The Straight Dope currently has a column about the golden ratio and a staff report about Fibonacci numbers. These have prompted a thread on the Straight Dope Message Board which raises a couple of mathematical questions to which I’d like to post my answers. But the SDMB is now a pay site, and I’m not about to pay $14.95 to share my knowledge.
A logical place to respond to the Straight Dope is alt.fan.cecil-adams, but I swore off posting there.
Well then, I’ll post here.
One of the questions is the relation between transcendental and irrational numbers. An algebraic number is a solution to a polynomial equation of the form a0 x0 + . . . + an xn = 0, where the coefficients ai are integers (whole numbers, not necessarily positive). If n=1 then x = -a0/a1, a rational number. Real numbers other than algebraic numbers are called transcendental. (Are there transcendentals which can be considered roots of integer polynomials of infinite degree?) As the rationals are a special case of the algebraics, it follows that a transcendental number cannot be rational.
The other interesting question is what it means to describe φ (phi) a.k.a. τ (tau) as the most irrational number. Any real number can be expressed as a continued fraction — and I see that that page says all that I was about to say on the subject.
