Monthly Archives: September 2006

Trigonometry: Unit Circle or Right Triangles

After a conversation with a colleague yesterday, I started doing some serious thinking.  In the several semesters that I have taught Trigonometry, which I think totals about 5 or 6 times, I have been using our adopted textbook which takes the unit circle approach to introduce students to trigonometric functions.  In other words, the functions are defined in terms of the coordinates of points on the outside fo the unit circle where an angle subtends the circle.

Now, I have varied when the right triangle approach is introduced, but it has always been after the unit circle definition.  In the current version of the text, it isn’t until the following chapter that the old familiar SOHCAHTOA (sine is opposite over hypotenuse, etc.) is even brought up.  Every time I teach Trig., I feel uncomfortable waiting this long, especially since, when I am calculating the value of these functions in my head, I am picturing right triangles.  Perhaps it is simply because it is the way I taught, but I have always been better able to make the students grasp how to calculate the trig. functions this way. 

Of course, as a mathematician who has used trig functions in all sorts of higher level courses, such as Differential Equations or Analysis, I see the validity in conveying to students the functional nature of the trig functions which is something they see much better through the unit cricle approach.  Nevertheless, the students just don’t seem to grasp the functions as well, this way.  That has been my experience.

In response, I started doing a little literature searching this morning.  It is really the first time I seriously made an effort in looking into research in Mathematics Education.  So, if any of you readers out there know of a good place to look or can quickly find any articles on the methodology in teaching trigonemetry, I would appreciate your help.  For the rest of you, how were you taught and did you prefer a particular method.

Champernowne’s Constant

During Calculus a few days ago, I covered an interesting little number called Champernowne’s Constant. We were in the middle of introducing the concept of infinite sequences of numbers and their convergence. We stated the theorem that states than any monotone, bounded sequence must converge. After review mathematical induction and proving the convergence of a couple example, I gave them the following example.

Let us construct a sequence in the following way:
[tex]a_1 = 0.1[/tex]
[tex]a_2 = 0.12[/tex]
[tex]a_3 = 0.123[/tex]
[tex]a_4 = 0.1234[/tex]
[tex]a_5 = 0.12345[/tex]
[tex]a_6 = 0.123456[/tex]
[tex]a_7 = 0.1234567[/tex]
[tex]a_8 = 0.12345678[/tex]
[tex]a_9 = 0.123456789[/tex]
[tex]a_{10}=0.12345678910[/tex]
[tex]a_{11}=0.1234567891011[/tex]

Notice that each time we simply append to the the previous value the next integer to the end of the decimal expansion. Now notice that because each term is appended at the end of the expansion, then each term is necessarily larger than the last. This implies that this sequence is increasing (thus, monotonic). Also, this sequence is bounded. For example, it will never be larger than 0.2 nor will it be smaller than 0. So, by the previously mentioned theorem it is convergent. So, it converges to something and we call that something the Champernowne Constant.

Now, the interesting thing about this sequence is that it contains every possible finite seqeuence of numbers. That is, eventually, any number will appear somewhere in the Champernowne Constant. For example, 32084701283472 will appear somewhere, because of the nature of the constant. So if one were to take any book and convert it to a number using the code A=1, B=2, etc., that book appears somewhere in the Champernowne Constant. This book could be already written, e.g. Hamlet is in there, Harry Potter, too. The book might not have even been written yet. In other word’s, the Champernowne Constant contains the future hidden somewhere in its sequence.

Now don’t get too excited, the information is not really there since along with every book every written it contains every possible ordering of letters so information is not discernible from the rest of the gibberish that is there as well. But, it’s still an interesting concept. Nostradomus has nothing on Champernowne.

Bad math humor

One day, Jesus said to his disciples: “The Kingdom of Heaven is like 3x squared plus 8x minus 9.”
A man who had just joined the disciples looked very confused and asked Peter: “What, on Earth, does he mean by that?”
Peter replied: “Don’t worry – it’s just another one of his parabolas.”

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Theorem. Every positive integer is interesting.

Proof. Assume towards a contradiction that there is an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But being the smallest uninteresting positive integer is interesting by itself. Contradiction!

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Math problems? Call 1-800-[(10x)(13i)2]-[sin(xy)/2.362x].

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Q: What’s purple and commutes?
A: An abelian grape

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Q: What is normed, complete, and yellow?
A: A Bananach space…

That’s enough for now. I hope you laughed at least once.

HT: Volker Runde

i.e and e.g.

During a lecture last week, I made use of the latin abbreviations, “i.e.” and “e.g.”, and while I know the correct usage of these I wanted to give them the Latin phrases that they abbreviated. My brain turned off at the moment and I couldn’t recall. Well, here they are.

The Latin abbreviation i.e., which stands for id est, means that is, that is to say, or in other words. The letters e.g. stand for the Latin phrase exempli gratia, which means for example.

Favorite Theorem #6: Pigeonhole Principle

In doing do preparatory reading for classes next term, I stumbled across this little interestingly named gem of a theorem. It is one that is frequently used in combinatorial analysis. It will also come in handy in our introductory Analysis class.

The reason for its name is that it can be used to say that if [tex]m[/tex] pigeons are put into [tex]b[/tex] pigeon holes and if [tex]m>n[/tex] the at least two pigeons must share one of the pigeonholes. To you non-mathematicians that see that as utterly obvious, just know that in the field of mathematics even the obvious concepts still need to be verified through rigorous proof.

So here it is, the pigeonhole principle:

Thm: Let [tex]m,n in mathbb{N}[/tex] with [tex]m>n[/tex]. Then there does not exist an injection from [tex]N_m[/tex] to [tex]N_n[/tex].

(Note: [tex]mathbb{N}[/tex], is the set of natural numbers, and for some [tex]kin mathbb{N}[/tex], [tex]mathbb{N}_k[/tex] is the set of all natural numbers less than [tex]k[/tex]

The only proof I’ve some up with is similar to the one that appears in Bartle and Sherbert’s Intro. to Real Analysis. It is below. If you have a more elegant argument, please share.

Proof:
Let’s use mathematical induction. If [tex]n=1[/tex] and if [tex]f[/tex] is any map of [tex]mathbb{N}_m (m>1)[/tex] into [tex]mathbb{N}_1[/tex] then it is clear that [tex]f(1) = f(2) = cdots = f(m)=1[/tex] so that [tex]f[/tex] is not 1-1, and thus, not injective.
Now, assume that for some [tex]k>1[/tex], we have that there is no injective map from [tex]mathbb{N}_m[/tex] to [tex]mathbb{N}_k[/tex] when [tex]m>k[/tex]. Let [tex]f[/tex] be a function that maps [tex]mathbb{N}_m[/tex] to [tex]mathbb{N}_{k+1}[/tex]. Consider two cases: either the image of [tex]g[/tex] is a subset of [tex]mathbb{N}_k[/tex] or it is not.
Case 1: If [tex]g(mathbb{N}_m) subseteq mathbb{N_k}[/tex] then we can consider [tex]g[/tex] as simply a map from [tex]mathbb{N}_m[/tex] to [tex]mathbb{N}_k[/tex] and by the induction hypothesis, it cannot be injective.
Case 2: Suppose that [tex]g(mathbb{N}_m)[/tex] is not contained in [tex]mathbb{N}_k[/tex]. If more than one element in [tex]mathbb{N}_m[/tex] maps to [tex]k+1[/tex], then [tex]g[/tex] is not an injection. Therefore, we assume that there is a single element [tex]pin mathbb{N}_m[/tex] such that [tex]g(p)=k+1[/tex]. Define
[tex]h (q) = left{ begin{array}{ll}g(q) & mathrm{if } q=1,ldots,p-1\ g(q+1) & mathrm{if } q=p, ldots, m-1 end{array}right. [/tex]
Now, since [tex]h:mathbb{N}_{m-1} rightarrow mathbb{N}_k[/tex], the induction hypothesis applies and [tex]h[/tex] is not injective. It is then easy to see that [tex]g[/tex] is not injective.
Q.E.D.

Interestingly this can prove that if two different people count the same set of objects, they will get the same number (assuming they count correctly). Stated mathematically:

Thm: [tex]S[/tex] is a finite set [tex]Rightarrow exists ! n in mathbb{N}[/tex] such that [tex]exists[/tex] a bijection from [tex]mathbb{N}_n[/tex] to [tex]S[/tex].