Off on a Tangent
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
November 11, 2016 Vol. 15, No. 5

Algebra will be explored in next week's colloquium

Title: What is Algebra?
Speaker: Victor Piercey, Ferris State University
Time:  Thursday, November 17 at 4 pm 
Place:  VanderWerf 102

Do we know what algebra is?  Sure, we took it in high school.  But in college, a course on groups and rings called “abstract algebra” appears to have nothing to do with high school algebra.  We also take a course called real analysis which is the study of functions, but functions are part of high school algebra.  So what gives?  Is algebra a collection of topics, or is there something deeper? In this talk, I will share my evolving perspective on what algebra really is and what it means to “do algebra.” I will also discuss how this perspective has impacted my teaching.

Statistics will be the focus of the Thanksgiving week colloquium

Title:  The simulation-based, standardized z-statistic: When and Why it works
Speaker: Yew Meng Koh, Hope College
Time:  Tuesday, November 22 at 4 pm 
Place:  VanderWerf 102

Simulation techniques provide intuitive paths for statistical inference. A quick and easy measure of the strength of evidence against the null hypothesis of a claimed value for a population proportion would be the simulation-based, standardized z-statistic. A brief introduction to this statistic and also some different modes of convergence in Probability Theory will be given, providing the necessary background for understanding when one could use this z-statistic and why it would be valid to do so.

Upcoming Colloquia

The  following additional colloquium is on the schedule for this semester.
  • Thursday, December 1, Hope Research Students

Students participate in MATH Challenge

Hope again had a great turnout of students participating in the Michigan Autumn Take-Home Challenge on Saturday, November 5 this year. Students competed with other students around the state (as well as other states) working in groups on ten interesting problems. We look forward to hearing the results in the near future.

The following students participated: Emily Joosse, Ben Pederson, Will Henkel, Chris McAuley, Alex Dolehanty, Erik Johnsen, Philip LaPorte, Eric Krzak, Jimmy Cerone, Russell Houpt, Richard Edwards, Michelle Plaver, Ashley Stegenga, Aleah Hahn, Kent Schwartz, and Graham Littlejohn.

 Math Club is going to the movies

The Math Club will gather on Friday, November 18 for the 7:30 pm showing of  The Man Who Knew Infinity in the Knickerbocker Theatre at 86 E 8th Street.

This movie is about one of the world's most creative mathematicians, Srinivasa Ramanujan, who grew up in India and was not formally trained in mathematics.  His meteoric rise from obscurity to the mathematical limelight is one of the greatest success stories in the entire history of mathematics. 

Please join us for this critically acclaimed film!  Students get in free with their student ID, while faculty pay $6.  Please arrive by 7:20 to get a good seat!

Problem Solvers of the Fortnight

In our last Problem of the Fortnight we presented the following conversation between Math Man and Vectoria.

"Hey, M's.  What'cha doin'?"
"Oh, hi, Vectoria." 
Math Man hands Vectoria the piece of paper he was looking at and says, " I'm trying to figure out what this is."
After a few moments, Vectoria says, "Oh, I get it!  It's a proof!"
"Really?  A proof of what?"
"It's a trig identity involving an inverse trig function."
"Which one?"
"Well, it could be any one, I suppose."
"No kidding. . . .  I still don't get it.  Which triangles do you use to form the identity?
"All three of them."
"Hmmm. . . .  This is really hard."
"No it's not!  It's as easy as. . . ." 
Glancing at her watch, Vectoria stops in mid-sentence as she realizes that she's late.  "Whoa!  Sorry, M's.  I gotta run.  I'm late!"
"Wait, wait. . . .  easy as what?  Can't you give me a clue?"
As she sprints straight down the hall, Vectoria says, "I think I just did!  Maybe even two!"
Vexed, Math Man sighs and stares at the figure again.  "I still don't get it.  I think I'm more confused now than before she came."

The Problem of the Fortnight is to help Math Man discover the trig identities lurking in the figure.  You don't have to list every conceivable trig identity that this figure contains, but giving three trig identities ought to help Math Math figure out what this figure is communicating.  Write out three trig identities that use all three triangles contained in this figure and explain your reasoning carefully so that you can give Math Man the best chance of understanding what Vectoria saw in this figure.

Congratulations to Philip LaPorte and Daria Solomon for solving the Problem of the Fortnight in the last issue of Off on a Tangent.

Problem of the Fortnight

"Hey, Double M!  How's it going?"
"Oh, hi, Vectoria.  Pretty well, I suppose.  Daria and Philip helped me out with my last conundrum, which was great -- especially since you weren't of any assistance."
"Oh, c'mon, M's.  I was running late and had to jet.  Besides. I gave you some hints."
"Well, I suppose you did.  But they didn't help me any. . . .  And now I'm stuck again."
"Well, what'cha got?"
"Oh, it's a math problem."
"Uh, yeah.  I could've guessed that.  What is the problem?"
"Well, the problem is that I can't solve the problem!"
"Uh, right.  You already told me that, though, Math Man."
"No, I didn't.  I said I was stuck."
"Same diff. . . .  Let's try this again.  Hey, M&M, what's the math problem you're working on?"
"Oh, it's a neat one, but I can't seem to get it.  The problem is:
Find a formula for the function f(x), knowing that 0 < x < 1, that f(x) is positive for such x, and that

[f(x)]2 - x[f(x)]2 - x3[f(x)] = x4."

"Hey, that is a cool one!  How in the world could somebody find the formula for a function knowing so little about it?"

"I know!  That's my predicament exactly," says Math Man, scratching his forehead just under his helmet.
After a while, Vectoria says, "Sorry, M's.  Don't think I can help you on this one.  It's a stumper!  I gotta get going.  See ya around."
"Uh, okay.  yeah, see ya, Vectoria.  Have a nice day"
Just as she's about to head out the door, Vectoria exclaims, "Oh!"
"What?  Did you solve it?"
"No, but I think I have an idea for how to go about it."
"What is it?"
"Aw, M's.  I don't want to spoil it for you.  And besides, it's not that hard. Any of the math students you see roaming these halls can help you.  See ya later!  Happy thinking!" Vectoria says as she disappears out the door.
Absorbed in thought, Math Man distractedly says, "Um . . . yeah . . .  See you later. . . ."

Help Math Man by finding a formula for the function and explaining your reasoning carefully.  Submit your solution to the Problem of the Fortnight slot outside Professor Mark Pearson's office (Vander Werf 212) by 3:00 p.m. on Friday, November 18 -- OR -- bring it with you to the Knick for the screening of The Man Who Knew Infinity that Friday night and hand it to any one of the math professors in attendance.  As always, be sure to include your name, as well as the name(s) of your math professor(s) -- e.g. Pat Anser, Professor Manny I. Diaz -- on your solution.  Good luck, and have fun!     

But McBean was quite wrong. I’m quite happy to say.
That the Sneetches got really quite smart on that day.
The day they decided that Sneetches are Sneetches.
And no kind of Sneetch is the best on the beaches.
That day, all the Sneetches forgot about stars and whether
They had one, or not, upon thars.

Dr. Suess

Off on a Tangent