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Impossible Numbers

MAT 320 Spring 2008. Impossible Numbers. Old Problems. You may remember from geometry that you can perform many constructions only using a straightedge and a compass These include drawing circles, constructing right angles, bisecting angles, etc.

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Impossible Numbers

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  1. MAT 320 Spring 2008 Impossible Numbers

  2. Old Problems • You may remember from geometry that you can perform many constructions only using a straightedge and a compass • These include drawing circles, constructing right angles, bisecting angles, etc. • But there are other problems that the ancient Greeks wanted to try to solve with this method

  3. Famous Problems • The Greeks wanted to know if any of the following were possible • Trisecting the angle: Given an angle, divide it into three congruent angles • Doubling the cube: Given a cube, construct another cube with exactly twice the volume • Squaring the circle: Given a circle, create a square with the same area

  4. Impossible!? • It turns out that all of these constructions are impossible • In order to understand why, we need to think about how constructions really work • We start with two points, (0, 0) and (1, 0) • We say that we can “construct” a point (x, y) if we can find that point as an intersection of lines or circles that we can construct

  5. The Rules • The things we can construct are • Lines: We can use our straightedge to construct a line between any two points • Circles: Given two points, we can construct a circle with the center at one point and which passes through the other • Perpendiculars: Given a line and a point, we can construct a perpendicular line that passes through the point

  6. “Constructible” Numbers • We say that a number is “constructible” if it is the x or y-coordinate of a constructible point • For example, all of the integers are constructible

  7. Square Roots • The number is also constructible, since the point is the intersection of the first two circles on the previous slide • In fact, the set of constructible numbers is closed under addition, subtraction, multiplication, division, and square roots

  8. The Field of Constructible Numbers • The set of constructible numbers forms a field that contains the rational numbers • This field contains only those numbers that can be obtained from (possibly repeatedly) extending Q with the roots of quadratic polynomials

  9. A More Complex Example • For example, Gauss showed that • Since this number is constructed out of rational numbers and square roots, this number must be constructible • We can use this fact to construct a regular 17-sided polygon

  10. The Original Problems • Let’s think about trisection of an angle, specifically a 60-degree angle • 60-degree angles are constructible: cos(60) and sin(60) are both constructible numbers • What about 20-degree angles? • Using trig identities, it’s possible to show that cos(20) is a root of the polynomial x3 – 3x – 1

  11. cos(20) is Not Constructible • Since the polynomial for which cos(20) is a root has degree 3, that means that cos(20) will involve cube roots, which aren’t allowed • So cos(20) is not a constructible number, and 60-degree angles are just one example of angles we cannot trisect with straightedge and compass

  12. Doubling the Cube • Given a 1 x 1 x 1 cube, we would need to construct a x x cube to have exactly double the volume • But is not a number we can construct, so we wouldn’t be able to create a segment exactly units long to create our cube

  13. Squaring the Circle • Given a circle of radius 1 (and area π), we would need to construct a square whose sides have length the square root of π • Even though square roots are allowed, π is not a rational number • It turns out π is a transcendental number, which means it’s not the root of any polynomial with rational coefficients

  14. Other Impossibilities • Another famous impossibility that is related to these ideas is credited to Niels Abel (1802-1829) • He proved that there is no way to solve a generic fifth-degree polynomial using radicals (even allowing 5th roots!)

  15. Wait a Minute… • Of course, some quintics are solvable using radicals • An example is , whose roots are 1 (twice), -1, i, and –i • But what Abel proved is that there is no analogue to the “quadratic formula” for quintics

  16. The Ideas Behind Abel’s Proof • Abel’s proof is beyond what we have learned in this course, but here are some related ideas • Have you ever noticed that roots of polynomials tend to come in groups? • For example, if you know that is the root of a quadratic, you can be sure that is also a root

  17. Roots are Related • It turns out that this is no accident • The roots of higher degree polynomials are related in more complicated ways, but they are still related • Once the degree reaches 5, the relationships become so complicated that there is sometimes no way to “unentangle” the roots from one another

  18. “Impossible” Is a Strong Word • Keep in mind that we can still solve quintic equations using numerical methods • The issue is that some quintic equations have roots that we cannot express with our normal radical notation • One example is x5 – x + 1 • This does not mean that the roots don’t exist as complex numbers

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