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john iskra
Works at Emory and Henry College
Attended University of Michigan
Lived in Bristol VA
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Category Theorist Eugenia Cheng (http://www.eugeniacheng.com/)   will be on the Colbert Show tomorrow night!﻿
Welcome to Better Know a Guest, your weekly round-up of guests on 'The Late Show with Stephen Colbert' for November 2 – 6, 2015.
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Good article. Both incisive and a much needed counterpoint to what has become a too little questioned conventional wisdom.

For many students, quiet time is key for the learning process.
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It seems like there were a whole lot of buzz words in that article that are currently buzzing around our fair institution. ﻿

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I'd never seen this before.  Very clever!

A picture proof that √2 is irrational

This Friday I was hanging out with some philosophy professors.  This is always fun, because they think sort of like me, but different.  They seem more optimistic about our ability to solve all sorts of puzzles just by talking.

To annoy them a bit, I said that philosophers are great at verbal  reasoning, but mathematicians should be good at three kinds of reasoning: verbal, symbolic and visual  reasoning.

In response, one of them showed me this picture proof that √2 is irrational.

We just need to show that it's impossible to have

a² = b² + b²

for whole numbers a and b.  So let's do a proof by contradiction.  We can assume a is the smallest whole number that obeys this equation for some whole number b.  We'll get a contradiction, by finding an even smaller one.

We do it by drawing a picture.

The big square in this picture is an a × a square.  The two light blue squares, which overlap in the middle, are b × b squares.

The area of the big square is the sum of the areas of the light blue squares.  But there are two problems.  First, the light blue squares overlap.  Second, they don't cover the whole big square!  These two problems must exactly cancel out.

So, the area of the overlap - the dark blue squares - must exactly equal the area that's not covered - the two pink squares.

So, the area of the dark blue square is the sum of the areas of the pink squares!   But the lengths of the sides of these must be whole numbers, say c and d.  So we have

c² = d² + d²

But c is smaller than a.  So, we get a contradiction!

Actually this proof uses a mix of verbal and visual reasoning, with just a tiny touch of symbolic reasoning.  I wrote the formulas like a² = b² + b² just to speed things up a bit and reassure you that this was math.  I didn't really do anything with them.

The philosophers who told me about this are Mike Pelczar and Ben Blumson.  The picture here comes from a website Mike pointed me to:

• Dave Richeson, Tennenbaum’s proof of the irrationality of the square root of 2, http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrationality-of-the-square-root-of-2/

Richeson says:

Apparently the proof was discovered by Stanley Tennenbaum in the 1950’s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).

On the other hand, Ben says John Bigelow published the proof in his book The Reality of Rumbers in 1988, without citing anyone.

We wondered if it was known to the ancient Greeks.

You can do similar proofs of the irrationality of √3, √5, √6 and √10:

• Stephen J. Miller and David Montague, Irrationality from the book, http://arxiv.org/abs/0909.4913.

And this particular style of proof by contradiction is famous!  It's called proof by infinite descent.  You assume you have the smallest whole number that's a counterexample to something you want to prove, and then you cook up an even smaller one.  It's really just mathematical induction in disguise, but it's more fun.  It was developed by Pierre Fermat - who, by the way, was a lawyer.

If you want to take all the fun out of the proof I just gave, you can do it like this.

Assume a is the smallest whole number for which there's a whole number b with

a² = b² + b²

Let

c = 2b - a

and

d = a - b

Then c and d are whole numbers and

c² = d² + d²

(You can do some algebra to check this.)  But c < a, so we get a contradiction.

Wikipedia shows you how to prove by infinite descent that whenever n is a whole number, either √n is a whole number or it's irrational:

https://en.wikipedia.org/wiki/Proof_by_infinite_descent

Fermat did a lot more interesting stuff with this method, too!

#spnetwork arXiv:0909.4913 #geometry﻿
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Light rain at Steele Creek Park. ﻿
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Seen (or, I guess not...) while school shopping today:﻿
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Inspiring the next generation!

CHARLOTTESVILLE, VA—In an effort to help his students develop inaccurate perceptions of their talents, University of Virginia creative writing professor Alan Erickson told reporters Monday that he takes the time to provide each and every one of them with personalized false hope.
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From the "you may be a nerd if" list:
You get a little verklempt after using gmailTeX with your child for the first time to discuss a calculus question﻿
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Wow!!!!﻿
Scientists looked to dark streaks that form each summer on the slopes of Martian mountains, craters and canyons.
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Ha!﻿
PASADENA, CA—Upending the conventionally held assumption that the United States must exclusively be moving along a single good or bad path forward, quantum political scientists at the California Institute of Technology published a paper Thursday hypothesizing that the country is, in fact, headed in both the right and wrong directions simultaneously.
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SAN JOSE, CA—Noting how lucky she was to have finally found the one she was meant to be with forever, local woman Julie Winters told reporters Tuesday she was relieved that her true soulmate also happened to have the same socioeconomic status as she does.
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In his circles
670 people
Have him in circles
409 people
Education
• University of Michigan
Math and Philosophy, 1982 - 1986
• University of Oregon
Education, 1989 - 1990
• University of Tennessee
Mathematics, 1996 - 2004
• Portland State University
Mathematics, 1994 - 1995
Basic Information
Gender
Male
Relationship
Married
Work
Occupation
Chair, Department of Mathematics, Emory and Henry College
Employment
• Emory and Henry College
2004 - present
• Emory University
2003 - 2004
• University of Tennessee
1996 - 2003
• Portland State University
1994 - 1996
• 4J Schools
1991 - 1994
Places
Previously
Bristol VA - Atlanta GA - Knoxville TN - Portland OR - Eugene OR - Ann Arbor MI - Detroit MI - Sydney Australia