The Three Greek Problems

1. The Three Greek Problems

2. The Three Greek Problems 1.) Doubling the Cube 2.) Trisecting the Angle 3.) Squaring the Circle

3. History of the Three Greek Problems These three problems have an interesting history, especially “doubling the cube”. According to legend, when ancient Athens was faced with a plague, they went to the oracle of Apollo at Delos for advice. The answer they received was to double the cubical alter to Apollo. However, instead of doubling the alter they doubled the length of each edge, making it eight times the original size. This has been called the “Delian problem” ever since.

4. Constructible Lengths The rules of constructible lengths are: 1.) Given 2 points, we may draw a line through them, extending it indefinitely in each direction. 2.) Given 2 points, we may draw the line segment connecting them. 3.) Given a point and a line segment, we may draw a circle with center at the point and radius equal to the length of the line segment.

5. Lemma 1 Given segments of lengths 1, a, and b, it is possible to construct segments of lengths a+b, a-b (when a > b), ab, and a/b (when b ≠ 0). a 1 ab b

6. A Field, F, is a set of numbers that is closed under addition, multiplication, and division. For example, rational numbers and real numbers are in a field. When two rational numbers are added, multiplied, or divided the answer is a rational number. The same is not true for integers.

7. If F is a field, a quadratic extension of F is the set such that a, b, and k are elements of F. EX. is an element of a quadratic extension of the rational numbers. This is because 2 and 3 are rational numbers.

8. WHEN IS A NUMBER CONSTRUCTIBLE ? A number, a, is constructible if and only if a Fⁿ where Q  F F² …  Fⁿ F¹ is a quadratic extension of Q, like Fi+1 is a quadratic extension of Fi.

9. Consider: Volume = L = 1 = 1 1 1 1 Volume = L = 2 Side =

10. LEMMA 2 If is in a quadratic extension of a field, then must lie in the field itself. If this is true, looks like or . Where a, b, and k are in the field.

11. If is in a quadratic extension of a field F, then it will look like this: =0 or =0

12. By the previous equations, can not be constructed because there is only one real root of . Since there is only one real root of , then b=0. When b=0, must equal a, which would mean that since a is in the field must also be in the field. That means that is a rational number, which is not true. This shows the contradiction.

13. Work Cited • Hadlock, Robert. Field Theory and Its Classical Problems. The Mathematical Association of America. 1978.