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/
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