6 Divided By Pi^2 and Relatively Prime Integers - A Video Derivation

Jaime didn't just teach math. Like all great teachers, he changed lives

– Edward James OlmosSOURCE: THE SAN JOSE MERCURY NEWS5 DAYS AGO

The current post presents a non-rigorous video derivation of a formula in mathematics which might create 'shock and awe' in kids of all ages. Oh, alright --  in people like me! This 3-part video is the followup to the post from March `16th -- Pi Day, More Videos on Counting, etc...



The following description comes from the YouTube Channel, MathNotationsVids:

Designed for anyone who has a passion for mathematics, this derivation of a classical result in math is suitable for advanced middle schoolers through undergraduate math. Further, teachers may want to show this to Math Clubs/Teams. This 3-part video builds on results from previous videos, is related to a post on MathNotations and is dedicated to the "Greatest Teacher in America" -- Prof. Jaime Escalante who passed away a few days ago.

Part I




Part II



Part III




Comments, Notes:

• I hope you will see in these videos a central theme beyond the content involved -- a fundamental heuristic in teaching mathematics: When introducing an abstract concept or in deriving a formula or theorem or rule, avoid heavy symbolism and work with simple concrete numerical cases before generalizing results. I believe this has validity at all levels of math instruction.
• This topic ties together so many apparently unrelated topics in mathematics in a wondrous and surprising way. Perhaps it will inspire a budding mathematician as it did me...
• Visit my YouTube channel (see above) and please comment on these if you feel you want to see more. They are fairly labor-intensive (do you really think I'm speaking extemporaneously!) but they are worth it if someone enjoys them. The quality of the videos can still improve much more but this is just a beginning...

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

Pi-Squared Over 6: The Algebraic Genius of Euler

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Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

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Yes, Pi Day 2010 is "history" unless of course you celebrate July 22nd! Then again, π is so universal in our world that "All π, All The Time" seems appropriate to me. That was a long-winded way of motivating a post about one of the most famous formulas in mathematics:

1/1² + 1/2² + 1/3² + 1/4² + ... + 1/n² + ... = π²/6

The videos below were inspired by one of my most faithful readers, Prof. Jablow. He brilliantly outlined Euler's derivation of the above formula in one of his comments.  I decided to develop it in more detail and provide some background for the younger student. Advanced middle schoolers through undergraduates in college may find this interesting. You might also want to share it with your math team/club.  The 2-part video presumes strong algebraic background and some knowledge of calculus although the latter is not necessary if you simply accept the well-known series expansion for sin(x). You may also find background and details in the excellent Wikipedia article, The Basel Problem.

As always, I add my disclaimer that I am solely responsible for any errors. I know there are a couple of errors in Part II, towards the end. They're pretty obvious and not serious, so I hope they won't ruin it for you! I invite you to comment on these videos both here and on my new YouTube channel, MathNotationsVids.  Of course, as I am finishing this post on 3-25-10 in the AM, YouTubew is down apparently worldwide, so I cannot embed these videos yet!!

Part I of Euler Video



Part II of Euler Video





FURTHER INVESTIGATION
Although the material on infinite series seems quite advanced, middle schoolers can use their graphing or scientific calculators to compute the sum of the first 10, 20, or even 50 terms of the series above. A simple program can also be written on the graphing calculator for summing the first n terms up to, say, n = 500 or 1000. Challenge them to see how "close" they can get to the decimal value of π²/6...
---------------------------------------------------------------------------------------------------------------------

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

PI Day, More Videos on Counting, "Odds and Evens"

Since pi day fell on a Sunday this year, we should still be celebrating it today. Besides, March should be declared pi-Month!


It is always fascinating to see how readership (or should I say one-time viewership) always picks up around March 14th every year! I feel obligated to add another pi Day activity or exploration in addition to those I've posted the past three years. By the way, the pi Day Scavenger Hunt is the most popular post by far and I'm not even the one who thought of that idea!


Despite the title of this post, I did not upload a video for this activity. However, there is another video on the MathNotationsVids Channel on YouTube.

Here is an investigation/exploration/activity for middle and secondary:

Part (A)
(i) List all ordered pairs of positive integers (m,n) such that
(1) 1 ≤ m ≤ 10 and 1 ≤ n ≤ 10
(2) m and n are divisible by the same prime p

For example, (m,n) could be (6,9) since 6 and 9 are each divisible by the prime 3.

(ii) Should (9,6) also be counted?

(iii) Another way of expressing Condition (2) is:
The _______________ of m and n is ________  one.
Answer: gcf; not equal to or greater than

(iv) If you listed and counted correctly, you should have found there are 37 ordered pairs which satisfy both conditions. If not, have a partner check your list. Each of you should be checking each other's lists routinely.

Part (B)
(i) Explain, using the multiplication principle, why there are 100 ordered pairs which satisfy Condition (1) above.

(ii) ) What % of all the possible ordered pairs from Condition (1) are relatively prime. If you have immediate access to the internet, research this term before asking your teacher what it means!

(iii) In probability terms, you could say:

If one of the 100 ordered pairs (m,n)  from Part (A) is selected at random, the probability that
m and n are relatively prime is ____%.

Part (C) (more advanced)

If you have access to a graphing calculator, such as the TI-84 or TI-Inspire, enter the following program into memory (call it RELPRIME):

:ClrHome
:Prompt N
:0 → K
:For (X,1,N)
:For (Y,1,N)
:If gcd(X,Y) ≠ 1
:K+1 → K
:End
:End
:Disp K
:Stop

Using this program, complete the following table:

N..........Total # ord. prs..........# of not rel prime prs........% rel prime prs

10.........100.............................37....................................63%

20.........400............................ 145.................................

30

40

50

100

Notes:
K represents the count of ordered pairs which are not relatively prime
N represents the greatest value for the integers
gcd is found by going to MATH, then NUM, then 9:gcd(
The program slows down considerably as N increases. For N = 10, it checks 100 ordered pairs which may take only 2-3 seconds. For N = 100, it checks 100^2 pairs, which could take up to 4-5 minutes. Be patient!!

Conclusion: So what does all of this have to do with π ?
Well, as N increases without bound in the program, the probability that a randomly chosen ordered pair of positive integers (with values up to an including N) will be relatively prime approaches 60.7% rounded.

From out of the blue, compute 6/π²...
Want to know why? Well, that requires some advanced machinery involving infinite products, infinite series, and the Riemann Zeta Function! Perhaps, I'll do an informal development in a video. I love this stuff...


----------------------------------------------------------------------------------
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

"Multiple" Pi Challenges for MS and HS

Consider the following table displaying the first four positive integer multiples of π rounded to 4 places:


So what's the challenge here? You will be a π-multiple investigator!

(1) Determine the EIGHT positive integer multiples of π, up to 1000π, which, when rounded to THREE places, produce an integer result. For example, the decimal 17.9996 when rounded to 3 places results in the integer 18.000, which we will regard as an 'integer'. Unfortunately, this decimal is not an integer multiple of π so have fun searching! Do you notice any pattern in your results? Describe it! Can you explain this pattern? [This requires more than a Yes/No response!]

(2) In fact, the "pattern" you may have found in (1) continues beyond 1000π. Show that you can go up to SIXTEEN multiples of π before the pattern breaks down. Why do you think the pattern eventually ends?

(3) Which of your results in (1) would produce an integer when rounded to FOUR places?

(4) Can you think of any practical application for finding multiples of π which are very close to an integer? Be creative! Responses may depend on how advanced your math background is.

Comments:

• Do your students know how to program their graphing calculator to do the search? OR in Java or C++ or Python or Perl?? This would certainly facilitate the search! I may display the code for the TI-83 or -84. However, students may also find a creative way to use the TABLE feature on their graphing calculators to save time without programming. Have fun!
• Please post feedback if you use this in your classes. I will not post answers yet in case students find this post in their 'searching'!
• For most students, the full significance of this innocent looking search problem will not be apparent. You might want to give them a hint. Perhaps we should call this post:
  
  IS π ALMOST RATIONAL?
  

The Best Pi Quiz on the Web?

I strongly encourage you or your students to try Eve Andersson's Pi Trivia Game.

I tried the quiz on Tue and got 21/25 right. I'll bet some of you get a Pi-Fect score the first time! Some of the questions are easy but many require technical or historical knowledge of pi. Many of the questions involve fascinating facts about π, so it is both educational and fun. The multiple guess format makes it less intimidating but the distractors are tough to crack! As soon as you submit your answers, you get immediate feedback.

What is really nice is that the quiz changes when you refresh the screen, since the questions are randomly generated from her large database. There will be some repetitions of course, but there are enough questions to keep it interesting and any duplication will test your recall!

Also visit her main site, Pi Land, to get more background and some excellent book references for the mathematics and history of Pi.

This is probably not intended for middle schoolers, although I'm guessing one could do some web searches to find most of the answers. Might be a fun activity for your students. If you have a bank of computers in the classroom, tell your group they can try the challenge in the last 10-15 minutes of class after they've completed their work. Highest 3 scores get a Pize - ugh...

Poe, E.: Near a Raven - "Poe"-tic Justice and a Tribute to Pi and Mike Keith

To kick off Pi Week, I cannot ignore the most famous Pi Poem I have ever read - Mike Keith's web site is worth visiting. He is a unique and talented individual who has written several excellent books. On his home page, he refers to his creations as "mostly original diversions in mathematics and word play." I agree!

Here are the first couple of verses of his poem (he refers to the style as constrained writing) he wrote over a dozen years ago. Again, go here to see the entire opus! I'm sure most of you are thoroughly familiar with this, but it's always worth seeing it again. If you're not familiar with the code beneath this, just count the number of letters in each word, starting with the title:
Poe:3
E:1
Near:4
a:1
Raven:5

Poe, E.
Near a Raven

Midnights so dreary, tired and weary.
Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap - the weirdest tap!
An ominous vibrating sound disturbing my chamber's antedoor.
"This", I whispered quietly, "I ignore".

Perfectly, the intellect remembers: the ghostly fires, a glittering ember.
Inflamed by lightning's outbursts, windows cast penumbras upon this floor.
Sorrowful, as one mistreated, unhappy thoughts I heeded:
That inimitable lesson in elegance - Lenore -
Is delighting, exciting...nevermore.

...

My 2nd favorite Pi-Poem is from Liz:
She has an excellent website -visit it!
Here is an excerpt of her ode to π:

Why, π! Stop, π! Weird anomalies do behave badly!
You, madly conjured, imperfect, strange, numerical,
Why do you maintain this facade?
In finite time you are barbaric!
You do wonders, mesmerize minds!
....

You may want to start your own Pi Poem contest this week, although this idea is coming very late. Somewhere out there is a hidden talent, perhaps another genius like Mr. Keith.
If you do this, please share some of your favorites with MathNotations....

The Genius of Archimedes: Parabolas, Tangents...

Pi day is over, but it seems fitting to continue exploring. Archimedes did more than develop an approximation procedure for pi! There are many excellent websites that explain the following in greater detail and discuss many more of Archimedes' theorems about parabolas and tangents. I attempted to draw a diagram using Draw in Word. It's crude but you'll get the idea. The object is to share this extraordinary piece of history of mathematics and have your students finish the proof that a light ray from the focus that strikes a parabolic surface is reflected in a ray that is parallel to the axis of the parabola. This is equally interesting in reverse: External light rays and other forms of electromagnetic radiation that are parallel to a parabola's axis are reflected to the focus, very useful for radar and other 'collection' devices.
Considering that Archimedes' proofs used only geometric properties makes his work even more astounding (now of course we can use coordinate geometry, calculus, etc.). This type of investigation is usually deferred to College Geometry courses, but I believe we can deliver it to motivated geometry, 2nd year algebra or precalculus students. If nothing else, it makes for a wonderful long-term project!

Ok, here goes...

In the diagram below, I've gone out of my way to make the reflecting ray NOT look parallel to the axis, even though we're trying to prove it is. This is to help students avoid assuming collinearity, when, in fact, that needs to be proved!

The two angles marked X are equal by a reflection principle (angle of incidence equals...). The two angles marked Y are equal because it can be proved that the tangent line at P is the perpendicular bisector of segment FP', where F is the focus and P' is the foot of the perpendicular from P to the directrix. I chose not to derive Archimedes' very subtle argument, but it is worth studying the proof. The proof starts by constructing the perpendicular bisector and showing that this line passes through P but no other point of the parabola, thus it is tangent. Alex Bogomolny's excellent and in-depth treatment (with java applets) of this topic (on cut-the-knot) is very worthwhile reading.

The student is being asked to prove that the reflecting ray is parallel to the axis. This is equivalent to showing that the line containing PP' and the reflecting ray are one and the same. The argument is straightforward, but students may want to continue learning more about the genius of Archimedes.

[Good luck copying this diagram (jpg). Some of you may find errors in my argument or an extremely simply argument for the parallelism, so pls share!!]

Wed 3-14 A Pi-Fect Day!

3.141592653589793238462643383279502884197169399375105820
97494459230781640...

Don't forget to celebrate Pi Day on 3-14-07 at 1:59 PM!! Considering how many out there are searching for information about Pi, this has become a huge event. Although we may not be quite ready for an official national holiday, many now refer to 3-14 as National Pi Day!

Here are three of the best Pi-Links I have found. You may want to visit them to learn more about one of mathematics most fascinating numbers - more than you may ever have wanted to know! The Wikipedia Pi article is also wonderful.

Exploratorium

A History of Pi

The Pi-Search Page

Enjoy the day but don't forget that in many other countries, Pi Day is celebrated on July 22nd in honor of the approximation 22/7! Personally, I've always been 'partial' to 355/113.

Pi Day Activity Part Two

It may not display well but I'm posting an image of the second part of the Pi activity I posted the other day. This kept the group engaged for over a period. It may not help them to understand the deep meaning of pi or its relation to a circle but they definitely got a sense of the decimal representation of pi and struggled with some of the birthdates, since a simple Google search will not suffice. Do they now appreciate pi more? How will I assess the learning from this? I hope you will see that this was more than just meaningless busywork to keep them quiet! See if you can find all of the mathematicians in less than 10 minutes!

An Online Pi Day Challenge

Even though pi day is about 2 weeks away, here's an online challenge I gave to all the students in my school back in 2005. Not very 'mathematical' but it engaged them. I posted the problem at 6 PM on March 14th and one student found a web site and submitted her solution in under 7 minutes! Get the feeling that this generation is able to learn a slightly different way!! For those offended by the fact that pi day is celebrated in other countries on July 22nd (22/7), I apologize! I am well aware that many teachers, math departments and schools have created wonderful pi day activities and there are abundant examples of these on the web. I would like others to share their favorites as well. Learning more about pi from MathWorld or Wikipedia is well worth our time, although it does get very technical. The nested radical problem from the other day (with the square roots of 2) relates to pi! Look it up!!


Background: The first 5 places (decimal digits) in pi, after the decimal point, are 14159. Find a web site that will allow you to search millions of places in pi.

The Challenge:
(1) Write down the seven decimal places in pi starting with the digit in the 5,191,306th place.

(2) These digits are a clue to the birthday of a famous mathematician (sorry, it's not Einstein!).
Give the full name of this mathematician and how he is connected to the number pi! Your submission should include the full birth date, the name and the connection.

EXTRA: Now challenge us - find another famous mathematician who is associated with pi and ask your own 'pi' challenge question!