How I Lost My Faith In Mathematics

I like to read math books and work math problems while I have lunch in my office.  I tried to solve this problem from The Contest Problem Book IV [1].

Problem 35 1973 Examination The Contest Problem Book IV

Since I didn’t make much progress save noticing the implicit decagon, I checked the answer in the back of the book.  It looked too complicated so I convinced myself that it was sufficient to find \sin 36 ^\circ and \cos 36^\circ.  I used this drawing to find \cos 36^\circ.

Diagram Used to Find Cos 36 degrees

Here \angle CAE is constructed to be 36 degrees.  Then use the fact that \triangle CAE is similar to \triangle CAD to get length \|CE\| which in turn yields \cos 36^\circ = \frac{1}{\sqrt{5}-1}.  Isn’t  this a beautiful looking result?

Now all I had to do was use the Pythagorean formula to find \sin 36^\circ = \sqrt{\frac{5-\sqrt{5}}{8}}. This result is not nearly so pretty.  I made it my lunch time objective to express \sin 36^\circ without the double root. (Technically this is called a nested root.)  I tried all the algebra tricks I could come up with and I noodled around with the diagram to no avail.  When I discussed my problem with the fellow next door, Nick Chura, he showed that the TI-Inspire gave a nested root.  So did Wolfram|Alpha.  My suspicions aroused, I went looking for a proof that denesting this particular expression was impossible.  A theorem in a paper by Allan Borodin, Ronald Fagin, John E. Hopcroft and Martin Tompa [2] applied.  The essence of Theorem 1 in their paper states that \sqrt{a+b\sqrt{r}} denests if and only if \sqrt{a^2-b^2r} is rational.  My expression failed this test.  The impetus for Borodin et al’s paper was the need for computer algebra system to simplify complex nested expressions if possible.  The authors used extension fields to prove their results and provided a denesting algorithm.

So the result of my investigations as still written in a corner of my office whiteboard is

\cos 36^\circ = \frac{1}{\sqrt{5}-1}.  Find \sin 36 ^\circ, no double nested roots allowed.

Proved impossible.

This is where I lost my faith in mathematics. Why should the numerical expression for \sin 36 ^\circ differ so significantly from the numerical expression for \cos 36 ^\circ?  Isn’t one function just a phase shift of the other?  Doesn’t  the difference between opposite over hypotenuse and adjacent over hypotenuse just depend on how you look at a right triangle? The beauty and symmetry that I had expected was proved impossible.  I experienced a sense of existential disappointment.

I looked for ways out of my dilemma.  Maybe the way we write radical expressions is flawed.  Maybe our number system obscures some type of fundamental symmetry.  The realistic solution however is that my limited abilities and lack of experience with extension fields excludes me from the pleasures that a more knowledgeable person would have contemplating the problem and the result. – Jim Hatton

[1]  R. Artino, A. Gaglione and N. Shell, The Contest Problem Book IV, Annual High School Examinations 1973-1982, The Mathematical Association of America

[2]  A. Borodin, R. Fagin, J.E. Hopcroft and M. Tompa, Decreasing the Nesting Depth of Expressions Involving Square Roots, J. Symbolic Computation, 1 (1985) 169-188.

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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2 Responses to How I Lost My Faith In Mathematics

  1. Gerard Jones says:

    If you stick “jim hatton” in the search thing at wordpress you get you. G.

  2. Fascinating thoughts, Jim; they sure get me thinking. Actually, when I ponder it a bit, I find it all quite poetic. I feel inclined to defend math here. If you’ll indulge me, let me give it a shot.

    The sneaky thing here, in my mind, is that the quantity of the phase-shift is pi/4, and pi is one of the most fabulously intriguing things I know. Indeed, shift the line f(x)=x by pi/4 and imagine how screwed up rational values, indeed ANY algebraic values become! So part of me thinks it’s lucky that nice values for one of sin/cos EVER turn out to be nice for the other.

    But then, gloriously, the Pythagorean Theorem is there, like a knight in shining armor, battling the dragon of that mysterious transcendental number and imposing a bit of order. From the dust and clamor of a furious battle, there emerges, at least, the double root solution. It’s almost as if the solution is just barely obeying, reluctantly, the laws of universe.

    It’s beautiful, really.

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