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Pi day
Message #1 Posted by Don Shepherd on 11 Mar 2007, 1:10 p.m.

OK, Pi day is next Wednesday. Before my students ask me the obvious question, what is my answer?

The question is: Mr. Shepherd, you said that Pi is the ratio of the circumference of a circle to its diameter, but you also said Pi is an irrational number, which means it cannot be expressed as a ratio of one number to another. What gives?

      
Re: Pi day
Message #2 Posted by Gerson W. Barbosa on 11 Mar 2007, 1:34 p.m.,
in response to message #1 by Don Shepherd

Quote:
Pi is an irrational number, which means it cannot be expressed as a ratio of one number to another.

Perhaps rephrasing this as "...a ratio between two integer numbers" would be better. A textbook might give a more rigorous definition for rational and irrational numbers.

Regards,

Gerson.

P.S.:

By the way, in this part of the world, Pi day falls on my birthday, 22/07 (Jul 22) :-)

Edited: 11 Mar 2007, 1:35 p.m.

      
Re: Pi day
Message #3 Posted by Karl Schneider on 11 Mar 2007, 1:37 p.m.,
in response to message #1 by Don Shepherd

Hi, Don --

Quote:
OK, Pi day is next Wednesday. Before my students ask me the obvious question, what is my answer?

The question is: Mr. Shepherd, you said that Pi is the ratio of the circumference of a circle to its diameter, but you also said Pi is an irrational number, which means it cannot be expressed as a ratio of one number to another. What gives?


It's straightforward: An irrational number cannot be expressed as a ratio of two integers.

Even a ratio of two floating-point values, each having a finite number of decimal digits, can be expressed as a ratio of integers, one of which may have trailing zeroes.

-- KS

Edited: 11 Mar 2007, 1:39 p.m.

      
Re: Pi day
Message #4 Posted by Dave Shaffer on 11 Mar 2007, 4:05 p.m.,
in response to message #1 by Don Shepherd

Of course, the "ratio of integers" is a good response, but I think you could also add that the radius and diameter are not "numbers" (let alone integers) but concepts.

While we humans insist on trying to give a value to everything, and one can measure the value of the diameter and radius, the values, at least in this case, are not exact, but only the best approximation that can be made with the measuring equipment at hand. (This is perhaps a good time to introduce the concept of significant figures and measurement/experimental errors. Even after a semester of beating on them, my physics students often screwed up their answers with regard to "sig. fig." (the notation from my red pen). Since our favorite calculators give results with 10 or 12 digits, they find it hard not to write them all down!)

      
Re: Pi day
Message #5 Posted by Valentin Albillo on 11 Mar 2007, 8:06 p.m.,
in response to message #1 by Don Shepherd

Hi, Don:

    You might consider offering the following very easy Pi-related teaser to your students:

      "Without using a calculator or actually computing the values, can you say which is greater, e^Pi or Pi^e ?"

    A very simple argument, which involves only trivial arithmetic with integers and a little geometrical visualization, is sufficient to decide the question. They might enjoy the reasoning and also get the idea that calculators aren't everything there is and some logical thinking can get you very far.

Best regards from V.

            
Re: Pi day
Message #6 Posted by Don Shepherd on 11 Mar 2007, 8:43 p.m.,
in response to message #5 by Valentin Albillo

Hello Valentin!

Gosh, as much as I would like to use "Pi^e" on Wednesday, my sixth-graders won't know what e is. In fact, they may not learn that until high school (9th grade here). Most of them have heard of Pi, and many will know that it is (approximately) 3.14, but I know that few of them know its relationship to a circle's diameter and circumference. So I'm going to bring in a bunch of round objects and let them measure diameter and circumference and see how close they get to 3.14.

My goal is for them to learn that diameter and circumference are related, no matter how big the circle, and that relationship is pi.

Interestingly, some web sites (www.brainpop.com in particular) claim that 22/7 is not a valid approximation for pi (presumably because, being irrational, pi cannot be expressed as the ratio of two integers), but 3.14 is. If a sixth-grader can come to understand that pi relates circumference to diameter, I don't think it is important whether he/she uses 22/7 or 3.14 or 3.14159.

Thanks V.

                  
Re: Pi day
Message #7 Posted by htom on 12 Mar 2007, 4:43 p.m.,
in response to message #6 by Don Shepherd

You could try Count Buffon's Method, too. That's the one where you drop toothpicks (or pennies or ...) onto spaced lines and count intersections.

http://www.worsleyschool.net/science/files/buffon/buffon.html

                        
Re: Pi day
Message #8 Posted by Crawl on 15 Mar 2007, 3:59 a.m.,
in response to message #7 by htom

No, pennies wouldn't work. The reason pi comes up in that calculation is because the possibility of having an angle with the toothpick. But a penny is round, and its angle of orientation has no effect on whether it'd hit a line. It's pretty obvious that if the line spacing was L, and the penny's diameter was D, then the probability of hitting a line would be D/L. No pi term.

                              
Re: Pi day
Message #9 Posted by htom on 15 Mar 2007, 12:10 p.m.,
in response to message #8 by Crawl

Huh. Right. Why did I think that pennies would work?

If you drop pennies on a regular square grid with the square sides being diameters (d) of the penny? Imagine circles centered at the intersections of the grid; the center of the penny falls either inside a quarter circle covering a part of the square or towards the center of the square, between the four circular arcs. If we take the unit to be the radius r rather than the diameter, the area of the square is 4r^2, the circle is pi*r^2, the excluded area is 4r^2-pi*r^2.

There has to be a way to recover pi from that, but I'm stuck. Seventh grade was a long time ago.

--- later ---

If you randomly drop 400 coins, 314 of them should cover an intersection point, 85 should not, and the other should cover sometimes.

Edited: 16 Mar 2007, 12:28 a.m.

      
Re: Pi day
Message #10 Posted by Chuck on 12 Mar 2007, 6:37 p.m.,
in response to message #1 by Don Shepherd

Pi-day is much celebrated, as is e-day (Feb 7th at 6:28 PM). There are numerous math days, but my favorite is i-day. You missed it!! It was February 29th. It's one of the quarternion days since it's real only once every four years! :)

Cheers.

CHUCK

            
Re: Pi day
Message #11 Posted by J. Mosand on 13 Mar 2007, 7:38 a.m.,
in response to message #10 by Chuck

One of my favorite numbers is '0', which also has its 'day', namely Feb. 29 in a non-leap-year :-)

(Right now I'm reading Charles Seife: "Zero".)

I have 'e', 'pi' and 'i' on a menu on my HP48SX, and by golly there is a '0' on one of the keys :-))

John

                  
Re: Pi day
Message #12 Posted by Les Wright on 15 Mar 2007, 4:22 a.m.,
in response to message #11 by J. Mosand

i am rather found of Euler's gamma, myself....

      
Re: Pi day
Message #13 Posted by Bruce Bergman on 13 Mar 2007, 10:59 a.m.,
in response to message #1 by Don Shepherd

Interesting side note: here's a guy who has memorized pi to over 12,000 digits. THAT's dedication! :-)

Link to San Diego UnionTribune article

thanks, bruce

            
Re: Pi day
Message #14 Posted by Ron on 13 Mar 2007, 2:39 p.m.,
in response to message #13 by Bruce Bergman

Daniel Tammet got it to 22,514 digits! See this, or just google him.

      
Re: Pi day
Message #15 Posted by Antonio Maschio (Italy) on 13 Mar 2007, 1:29 p.m.,
in response to message #1 by Don Shepherd

Well, for tomorrow:

3.14 PI day la US way

the PI Fraction Day is

22.7 (22/7) (or July 22) la EU way.

            
Re: Pi day
Message #16 Posted by Eduardo on 13 Mar 2007, 3:21 p.m.,
in response to message #15 by Antonio Maschio (Italy)

Elaborating on the 22/7 approximation to Pi. This is the partial fraction

[3;7] = 3+1/7

Anybody here know anyone born 16 July '03? (Either 1903 or 2003)

pi = [3;7,16,...]

The next term in the fraction expansion of pi is 16, giving pi=3+1/(7+1/(16+...)). Truncating, one gets the fraction 355/113, which is within 3x10^(-7) of the correct value.

Eduardo

Edited: 14 Mar 2007, 12:37 a.m.


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