Tuesday, March 16, 2010

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):

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

Using this program, complete the following table:

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


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





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


Eric Jablow said...

This is a marvelous teaching opportunity. First, you have Euler's infinite product for the zeta function. Then, take that product, allow the exponent s to be 1, and show that the divergence of the harmonic series implies the infinitude of the primes. Then, you can discuss Euler's initial 'proof' of the Basel problem, that ζ(2) = π²/6:

Consider the function sin x / x. By substitution, its MacLaurin series is:

sin x / x = 1 - x²/6 + x⁴ / 120 ...

Its roots are nπ where n is any nonzero integer. Euler reasoned that it could be factorized into

sin x / x = (1 - x/π) (1 + x/π) 1 - x/2π) (1 + x/2π) ...
= (1 - x²/π²) (1 - x²/4π²) (1 - x²/9π²) ...

He assumed this just as 1 + x/2 - x²/2 has roots -1 and 2 and equals (1 - x/(-1)) (1 - x/2).

But then, equate the coefficients of x² to see

1/6 = 1/π² + 1/4π² + 1/9π² + ...

Of course, math was a little wild and wooly in the 18th Century. This is certainly not a rigorous proof.

You can continue from there to discuss the Bernoulli numbers and the formula for ζ(2k). Finally, because the Riemann Hypothesis is not suitable for your students, you can end with Apery's theorem on the irrationality of ζ(3).

Dave Marain said...

Thank you, Eric, for "opening the door" to some wondrous marvels. Your explanation of Euler's "proof" is a clear demonstration of the manipulative brilliance of 18th century mathematicians, Euler in particular. It was all about "form" for them, issues of convergence notwithstanding!

My intent was to show how a middle schooler could begin to make sense of how relative frequencies can approach a theoretical probability.

Using a statistical approach with younger children always seemed to make sense to me.

Dave Marain said...

By the way, I should also add the obvious:
What young thinker would not be tantalized by the omnipresence of pi in so many apparently non-geometric contexts. The first time I saw pi appear in the sum of squares of the reciprocals or in the probability of relatively prime pairs, I was hooked...

Eric Jablow said...

Try the Buffon's Needele experiment then, Dave.

Lindsay H. said...

I was just teaching my students about pi and we celebrated pi day! This is a great teaching/investigation tool. I am stuck for what to do with four days before spring break (I don't want to start a new chapter), I think I will try this! Thanks!

Dave Marain said...

Now you've given me another video project to work on!

Lindsay --
Thanks for the nice words! Actually, the videos are fairly crude. I am still talking too much which turns people off. The 2-3 careless errors I make are frustrating but I only have so much time to re-do these. However, the math content itself as you noted is compelling and that transcends everything.

I am working on the probability of two randomly chosen positive integers being relatively prime -- the result is 6/(pi^2), the sum of the reciprocals of the squares problem. Let me know what you think of the latest videos I put up, particularly Part II.

Again, thanks, and let me know what your students think about Euler and pi!!