Thursday, April 6, 2017

Lesson 13-3: Ruling Out Possibilities (Day 133)

This is what Theoni Pappas writes on page 96 of her Magic of Mathematics:

(no text)

That's because page 95 was the final page of Chapter 3, "Mathematics & Art." Page 97 starts Chapter 4 of the book, "The Magic of Numbers." So technically, page 96 is the "blank" page that's inserted in between the chapters.

At least, it would be blank except that Pappas usually places a drawing on the page that's facing the new chapter. For Chapter 4 though, instead of a drawing, she lists several expressions that equal 1. In case you're curious, here's the list of expressions:

1 = 1/2 + 1/2 = sqrt(1) = -e^(pi i) = 1/1 = sum_(n=1)^inf (1/2)^n = 1 * 1 = googol^0 = 1^99
= (1000 - 1)/999 = int_0^1 (3x^2)dx = sin(pi/2) = -(x - y)/(y - x) (with x =/= y)

Unfortunately, many of these expressions don't look good in ASCII. In particular, the summation and integral are hard to read. The ASCII tradition is to follow _ with the lower limit and ^ with the upper limit of the sum or integral. This is consistent with the use of ^ to mean superscript, as even the exponents in googol^0 and 1^99 are superscripts.

We'll start Chapter 4 tomorrow, but this list of expressions equal to 1 gives us some idea regarding the content of this chapter.

The fact that all of these complicated expressions equal 1 reminds me of Vi Hart, who created a video about the properties of this most amazing number she calls "wau":

The punchline is that wau = 1. I know that April Fool's Day has already past, but this would have made a great April 1st prank.

Lesson 13-3 of the U of Chicago text is "Ruling Out Possibilities." I tried to find the Lesson 13-3 worksheet from last year, but I simply never posted it last year. I'd chopped Chapter 13 into parts and included them throughout the year, but most likely I never found 13-3 important enough to include.

So instead I post Lesson 13-3 from two years ago. Again, Lesson 13-3 was combined with 13-4 that year, so I only include one side of that worksheet.

This is what little I wrote about Lesson 13-3 two years ago:

Here are a few things that I want to point out. First of all, some texts refer to the Law of Ruling Out Possibilities in Section 13-3 by another Latin name, modus tollens. Here is a link to the Metamath reference to modus tollens.

As we can observe in the proof at the above link, modus tollens is essentially modus ponens (The Law of Detachment) applied to the contrapositive (Law of the Contrapositive, or contraposition.)

Section 13-3 is another section that lends itself to an activity, since many of its questions are actually logic problems, like the ones that often appear in puzzle books.

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