Todd's Place
I don't do these:
  • Last Seen
  • Blog
  • Contact
  • Air Quality

Integers can't have rational roots

3/25/2017

0 Comments

 
I've been reading a lot of number theory. Mostly it's recreational, and I'm always looking for simple proofs which are mind-blowing. This is one of them. I've copied it below.

This proof shows that the square root of an integer, such as $\sqrt{m}$, must be either an integer (i.e. $m$ is exactly a "perfect" square) or irrational. Another way of saying it is that almost all roots of integers are irrational. Yet another way of saying that is that $\sqrt{m}$ either has no decimals at all, or $\sqrt{m}$ has an infinite, non-repeating decimal expansion!

Consider that Pythagoras and Euclid had proofs that $\sqrt{2}$ and $\sqrt{3}$ are irrational, but these proofs were one-offs. They rely on contradictory inequalities where one side is even and the other is odd, or that irreducible fractions must be reducible. Euclid had a proof that square roots of primes, $\sqrt{p}$, are irrational. However, these are just "special" integers. None of these proofs addressed the vast majority of integer roots.

In a sense, it took two thousand years for the brilliant mathematician, Theodore Estermann, to come along and illuminate the entire landscape. This two-line proof is valid for every integer! Think about that. I learned from an obituary published in a math journal, that Estermann's work formed theoretical bridges for some of the greatest number theorists in history. While few modern theorems are named after him, still there are tons of his students and contemporaries who created "named" theorems, based on his insights. 

Since my blog is configured to display equations, and the proof is so simple, I've copied it here. This simple generalization from $\sqrt{2}$ to $\sqrt{m}$ was shown by Harley Flanders:


THEOREM:  Suppose $m$ is not a perfect square. Then $\sqrt{m}$ is irrational.

Proof.    Let $n$ be the integer with $n < \sqrt{m} < n+1$. It suffices to prove that $\alpha = \sqrt{m}-n$ is irrational. Suppose not. As $0 < \alpha < 1$, we have $\alpha=\frac{p}{q}$ where $0<p<q$. Assume that $q$ is as small as possible  (Estermann's key idea). Then we have
\[ \frac{q}{p} = \frac{1}{\sqrt{m}-n} = \frac{\sqrt{m}+n}{m-n^2} = \frac{\alpha+2n}{m-n^2} . \]
We solve for $\alpha$:
​\[ \alpha = \frac{(m-n^2)q}{p}-2n = \frac{r}{p} \]
where $r = (m-n^2)q-2np$. ​Thus $\alpha$ is a fraction with even smaller denominator, a contradiction.


0 Comments

Vivi Dressing Up For School

3/10/2017

0 Comments

 

Rocking some new jewelry. Accessories first!

​Add a nice floppy hat!

All ready for lessons, with her tiny scarf.
Subtitles (and rough translations):

How are you?         Fine. 

How's the Weather?        Nice.

Guai, mm-Guai?  (Are you good or bad?)                Guai.  (I'm good)

Let, mm-Let?  (Are you smart or not?)             mumble mmmumble  Let Let Let. (I'm quite smart)

Ho Mei? (Is it tasty?)                Ho Mei! (Yes, delicious!)

0 Comments

    Author

    I'm an applied-math-research Ph.D. and serial startup founder. I am a recognized computer security expert, fortunate to join the ranks of many, great CTO's. I've founded and seed-funded multiple, successful, VC-backed companies. I'm still at it!

    My wonderful wife and I moved from New England to near the Portland Oregon area. We LOVE the Pacific Northwest, and we've been here a few years now. We have an adorable baby girl, Vivi.

    People here are nice and smile a lot. Vegetables are insanely delicious. Driving is not like Mad Max.

    This blog is very Vivi-centric. Our family just can't resist. :) Also, there are some stupid hacking and geek tricks.


    Archives

    December 2020
    November 2018
    October 2018
    September 2018
    April 2018
    March 2018
    February 2018
    January 2018
    December 2017
    November 2017
    October 2017
    September 2017
    August 2017
    July 2017
    June 2017
    May 2017
    April 2017
    March 2017
    February 2017
    January 2017
    December 2016
    November 2016
    October 2016
    September 2016
    August 2016
    July 2016
    June 2016
    May 2016
    April 2016
    March 2016
    February 2016
    January 2016
    December 2015
    November 2015
    October 2015
    August 2015
    July 2015
    June 2015
    May 2015
    April 2015
    March 2015
    February 2015
    June 2014
    April 2014
    March 2014
    January 2014
    October 2013
    April 2013
    March 2013
    February 2013
    January 2013

    Categories

    All

    RSS Feed

Copyright 2012-2021  Todd Brennan