Re: [Phys-L] math +- visualization etc. etc.

[John Denker <jsd@av8n.com>]

On 12/17/2013 11:19 AM, LaMontagne, Bob wrote:
> I envisioned a line divided into 27 parts and one divided into 17
> parts. I could "see" that 25 parts were closer to the right end of
> the line than 15 parts - it was all purely visual - no deliberate use
> of formal logic.

a) I know some people who are intensely visual.

b) I also know some who are not, including some very fine 
physicists who find geometrical arguments annoying and
just want to get to the equations as quickly as possible.

Approach (b) is guaranteed to work, if you push it hard enough.
I'm not sure that approach (a) /by itself/ is sufficient for
really hard problems.

In any case, I strongly recommend using *both* approaches together.
The picture tells you what equations to write down, and the equations
tell you what pictures to draw.

The grandmaster of this approach was Feynman.  He could visualize
all sorts of stuff, and he also had formidable mathematical skills.
He used them both *together* not separately.  The famous "Feynman 
diagrams" are the perfect example of this:  each one is a picture of 
something happening in spacetime ... and each one corresponds to an 
integral that needs to be evaluated.

The "number line" approach Bob mentioned is isomorphic to the
"rearrange the math" approach that I mentioned, i.e. 1 - 2/27 
versus 1 - 2/25.

===========

The main problem with the visual approach is that visual communication
is sometimes very expensive.
 ++ I can write down a four-vector equation in a few seconds.
 ++ I can visualize the corresponding spacetime diagram in my
  mind very quickly.
 -- Actually drawing the spacetime diagram so that somebody other
  than me can see it might take a couple of hours.  Ouch.

==================================

If you want to know how I really solved this problem....  I solved it 
three or four times in rapid succession.

1) At first glance, it was obvious that both numbers were close to 
 unity.  So write them as 1 - ε and 1 - δ.  This immediately makes
 me think of Taylor series.  I'm not going to mention Taylor series
 in front of high-school students, but you asked how *I* approached
 it.

 I am accustomed to solving much harder problems than this, and I
 like to collect /systematic/ approaches that work for more than
 one problem.  The  Taylor series is overkill for this particular 
 problem, but I can apply the Taylor series in less time than it 
 takes to tell about it, so why not?

 If I'm talking to people who know about Taylor series, this answer
 is super-easy to explain.  It's obviously correct.  For the more
 devious approaches mentioned below, the proof of correctness is
 more elaborate.

2) Then I thought of the number-line picture.

3) Then I immediately formulated that picture in terms of regrouping
 the math.

4) Then I classified it as an example of a more general notion of 
 "regrouping", which is a tool that can be applied to a huge class
 of problems.  The value of solving this particular problem is small, 
 whereas the value of learning the general-purpose tool is large.

 This generalization protects against the well-nigh-inevitable student 
 complaint that the question is not fair "because we never studied this". 
 Actually they should have studied it.  It's on the second- or third-
 grade syllabus.  The educationalese term is "compensation" but that's
 just jargon for any indirect, devious, outside-the-box approach.


On 12/17/2013 11:37 AM, Richard Tarara wrote:
> > I look at the percentage change of the numerator versus the
> > percentage change of the denominator.

That's clever.  That implicitly depends on the fact that the
logarithm is a monotone function.

It is characteristic of the clever solutions that they rely on
facts that might at first glance seem unrelated to the original
question, such as monotonicity of the logarithm, monotonicity of
the inverse, et cetera ... plus the not-entirely-obvious fact 
that monotone functions commute with inequalities.

This makes the solution hard to explain.  When we solve the problem,
these facts are ten layers deep in the subconscious.  It would be
an exaggeration to say we use these facts without thinking; rather 
we use these facts without /conscious/ thought.  I reckon 99.999%
of all thought is subconscious.