Is the squaring of a circle just a riddle that when solved to a great degree of accuracy then thats it..... riddle solved. Or there more to it then that?
Welcome to PDF; You mean this: http://en.wikipedia.org/wiki/Squaring_the_circle ??? If I ask you to build a house according to a set of instructions I write down, is that a riddle? "Squaring the circle" is a task, like building a house, that has been proved impossible. The task is not to construct a square with the same area as a given circle to some "great degree of" accuracy, but to perform the task exactly in a finite number of steps. If we treat it as a "riddle" then the riddle was solved when it was proved impossible. "It can't be done" is an acceptable "solution" in mathematics. But you can treat it as a kind-of koan if you like?
Hello Simon, Thanks for replying. How close to exact could one expect to get in less then 20 steps using only a compass and straight edge; Resulting in a square and circle of equal area accurate to the thickness of a pencil line. I was wondering what areas of our understanding would be impacted if it was possible at all....which its not. I understand it is speculative but would greatly appreciate your comments.
The accuracy to a pencil line is arbitrarily accurate - it depend only on the scale at which the diagram is being drawn. So you pick your accuracy and work out the scale. I don't know how accurate 20 steps can be ... you'd have to look up the different approaches. If it were possible, then it would mean that pi is merely irrational rather than transcendental. The importance is usually discussed at length in historical treatments. But since it is not possible, the speculation is pointless. Have you read the reference I gave you? Have you looked at other references?
When they made that step in understanding, they had revealed that the transcendental numbers fall between members of the set of irrational numbers. SO how can there be 'spaces' between a continuous array? Creepy. Those Mathematicians are full of stuff like that - take infinity(ies) for instance. I just let them get on with it and accept that it's too hard in the end.
In what end? That's the whole point of infinities, that there is no end. That is ambiguous. My first reaction is to assume that you are saying that the transcendental numbers are a subset of the irrational numbers. [An important subset: the set of transcendental numbers is uncountable, but the set of algebraic (non-transcendental) irrational numbers is only countable.] Or perhaps you mean something else? What, and you don't find that fun? Not even the Skolem paradox? Ah, these humourless physicists
I may not have put it as well as I should have. (But I am not a Mathematician) Irrational numbers (not a ratio of integers) are, afaik, the roots of integer polynomials. I think that transcendental numbers, although they are not ratios, are not the roots of an integer polynomial so there will always be a space between the irrational numbers in which you can fit π, e and all the rest. So transcendental numbers must fall in between (not being a subset of) the sequence of irrational numbers. Edit: I just read about the Skolem Paradox. I believe I heard it from a stand up comedian at the last stag night I attended.
Ah, here lies the rub. No, algebraic numbers are the roots of (finite) integer polynomials. Your parenthetical phrase got it right: irrational numbers are all numbers which (surprise!) are not rational, an irrational number is any number which cannot be expressed as a ratio of two integers a/b, where b ≠0, but lots of roots of integer polynomials cannot be expressed as such a ratio. Good ol' square root of two, and so forth. So, transcendental numbers are all irrational (but not conversely), and a lot , but not all, of algebraic numbers are also irrational. π, e, and the other transcendental numbers are all irrational, so there is no "crack" between irrational numbers to put transcendental numbers, although there are lots of (countably infinite) "cracks" between algebraic numbers in which to put the transcendental numbers. In fact, between any two algebraic numbers, there are as many transcendental numbers as there are real numbers. Oh, and it is not true that contemplating different levels of infinity is what drove Cantor insane .....
Irrational numbers are real numbers that are not the ratio of integers. There is no further qualification. That wording makes me cringe. But it is true that there are holes between the algebraic numbers into which one can fit the transcendentals. An algebraic number is one that is the root of a non-zero polynomial with integer coefficients. This includes the integers (roots of monic integer polynomials of degree 1) and the rest of the rationals (roots of integer polynomials of degree 1)] "Monic" just means that the high order coefficient is 1. [Edit: Beaten to it by nomadreid] I am not a mathematician either. But I like to pretend sometimes.
Sorry about the Cringe aspect but I think the idea of 'fitting in' is a valid one. That is, if you present the rationals in order of magnitude. @nomadreid: Thanks for putting me straight on the precise definition. It was the term 'algebraic' that had escaped me.