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gnikm
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Is the squaring of a circle just a riddle that when solved to a great degree of accuracy then that's it... riddle solved. Or there more to it then that?
In what end? That's the whole point of infinities, that there is no end.I just let them get on with it and accept that it's too hard in the 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.]the transcendental numbers fall between members of the set of irrational numbers.
nomadreid said: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![]()
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.Irrational numbers (not a ratio of integers) are, afaik, the roots of integer polynomials.
Irrational numbers are real numbers that are not the ratio of integers. There is no further qualification.sophiecentaur said: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.
That wording makes me cringe. But it is true that there are holes between the algebraic numbers into which one can fit the transcendentals.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.
"Squaring the circle" is an ancient mathematical problem that involves constructing a square with the same area as a given circle, using only a compass and straightedge. It is important because it represents a fundamental challenge in geometry and has been attempted by many great mathematicians throughout history.
No, it is not possible to exactly square the circle using only a compass and straightedge. This was proven by the ancient Greeks and has been accepted by mathematicians ever since. However, it is possible to get very close to squaring the circle using other mathematical methods.
Squaring the circle is considered impossible because it requires the construction of a line segment that is both a straight line and a curve, which is mathematically impossible. Additionally, pi, the ratio of a circle's circumference to its diameter, is a transcendental number, meaning it cannot be expressed as a fraction, making it impossible to construct a perfect square with the same area as a given circle.
Some modern approaches to "squaring the circle" include using calculus and other advanced mathematical concepts to get closer and closer to the exact area of a circle, as well as using computer algorithms and simulations to approximate the solution.
While there are no direct practical applications of "squaring the circle," the pursuit of this problem has led to advancements in mathematics and geometry, which have numerous real-world applications. Additionally, attempting to solve impossible problems has often led to groundbreaking discoveries and innovations in other fields.