works also in the complex. Actually, one can use any field of numbers but since one has already given the multiplication, there is nothing new. The multiplication on elliptic curves on the other hand provides examples of new groups, which has applications in cryptology (for example to factor large integers).
- The geometric implementation of the multiplication group shows that commutativity of the multiplication is reflected by a symmetry of the parabola. We will talk about arithmetic in the next class and interpret the commutativity as a Euclidean symmetry, when seeing the product as an area. It is a fascinating question why we humans use an Abelian multiplicative structure at all. Nature tells us that fundamentally on the microscopic level, noncommutative multiplications and geometries are more natural and more powerful. Commutative structures are only an idealization, a shadow on the wall of Plato's cave so to speak.
- Here is a vine of the algebra part in the proof which we recorded in class (Chad did the computation).
- Janice knew the property of the parabola and even had a proof using square roots. The use of square roots is a bit more tricky because one has always to worry about the signs. But Janice got through.
- The different orientations of the parabola, sometimes opened upwards, or to the right
or to the left was a bit confusing at first. For teaching purposes, it is better to stick to
one orientation. The one which opens to the left y
^{2}= -x is the most natural one because the result will be on the real axes with the same sign. For the geometric proof it is better to stick to the inside of the parabola at first. The MoMath Museum chose a visualization with a paraboloid. - Clayton realized early that one can use the parabola also to do division.
- A proof suggestion by Greg is a brute force computation. Here it goes: the line
r(t) = (-x

connects the points (-x^{2},x)+t(x^{2}-y^{2},y-x)^{2},x),(-y^{2},y). It intersects the y-axes when x+t(y-x)=0 which means t=x/(x-y). Now getr(x/(x-y)) = (-x

This multivariable calculus proof uses a parametrized curve. It is probably the most straightforward and clearest computation because it works for all x,y if they are different and all computations are rational expressions in x,y. Interesting is still the case x=y, where we have a tangent. In that case the above computation does not work any more and I don't see any other way than to refer to^{2},x) + x/(x-y) (x^{2}-y^{2},y-x)=(xy,0)**calculus**for the computation of the tangent or to factor out (x-y) and**heal**the function to see that the limit exists and things work also in this case. - Jeff saw a connection with cross ratios. Indeed, the proof we followed used that. We used a geometric proof followed by some algebra. We had some short discussion whether there are relations with Pappus or Desargues theorems. There could be but we did not see any.
- A blog entry gives a reference for the parabola property to the book "Mathematics: A Human Endeavor" by Harold Jacobs. (Feb 2: I got the book from the bookstoke. Even after browsing through it twice, the parabola multiplication can not be found in the 1994 edition. Moreover, as often in high school textbook, there are also no references).
- Here is the front page as well as the page on the parabola from the book "Mathematics, An illustrated History of Numbers", edited by Tom Jackson.

Please send questions and comments to knill@math.harvard.edu