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Date: 07/31/99 at 14:13:51
From: kiki nwasokwa
Subject: Random questions 

Why was the Cartesian plane invented/developed/discovered?

Is the development of the more abstract, or "theoretical," branches of 
math just for fun? Are they merely an exercise in the application (so 
to speak) of human logic? 

What is chaos theory?

Why is Einstein's equation e = mc^2? Why not e = mc^3? This still 
yields a large amount energy from a small amount of mass.


Date: 08/01/99 at 15:39:31
From: Doctor Ian
Subject: Re: Random questions 

Hi Kiki,

> Why was the Cartesian plane invented/developed/discovered?

One story that many people have heard is that Rene Descartes was lying 
in bed one morning, watching a fly crawl around on the ceiling, when 
he realized that he could describe the path of the fly by assigning it 
coordinates relative to orthogonal axes.

Is the story true? Descartes definitely invented analytic geometry, 
and he certainly did a lot of his work lying in bed.

However, according to _Men of Mathematics_ by E. T. Bell, Descartes 
said that the idea was one of many that came to him in the second of 
three dreams that he had on November 10, 1619, although he apparently 
never told anyone just what the dream was about.

I suppose the moral is, if you ever have a really great idea, write 
down how you came up with it immediately and send it to the 
Smithsonian Institute, so people won't have to wonder about it later 
on.

>Is the development of the more abstract, or "theoretical," branches 
>of math just for fun? Are they merely an exercise in the application 
>(so to speak) of human logic? 

It depends on whom you ask. If you ask the mathematicians, the answer 
is 'yes'. If you ask the people who fund the mathematicians, hoping 
that even the more abstract fields will turn out to be useful for 
something, the answer is 'no'.

>What is chaos theory?

Biologist Arthur Winfree used to tell his students: "The basic idea of 
Western science is that you don't have to take into account the 
falling of a leaf on some planet in another galaxy when you're trying 
to account for the motion of a billiard ball on a pool table on earth. 
Very small influences can be neglected. There's a convergence in the 
way things work, and arbitrarily small influences don't blow up to 
have arbitrarily large effects."

Chaos theory is the study of situations in which this assumption turns 
out not to be true.

>Why is Einstein's equation e = mc^2? Why not e = mc^3? This still 
>yields a large amount energy from a small amount of mass.

Take a look at the cartoons by S. Harris on this page from Juergen 
Giesen's "Around Physics and Astronomy":

   http://www.jgiesen.de/Divers/Humor/Humor.html   

Seriously, though, if you look at the units in the equation, you will 
see that

     energy = mass * (distance/time)^2

            = mass * (distance/time^2) * distance

            = force * distance

which is one of the definitions of 'energy'.

If you added an extra 'c' in there, you'd end up with something other 
than 'energy' on the right side of the equation. Remember, c isn't a 
dimensionless number like pi, it's a speed.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
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