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Constructible Numbers. By Brian Stonelake. The Three Problems of Antiquity. Roughly 2500 years ago, the Ancient Greeks wondered if it is possible to: Square the circle : construct a square with the exact same area as an arbitrary circle
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Constructible Numbers By Brian Stonelake
The Three Problems of Antiquity • Roughly 2500 years ago, the Ancient Greeks wondered if it is possible to: • Square the circle: construct a square with the exact same area as an arbitrary circle • Double the cube: construct a cube with exactly twice the volume of an arbitrary cube • Trisect an angle: split a given angle into three equal angles
Constructible Numbers • To understand what makes these ancient tasks so interesting, we need to understand which numbers are “constructible.” • A number is constructible if it is possible to construct a line segment of length using only a compass and a straightedge.
Rules of the Game • The Bad News - You are given only a terrible compass, and the world’s worst ruler. • The Good News – You are the best artist ever; you are infinitely precise, have a perfect memory and all the time in the world. • So, which numbers can you construct?
Handy Tricks It may help to note that we are able to: • Construct a perpendicular bisector • Drop a perpendicular • Bisect an angle
First Important Construction • We want to be able to add, subtract, multiply and divide any constructible numbers. • Adding and subtracting is easy. • To add a and b, start constructing a at the end of b. • To subtract b from a, construct b in the opposite direction of a. What about multiplying and dividing?
If You’re Picky • If , double repeatedly until it is not. Use in the prior construction to get . Double that repeatedly to get . • If you want to multiply, finding is now easy. Just find , and then find .
So What is Constructible? • All of the rational numbers • Is that it? Presentation’s over? NO! • Other square roots? • Any square root? • Cube roots?
Second Important Construction • We can find certain square roots like . • We can find square roots of sums of constructible squares like , using The Pythagorean’s Theorem What about the square root of ANY constructible number?
Quick Timeout • Before we continue constructing numbers, it may be helpful to prove a quick lemma about angles inscribed in semi-circles.
If is inscribed in a semi-circle… What can be said about ?
If is inscribed in a semi-circle… The interior angles of triangle ABC must sum to 180. Thus So Therefore is a right angle.
Back to the Game • Before the timeout, we were wondering if it was possible to take the square root of an arbitrary, constructible number.
Quick Refresher on Algebraic Numbers • Algebraic numbers are all numbers that are roots of a polynomial with integer coefficients. • Examples of algebraic numbers and their minimal polynomials:
What do we have so far? Let’s call the set of constructible numbers . What is in ? - All rationals, i.e. - Square roots of rationals, i.e. • Sums of square roots of rationals, i.e. • Square roots of sums of square roots of rationals, i.e. • Lots of stuff. Put semi-succinctly, so far contains all integers, and finite iterations of sums, products and roots of any elements of . Note: So far, is a subset of the algebraic numbers, and (importantly) all “minimal polynomials” of elements of have degree a power of two.
What else?... Nothing! (hopefully) • Let’s switch from geometry to algebra. • To add numbers to , we really just intersect constructible lines and circles. • Recall from Math 111: • Equation of a line: • Equation of a circle:
Case 1: Intersecting two lines Let y = ax + band y = cx + dbe constructible lines (a, b, c and d are constructible). Solving for their intersection we substitute for y to get: ax + b = cx + d ax – cx = d-b x(a - c) = d – b so and . But if a, b, c and dare constructible, xand yalready were. Thus we can’t add anything to our set by intersecting lines.
Case 2: Intersecting two circles Let and be constructible circles (a, b, c, d, e and f are constructible). By subtracting one equation from the other, we get: (ax + by + c) – (dx + ey + f) = 0 (a – d)x + (b – e)y + c – f =0 (b – e)y = (d – a)x + (f - c) Which is just a line with constructible slope and intercept. Thus, intersecting two circles can’t give us anything that intersecting a line and a circle doesn’t.
Case 3: Intersecting a line and a circle Let y = ax + band be constructible (a, b, c, dand eare constructible) If we substitute for y in the second equation, we get: Which, if you expand and collect terms becomes Which is really just , for constructible numbers f, gand h. So we can solve for: and Which are already constructible. So our set can go no larger!
Back to the problems of antiquity • Task A: Squaring a circle: We can certainly construct a circle of radius and thus area . A square with area would have sides of length . Because is transcendental (not algebraic) so is . Thus is not constructible, and squaring an arbitrary circle is impossible. • Task B: Doubling a cube: We can certainly construct a cube with sides of length . If we doubled the cube, it would have volume and sides of length . The minimal polynomial of has degree 3, so is not constructible and doubling an arbitrary cube is impossible.
Back to the problems of antiquity Task C: Trisecting an arbitrary angle. I claim we can construct a angle (how?). If we could trisect constructible angles, we’d be able to construct a angle. Thus would be constructible. Using the triple angle formula, we have: Which has no rational roots because 1 and -1 don’t work (rational root theorem), so the minimal polynomial is 3rd degree, so we cant construct and thus can’t construct . This means we can’t construct a angle so trisecting an arbitrary angle is impossible.
Conclusion • Interestingly, the impossibility of these constructions wasn’t proven until the 19th century – Gauss credited with much of it. • The set of constructible numbers has interesting applications in Abstract Algebra, specifically Field Theory. • If you have further questions, my office is right over there (or you can email me at brianstonelake@gmail.com) Presentation available (under “links”) at http://webpages.sou.edu/~stonelakb/math/index.html
