yet another test post

January 4, 2011

Photo on 2010-11-27 at 16.25 #2

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February 14, 2010

z:= x+yi
w:= a+bi

zw = (xa-yb) + (xb+ya)i

u:= xa-yb
v:= xb+ya

f: \Bbb C \rightarrow M_2(\Bbb R)
f(p+qi) := \begin{pmatrix} p & q \\ -q & p\end{pmatrix}

f(z)f(w) = \begin{pmatrix} x & y \\ -y & x\end{pmatrix} \begin{pmatrix} a & b \\ -b & a \end{pmatrix} = \begin{pmatrix} xa-yb & xb+ya \\  -ya-xb & xa-yb \end{pmatrix} = \begin{pmatrix} u & v \\ -v & u\end{pmatrix} = f(zw)

f(z)+f(w) = f(z+w) trivially.

January 5, 2010

(:={
0:=}

()

{0 \choose 0} = 1

(),(a)

{1\choose0}=1, {1\choose1}=1

(), (a) (b), (a,b)

{2 \choose 0} = 1, {2\choose 1} = 2, {2\choose 2} = 1

(),(a)(b)(c),(b.c)(a.c)(a.b),(a.b.c)

{3\choose0}=1,{3\choose1}=3,{3\choose2}=3,{3\choose3}=

Archive

November 23, 2009

As I Was Saying
The Natural Log: Eli Maor plugged.
In The Arena: “inflexible knowledge” at KTM & Lefty on Devlin.
exp & log
Monday Linkage: CO9’s on ANT.
Math Newsbite: “Square root day”.
Introduction to Logarithmic Functions
The Veil of Maya: simple, compound, & continuous interest.
When There Is No Peace: links.
Stretching A Point: doubling time etcetera.
Why Was I Not Informed?: MT@P, Spirograph.
Catherine Nails It: 21st C. skills.
Remarks Toward The Dissolution Of This Blog: review.
Forests And Trees: links.
Herculean Sense-Making: Garelick on discovery learning.
More Review
Formula Scratchpad
3.14
Last Post

Last Post

March 17, 2009

It’s all over but the grading for me and my mind’s elsewhere. I’m doing a 103 and a 153 next quarter and have started work on those. Most of the blogging for the 148 classes now ending was in VME.

I’d like to thank my life-support system, Madonn’. Thanks for letting me hang on. Also all my students and anyone who commented here or even ever seriously considered it. Without you, no blog.

The world of mathblogs is expanding vigorously at all levels. It’s a great time to be alive.

3.14

March 16, 2009

JD’s Carnival of π.

Formula Scratchpad

March 14, 2009

Let P_1 = (x_1, y_1) and P_2 = (x_2, y_2). The distance between P_1 and P_2 is then
d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 +(y_2 - y_1)^2}.
Humans as opposed to computers would do well to understand that this is the square root of the sum of squares because of the Pythagorean Theorem; my habit in in-class computations is to write out the radical with the parens and superscripts first and fill in the differences after reminding everybody of this connection. Every now and then somebody gets it.
There’s a “formula” given in the same section for the midpoint of the line segment connecting P_1 to P_2: but I consider it foolishness to think of this as a formula. Average the x co-ordinates for the “new” x; average the y’s for the y.

One immediate consequence of the distance formula (or of the Pythagorean Theorem directly…) is that x^2 + y^2 = 1 is the equation of the unit circle; shifts-and-scalings lead to the circle equation
(x - H)^2 + (y-K)^2 = R^2
(for the circle with center (H,K) and radius R). One is prepared either to put an appropriate polynomial in x and y into this form (by completing the square) and “read off” the center and radius, or (what is much easier) to write out the appropriate equation for a given center and radius.

The financial section of the exp-and-log chapter is fraught with formulii. And typing ’em out is no walk in the park, as I might have mentioned from time to time. But let’s see. There’s the old A = P(1+{r\over n})^{nt} for compound interest; its limit as n grows without bound gives the continuous compounding formula A = Pe^{rt}. Dividing on both sides of these gives the present value formulas (for “P”, when “A” is known); these should not be understood as separate formulas and I’m sure as heck not gonna write ’em out here. I’ll gladly mention that the division I just mentioned is typically denoted in “negative exponent” style. The effective interest formulas are (1+{r\over n})^n -1 and e^r - 1. As is usual, our hope in presenting this is that having thought it through, you can forget it and remember instead that one is here letting P and t be equal to unity (by selecting units appropriately); the “multiplier” on P, which is one, for one year, minus the original “one” (one hundred percent of P), is the total interest earned during that year (with its n compounding periods). But wait. Say something once; why say it again. This “formula sheet” project isn’t going to be the affair of an hour and I was crazy to think it would be. Anyway, as usual, there are typically real good summaries in our or any such book.

Better still, we’re on the net. Google “precalculus formula sheet” and look around. Finding the right ones from the best few you find sounds like a darn good exercise. I should have started here.

More Review

March 12, 2009

I posted some remarks during the review for Exam II (about rational functions and inverse functions) in the old blog. Then there’s this post about the “simplest” Rational Function, some discussion of the Factor Theorem, the review problems of 1/30, and a summary of Transformations.

“Quadratic Formula Lore” was one of my most popular posts. I don’t know whether my decision about Transformation notation counts as “review” (since these notations are “unofficial” for the course and won’t Be On The Test). I made up some hints about using the net and the library. Near the beginning, before I hit my stride, I posted this short rant about defining “relation”.

Let the record show that this was a very productive quarter for me (and that I’m proud of it). What the hell, I’m feeling generous… let the record even show that I had the time of my life doing it.

Posting about the Final will cease by Monday at 8:00 AM (so everybody gets the same info from here). Go do some math problems.

Herculean Sense-Making

March 12, 2009

Barry Garelick’s (PDF) “Discovery learning in math: Exercises versus problems” at the Third Education Group‘s Review (I can’t quite read this site… there’s a sidebar covering up some text). Spotted in KTM.

Forests And Trees

March 10, 2009

Old fave Gilbert Highet noticed at KTM.

Dollars and sense at Social Maths.

Problem solved (probably), at Gowers’s.

Sez here I’m doing a 103 next quarter (8:00—9:50 AM MW) and a 153 (1:00—3:15). I’ve blogged about 153 before (badly) and probably will again real soon (it’s not for me to say how well).