Do you mean the obvious conclusion that sqrt(d) must be irrational also? Or is it something deeper?
Bob at PC
________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu on behalf of John Denker
Sent: Sat 1/31/2009 4:17 PM
To: Forum for Physics Educators
Subject: [Phys-l] amusing puzzle + remarks
I was recently reminded of the following puzzle. It makes
me smile:
As preconditions, we are told that a, b, c, and d are
rational numbers, but sqrt(b) is irrational. Given
that, prove or disprove the following proposition:
a + sqrt(b) = c + sqrt(d) ...
if and only if a=c and b=d.
===============
This amuses me because it works on a couple of levels.
-- At the lower level, it is a cut-and-dried piece of high
school algebra.
-- But there's another level. Something more open-ended.
Something you might wonder about. Do you see it? That's
the second part of today's puzzle.
This illustrates, yet again, why a high school teacher needs
a whole lot more than a high school education. The level a
naive outsider sees as "overqualified" is really just barely
qualified for doing a good job.
This makes it hard to talk about "overqualified" teachers.
The word means wildly different things to different people.
Returning now to the second part of our puzzle, here's a
huge hint. Part 1 of the puzzle appeared (without any
hint of part 2) in a textbook chapter right after complex
numbers were introduced. Hmmmm.
Credit: Uspensky.
Answer to part 2:
uggc://zngujbeyq.jbysenz.pbz/RkgrafvbaSvryq.ugzy