Unsolvable Equations, Feedback Loops, Existence Proofs...

Date: 04/14/97 at 07:19:22
From: Benjamin Goh
Subject: Looping functions and equations where x is unsolvable

I've recently come across these types of questions, but I can't make 
heads or tails of how to solve them.

One of them goes like this:

(9/4)^x = x^(9/4).

One solution is that x = 9/4.  There is another solution for x, but 
after trying logarithms in a variety of ways, I just can't solve for 
this second solution.  It DOES exist, for I've plotted the graph for 
it, but the closest answer is an approximate.  Do you know how to 
solve for this second x?

Another one goes like this:

x = sin x.

I've discovered that by substituting x into the equation again, I get:

x = sin sin x

and if I substitute again...

x = sin sin sin x and so on till x=sin^(Infinity) x.

x is also equal to the inverse sine of x.

This would give me:

x = sin sin^-1 x which is x = x. (identity)

(Of course, x = 0 would work, but I'm sure there are other values.)
How does one go about finding the exact value for this kind of stuff?

Date: 04/14/97 at 11:15:47
From: Doctor Mitteldorf
Subject: Re: Looping functions and equations where x is unsolvable

Dear Ben,

Congratulations!  You've just discovered, all on your own, an 
important result which nobody has bothered to tell you in all your 
studies so far!

Teachers like to concentrate on solving equations, and all the many
techniques we have for solving equations, and all the tricks we can 
play to make difficult equations easier, and ultimately to solve them.

What no one has yet told you is this: the vast majority of equations 
that you could write down using powers and exponents, sines and 
tangents are not solvable.

This is not to say they have no solutions, but the solutions, as you 
say, can only be gotten by successive approximation, getting closer 
and closer to the real answer by making smart guesses that get better 
and better.

Instead of talking about algebraic manipulation, we end up talking 
about existence proofs.  Prove that there are two solutions to 

   (9/4)^x = x^(9/4).

The field I find more interesting is the field of algorithms:  How can 
you find solutions to 

   (9/4)^x=x^(9/4) 

in practice on your computer?  Here's a method you can try on your 
calculator, or write a short computer program to try it.  Take the 
natural log of both sides of your equation.

   x ln(2.25) = 2.25 ln (x)
   x = 2.25 ln (x) / ln(2.25).

Now you've got a kind of feedback loop that takes you from one guess
for x to a better guess.  Start by guessing high, and keep doing the
process over and over:  Take the ln of your guess, multiply by 2.25, 
divide by the ln of 2.25.  The result is your next guess.  Repeat...

Try it!

You'll find that your answers get closer and closer to 3.37...  This 
may be the closest you're going to come to actually "solving" the 
equation.

Here's a philosophical question to ponder:  What does it mean to say 
that one equation has a solution, but another doesn't?  Take the 
equation

   sin(x) = 0.3.

You say - that's easy:  the solution is arcsin(0.3).  But suppose you 
had no such thing as an arcsin function.  Would the equation then be 
"unsolvable"?  Does our definition of the word "solvable"  depend on 
what functions we are allowed to use in our "solution"?  

Go a step further:  Suppose you had the equation 

x^2 = 5.  

You might say that's "unsolvable" if you didn't have a square root 
key on your calculator.

Or take a step in the opposite direction:  Suppose your function 

   x^y = y^x

came up so often in our society that there was a button on your 
calculator that computed a solution, using the successive 
approximation algorithm I described a moment ago.  Would you then 
say that  

   (9/4)^x = x^(9/4)

was a solvable equation?

You're obviously a deep thinker, and I'd like to hear your thoughts 
about these questions.

(By the way - I want you to think again about how many solutions there 
are to the equation x = sin(x).  Try drawing graphs of y = x and
y = sin(x) and you'll see what I mean.)
   
-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/