math question!?

Question

Why is x^3=2 impossible to construct using a compass and a straight edge, in euclidean geometry?

Answer 1

There are exactly three things you can do in the greek rules of construction --

#1: given two points, construct the line through them
#2: given two points, construct the circle centered at one point and passing through another.
#3: given two lines, or a line and a circle, or two circles, find their point(s) of intersection.

That's it. These are the only things you can do. Now, all constructible geometric objects are formed from two points, thus the points of intersection of any two lines or circles can be completely determined from the two points used to construct the first object and the two points used to construct the second object. Examining the equations for lines and circles, we see that the x and y-coordinates of every constructible point can be obtained from the x and y-coordinates of the four (not necessarily distinct) points used to construct it by a finite number of additions, subtractions, multiplications, divisions, and square roots. And if we use these new points to construct even more points, then the coordinates of these other points will also be expressible from the coordinates of the original points using a finite number of additions, subtractions, multiplications, divisions, and square roots. This means that, if we choose two points and declare the distance between them to be 1, and some pair of points can be constructed from them such that the distance between them is exactly ∛2, then ∛2 could be obtained from 0 and 1 using a finite number of additions, subtractions, multiplications, divisions, and square roots. However, in this case finding ∛2 requires taking a cube root, which is not one of the operations that you are allowed to do, so ∛2 cannot be constructed.

Of course, this is somewhat less than a complete proof, because you might say "well, obviously the straightforward method of finding ∛2 requires taking a cube root, but might there be some other method of obtaining that number which requires only taking square roots?" And the answer is no, but to demonstrate why that is true requires me to go into field theory. If you already know some, then it basically amounts to the fact that since the constructible numbers can be found through a finite number of field operations and square roots, then every constructible number can be found through some sequence of quadratic extensions over the rational numbers, and thus lies within a field extension over Q whose degree is an exact power of two. So if ∛2 were a constructible number, then Q(∛2) would be a subextension of a field extension whose degree is an exact power of two, and thus [Q(∛2):Q] would divide an exact power of two, requiring it to BE an exact power of two. This means that ∛2 would have a minimal polynomial over Q whose degree is an exact power of two, and since it is a root of a polynomial of degree 3, this means that the minimal polynomial of ∛2 would be of degree 1 or 2 -- in the first case, ∛2 is rational, and in the second case, ∛2 would be the root of a quadratic with rational coefficients that divides x³-2, and thus the third root of x³-2 (i.e. the one that would not be the conjugate of ∛2) would be rational. In either case, ∛2 is constructible implies that x³-2 has a rational root, and simple testing by the rational root theorem reveals that it does not, thus generating a contradiction. Therefore, ∛2 is not constructible. (Note that the same trick can be used on cos 20°, since it is also a root of a polynomial of degree 3 that has no rational roots. Since 60° angles are constructible, if you could trisect an arbitrary angle you could trisect 60°, and then using your 20° angle obtain cos 20°. This proves that it is impossible to trisect the angle as well).

If you don't know any field theory, the second paragraph was probably incomprehensible to you. In that case, you have two options -- either learn field theory, or have faith that modern mathematicians know what they're talking about when they say that ∛2 cannot be obtained by any finite sequence of additions, subtractions, multiplications, divisions, and square roots.

Answer 2

The graph of the given equation is a curve, not a circle or a straight line either, so it cannot be constructed using a compass or any straight edge :)

Answer 3

beacuse when e=mc^2 on the 3 sunday of april and the moon is allined with saturn the answer is recessive

Answer 4

i dont no the answer to dat 1