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Constructible Numbers
By Brian Stonelake
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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
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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.
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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?
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Handy Tricks
It may help to note that we are able to:• Construct a perpendicular bisector• Drop a perpendicular• Bisect an angle
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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?
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If and are constructible…(and )
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If and are constructible…
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If and are constructible…
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If and are constructible…(and b>1)
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If and are constructible…
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… then is constructible!
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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 .
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So What is Constructible?
• All of the rational numbers• Is that it? Presentation’s over?NO!• Other square roots?• Any square root?• Cube roots?
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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?
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Quick Timeout
• Before we continue constructing numbers, it may be helpful to prove a quick lemma about angles inscribed in semi-circles.
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If is inscribed in a semi-circle…
What can be said about ?
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If is inscribed in a semi-circle…
The interior angles of triangle ABC must sum to 180.Thus So Therefore is a right angle.
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Back to the Game
• Before the timeout, we were wondering if it was possible to take the square root of an arbitrary, constructible number.
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If is constructible…
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If is constructible…
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If is constructible…
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If is constructible…
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If is constructible…
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…then is constructible!
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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:
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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.
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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:
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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 + dax – cx = d-bx(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.
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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.
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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!
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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.
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Back to the problems of antiquityTask 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.
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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 [email protected])
Presentation available (under “links”) at http://webpages.sou.edu/~stonelakb/math/index.html