Mr. Finch and pi

March 14, Pi Day, is coming up. Is there any more to be said about π that hasn’t been said?



In the January 3, 2013 episode of the TV show “Person of Interest” (“you are being watched ... ”), computer genius Mr. Finch says that since π is an infinite non-repeating decimal, “contained within this string of decimals is every single other number. Your birth date, combination to your locker, your social security number ...”

Really? I didn’t think that is necessarily so. Mr. Finch doesn’t explain why.

There is a site that lets you search for a specific string of digits within the first 200 million digits of π, http://www.angio.net/pi/piquery, but of course just because your number doesn’t appear, doesn’t prove anything. 200 million is a long way from infinity.

Are the digits in $\pi$ truly random? There is something called a normal number, which is a real number whose infinite sequence of digits is distributed uniformly. See http://en.wikipedia.org/wiki/Normal_number which says it is believed that π is normal, but this has not been proven.

So we think Mr. Fitch was right, but we’re not 100% sure.

Counting with my fingers

We just did matrix multiplication, and I shocked my class by telling them that I think that is the perfect place to count with your fingers. I put my left pointer finger on the first element of the appropriate row, I put my right pointer finger on the first element of the appropriate column, do the multiplication, and move one element right and one element down to the next multiplication. Of course you need to mentally, or some other way, add these products, so that c_ij = ∑_k a_ik * b_kj. The other time I count with my fingers is when I do modular arithmetic, such as determining what is 7 months after October. I don't teach physics or three dimensional vectors, but I guess another application is the Right Hand Rule.

The class seemed shocked because it doesn't seem very sophisticated to count on your fingers. I can think of a few friends who would never lower themselves to do such a thing. That doesn't bother me. Let's just get the job done.

Does anyone have any other applications of using your fingers to count, or experiences in doing this?

Let's stop making excuses for Aunt Sally

Let's stop making excuses for dear Aunt Sally. You know who I mean: Please Excuse My Dear Aunt Sally. Many students can still remember the mnemonic PEMDAS for the rules of order of operations, but PEMDAS omits or confuses some situations. Most prominently, even my current college algebra textbook does not make it clear that in the M and D of PEMDAS multiplication and division are of equal priority, so a division is performed before a multiplication if the division appears to the left of the multiplication.

I like to invite my students to use all their calculators on expressions like 300 - 200/50*10 to see which calculators obey the rules of operations and which do not. I am always surprised when students first turn to the calculator on their cell phone to answer this.

PEMDAS doesn't address fraction bars, nor that with multiple grouping symbols you start with the innermost grouping symbols first. And I suppose it's too much to expect that PEMDAS address that in -3² the negation precedes the exponentiation; by the way, Excel disagrees with this.

Speaking of exponentiation, what do you do with x ^{y z} ? Exponentiation is not associative. Johnny Lott in A Problem Solving Approach to Mathematics says exponentiation is done in order from right to left (page 276), although Excel disagrees with this one too.

So there's more to the rules of order of operation than PEMDAS. But if you are getting a little tired of Aunt Sally, why not switch to Please Email My Dad A Shark, by the authors of xkcd.com

Anyone have any good ideas on teaching order of operations?

GFU2TT2BG3ZC

The color-challenged math student

If you are part of the 92% majority (that was not supposed to be a political comment), then you can see a number inside that circle.  But I can't.  The common description for my condition is being colorblind, although I prefer the more politically correct "color-challenged".  I am not truly colorblind.  I see some colors, just not as many as you see, and I get especially confused between red and green, shades of red and green, and colors containing either red or green (for example blue versus purple).

I will stop my car for the top (or left) traffic light, but I don't really think it is red. I would wear a red tie with a green-striped shirt if no one stops me, so I try not to own shirts with green.  Or to wear ties.  I'm never quite sure if the meat is cooked enough. You won't see me choosing colors of house paint or women's makeup shades, and I won't be disabling colored wires for the police bomb squad.  So generally I have learned to live with this minor inconvenience, and the accompanying jokes, which has not impeded my career choice.

But the world of math is not as black and white as it used to be.  A textbook author would never graph two curves in the same color.  But if y = .5e^x  is in red and y = x² is in green, I am going to be confused as to which curve is greater in 1 < x < 3.  I also get confused by multi-color pie charts where some of those colors start to blend together.  And if you really want to lose me, just show me one of those colored maps with ten or so different colors representing ten different levels of the amount of rainfall.

So if you are a teacher with control over such things, please be aware that some small percent of your students could be color-challenged and they may not even know it.  The best thing you can do is provide labels in addition to different colors.  Another possibility is to provide different kinds of patterns (see Excel's FORMAT, FILL, PATTERN STYLES).  Lastly, choose among colors we are more likely to recognize, and especially avoid red AND green together.

Have you had any difficulties in math with a color-challenged math student?

Beyond the scope of the course

Hi.  Occasionally a student will ask me a question, or will make a mistake, that leads to a topic that is beyond the scope of the course.  Recently this occurred in college algebra where students confused algebraic functions and algebraic numbers.  This entire subject is beyond the scope of this course - we don't prove a function is not algebraic, but we merely state it without proof;  and we really don't care for this course whether a number is algebraic or not.

My philosophy is that I always try to answer a student's question, or try to correct a mistake, even though  this gets us outside the scope of the course.  Students will not be tested on whether a particular function is algebraic or not.  But I always have this nagging feeling that I'm providing more information than anyone really cares about.

Thinking about this dates back to when one of my sons was in middle school and says he was taught that only rational numbers can be exponents.  I then dashed off a letter to my son's teacher, reminding him that while logarithms were not part of the middle school curriculum, did he really want to teach the kids something that they would have to un-learn in a future class?  I try not to teach something that a future teacher may have to correct;  my son's teacher felt an irrational exponent was too much information for that grade level.

Which of us is right?