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 and are constructible… (and )

If and are constructible…

If and are constructible… (and b>1)

If and are constructible…

… then is constructible!

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.

If is constructible…

…then is constructible!

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 + b and y = cx + d be 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 d are constructible, x and y already 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 + b and be constructible (a, b, c, d and e are 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, g and 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 3 rd 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 19 th 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 me at Presentation available (under “links”) at