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MAT 320 Spring 2008

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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

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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

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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

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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

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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

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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

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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

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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

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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 x 3 – 3x – 1

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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

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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

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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

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

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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

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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

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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

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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 x 5 – x + 1 This does not mean that the roots don’t exist as complex numbers

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