Natural Blogarithms

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.

written by SplineGuy

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:











…
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.

written by SplineGuy

Since I am now on a roll with these “wonderful” capsules of hilarity, try a few more that tickled my funny bone. By the way, most of these are not that funny, but just plain sad. And yes, the title of this entry should probably read, “Worse Math Humor.”

written by SplineGuy

I laughed out loud for 15 minutes when I read this one.

Click here if you don’t get it.

HT: CastingOutNines

written by SplineGuy

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

written by SplineGuy

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.

written by SplineGuy

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 pigeons are put into pigeon holes and if 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 with . Then there does not exist an injection from to .

(Note: , is the set of natural numbers, and for some , is the set of all natural numbers less than

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 and if is any map of into then it is clear that so that is not 1-1, and thus, not injective.
Now, assume that for some , we have that there is no injective map from to when . Let be a function that maps to . Consider two cases: either the image of is a subset of or it is not.
Case 1: If then we can consider as simply a map from to and by the induction hypothesis, it cannot be injective.
Case 2: Suppose that is not contained in . If more than one element in maps to , then is not an injection. Therefore, we assume that there is a single element such that . Define

Now, since , the induction hypothesis applies and is not injective. It is then easy to see that 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: is a finite set such that a bijection from to .

written by SplineGuy

As a result of a question asked about the history of the word “coefficient” I did a little personal research. Actually, I think I asked the question of myself in Intermediate Algebra and realized I didn’t know. Well, now I do.

From http://www.pballew.net/arithme9.html

Coefficient: A coefficient is a number, or variable, that is multiplies a variable term. In a common linear equation like 2x-3y=5 the 2 and 3 are the coefficients of the variables x and y. In the typical equation of a general quadratic polynomial we write Ax2 + Bx + C =0 but we call the letters A, B, and C the coefficients of the terms. Even though they are variables, the represent some constant, but unknown value unlike the variable x which is variable of the expression. The origin of the word reaches back to the early Latin word facere, to do. The prefix for out, ex preceding this gave the meaning of bringing about a result, and is the source of the intermediate word, effect, and its variation efficient. When two things were joined to make something more effective, we add co, the root for with, to form coefficient. The math historian Cajori credits 16th Century mathematician Francois Vieta for the creation of the word, but suggest that it did not become common until near the beginning of the 18th century.

I was also the motivated to learn a little about the source of this information, namely Florian Cajori. He authored a history of mathematics and a history of mathematical notation.

From Wikipedia (also see Florian Cajori Biography):

Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. He emigrated to the United States at the age of sixteen. He received a Ph.D. at Tulane University, where he taught for a few years before being driven north by his health. He taught at Colorado College, where he founded the Colorado College Scientific Society. He became one of the most celebrated historians of mathematics in his day (he was the author of “A History of Mathematical Notations” (ISBN 0-486-67766-4)). In 1918, he was appointed to a specially created chair in history of mathematics at the University of California, Berkeley. He remained in Berkeley, California until his death, August 15, 1930.

written by SplineGuy

[This entry is cross-posted on this blog and on Zone Defense]

This morning in Sunday School, a quote stood out to me and I wanted to pass it along. I didn’t find it online so this is a quote that was attributed to D.L. Moody Charles Spurgeon, and it may be a slight paraphrase as I am quoting from memory what was quoted to me:

“I looked at Jesus, and the dove of peace flew into my heart.
I looked at the dove of peace, and she flew away.”

It reminds me that the advantages and benefits I receive from my relationship with Christ is not the motivation for my relationship with Christ.

written by SplineGuy

I was posed with the question today of how secant lines are related to the secant function. Equivalently, it lead to the question of how tangent lines are related to the tangent function. I vaguely recalled seeing them illustrated with the unit circle but was unable to recall the exact relationship. After a minimal amount of searching, I came across this image in the public domain that helps to explain it.

You’ll notice from the picture that when you consider some angle, , you can draw the secant line passing through the point on the outside of the unit circle (which is the point in the picture) and through the origin. Now, it is not on the picture but you can then draw the line and the distance from the origin to where the secant line intersects is the value of the . This is equivalent to the length of the line in the image. You can also see from the image, how the tangent line at is related to the

You may be wondering, what in the world are versine and exsecant? Well, I didn’t know either, but it turns out that they were common historically (and appeared in the earliest tables), but are now seldom used:

There, I learned something new today.

written by SplineGuy