[Video] - Why Pi?

Why Pi? is the title of Don Knuth’s 16th Annual Christmas Tree Lecture. Don Knuth is a famous computer scientist and the designer and programmer of TeX. The first book I seriously self-studied was Concrete Mathematics by Knuth, Ron Graham and Patashnik with beautiful stuff on the Fibonacci series, the Pascal triangle, combinatorial identities, generating functions and numbers and divisibility. It's a beautiful book. I haven't watched the ( entire ) lecture yet, i parked it on my watch-queue.

[Exercise] - 3

Calculate, or otherwise determine $$\sum_{k=1}^{500}k! \mod{189}$$

Reminder 2011 course registration

Ref: 47911183 Reminder - changes to registration dates for undergraduate courses starting in February 2011 - 22 Dec 2010.

I still haven't decided yet... I really don't know at this moment. Tired, tomorrow...

[Exercise] - 2a: The coconut congruence equation

The coconut problem is what is called an algebra word problem. In a course in elementary number theory we could skip the words and ask to solve the following equation:

$$\frac{4}{5}(\frac{4}{5} (\frac{4}{5} (\frac{4}{5} (x-1) - 1) -1) -1) \equiv 1 \mod{5}$$

[Sign of the times] - Wikileaks: Yes, USA: No

Often heard: "What can I do about it, anyway?"

Well, you can close your MasterCard account for example. Or stop buying books from Amazon. Don't buy American if you have the option. You can also support Wikileaks directly of course. I think they need it and deserve it. - They belong to the good guys on this planet. Wikileaks is more than just Assange, although he is important.

Just my 0,02c.

[Exercise] - 1a : Hint

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

Hint: MST121 / MS221 math suffices to solve this one, we need to find a function $A(n)$ such that $${A(n)}^2 = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$
and
$$f(n) = A(n) \in \mathbb{N}$$
( To be continued. )

[Exercise] - 2

This is a famous problem. I found it on the Internet by searching for "mathematics, monkey, coconut, problem". - My version is in a Dutch book called Algebra by M. Riemersma.

Five men and a monkey were shipwrecked on a desert island, and they spent the first day gathering coconuts for food. Piled them all up together and then went to sleep for the night. But when they were all asleep one man woke up, and he thought there might be a row about dividing the coconuts in the morning, so he decided to take his share. So he divided the coconuts into five piles. He had one coconut left over, and he gave that to the monkey, and he hid his pile and put the rest all back together.

By and by the next man woke up and did the same thing. And he had one left over, and he gave it to the monkey. And all five of the men did the same thing, one after the other; each one taking a fifth of the coconuts in the pile when he woke up, and each one having one left over for the monkey. And in the morning they divided what coconuts were left, and they came out in five equal shares. Of course each one must have known there were coconuts missing ; but each one was guilty as the others, so they did not say anything.

How many coconuts were there in the beginning?”

To be continued ( i.e. answer and comment )

[Exercise] - 1

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

For $n=1$ to $5$ we have ($n, \ \sqrt{f(n)}, \ f(n)$):
$\begin{array}{lll}
1. & 3. & 9. \\
2. & 11. & 121. \\
3. & 41. & 1681. \\
4. & 153. & 23409. \\
5. & 571. & 326041.
\end{array}$

[News] - Math and industry

The European Science Foundation published the ESF Forward Outlook "Mathematics and Industry". Download a PDF copy from this page.

[Sign of the times] - New York /12-2010

From an article in the New York Times: LAPTOPISTAN

credit: Piotr Redlinski for The New York Times

 

( A new series of posts [Sign of the times] with mainly pictures. Common theme is of course the theme of this blog: "mathematics and / or study". ) Let me know if you think this series doesn't belong here.

Comment:
Take your laptop with you and you can go out on your own. Have coffee, work, study and enjoy the company of people around you. In the Netherlands public libraries are ( or already have ) transformed themselves into embassies of Laptopistan with free power-points and WiFi.

[Sign of the times] - Abacus still in use

The Corporation is optimistic about the potential of the abacus system of mental arithmetic, which it introduced in two of its primary schools on Friday. The system is aimed at improving the comfort of young students with numbers and the mathematical functions.

Source: express buzz

Link: Abacus ( see the comment )

Link: An introduction to the abacus

Link: Chinese Abacus + Manual

Video Lectures on Number Theory

Among lectures on Calculus I,II and III, ( Introduction to ) Linear Algebra and ( Introduction to ) Differential Equations from the UCCS ( University of Colorado and Colorado Springs ) Department of Mathematics you will find video lectures on Math 311 Number Theory by Professor Dr. Seung Son here. I have watched most of them earlier this year. This week I watched some of them again.

While watching a video on mathematical induction something amazing happened, not sure if I would call it a cognition, but it's close. Since I was able to do proofs by mathematical induction and thus understood it, I thought I was done studying mathematical induction. ( I mean both the MS221 and M208 exams included questions on induction). Well, I close-to-cognited that I didn't understand proofs by mathematical induction -at all-.

Do you? If so:
- state the first principle ( of mathematical induction ) using symbols only,
- state the second principle using symbols only,
- re-formulate: "Show that: ... $$\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$$ ..." using Set Terminology,
- can you explain the difference between the first and second principle?
- give an example of a proof using the first principle,
- give an example of a statement which can only be proved with the second principle.

I failed ( note: past tense ) all answers to the questions above. Post is To Be Continued ...