### Algebraic and Transcendental

A number is called algebraic if it is the root of a nonzero polynomial with integral coefficients (or, equivalently, rational coefficients); otherwise it is transcendental.

Item: there exists a nonintegral c > 1 such that c^n 'converges to the integers': that is, for any eps > 0 there's an N > 0 such that c^n is within eps of some integer, for all n > N. (I think the golden mean was an example, but I can't find or remember the reference book at the moment.)

Item: it is apparently open whether any such number can be transcendental.

***

Next up, two questions of my own on algebraic numbers. Possibly easy, possibly silly, but I don't know the answers.

Define the interleave of two numbers

b = 0.b_1b_2b_3... and c = 0.c_1c_2c_3...

(both in [0, 1], and using the 'correct' binary expansions) as

b@c = 0.b_1c_1b_2c_2...

1) Suppose b, c are algebraic. Must b@c be algebraic?

2) The other direction. Suppose b@c is algebraic. Must b, c be algebraic?

In both cases, I am inclined towards doubt.

***

Final thoughts (and these are observations others have made as well)... computational complexity theory seems to have certain things in common with transcendental number theory:

-an interest in impossibility/inexpressibility results;

-markedly slow progress on seemingly basic

questions--is (pi + e) irrational? does P = NP?

-attempts to use statistical and otherwise 'constructive' properties as sieves to distinguish simple objects from complicated ones.

So, could complexity theorists benefit from interacting and learning from transcendental number theorists? If nothing else, looking back on their long historical march might teach us patience and an appreciation for incremental progress.

Item: there exists a nonintegral c > 1 such that c^n 'converges to the integers': that is, for any eps > 0 there's an N > 0 such that c^n is within eps of some integer, for all n > N. (I think the golden mean was an example, but I can't find or remember the reference book at the moment.)

Item: it is apparently open whether any such number can be transcendental.

***

Next up, two questions of my own on algebraic numbers. Possibly easy, possibly silly, but I don't know the answers.

Define the interleave of two numbers

b = 0.b_1b_2b_3... and c = 0.c_1c_2c_3...

(both in [0, 1], and using the 'correct' binary expansions) as

b@c = 0.b_1c_1b_2c_2...

1) Suppose b, c are algebraic. Must b@c be algebraic?

2) The other direction. Suppose b@c is algebraic. Must b, c be algebraic?

In both cases, I am inclined towards doubt.

***

Final thoughts (and these are observations others have made as well)... computational complexity theory seems to have certain things in common with transcendental number theory:

-an interest in impossibility/inexpressibility results;

-markedly slow progress on seemingly basic

questions--is (pi + e) irrational? does P = NP?

-attempts to use statistical and otherwise 'constructive' properties as sieves to distinguish simple objects from complicated ones.

So, could complexity theorists benefit from interacting and learning from transcendental number theorists? If nothing else, looking back on their long historical march might teach us patience and an appreciation for incremental progress.

Labels: general math, puzzles