consider the field of all numbers. meaning, of course, the complex numbers. we say that a number is representable if we can describe it by a text (for example, by a binary coded string of 0′s and 1′s of finite length). the set of all representable numbers is countable, as there’s a surjection of the set of binary strings (which is countable) onto it. moreover, it is a field, as if a and b are representable numbers, we have that a + b, a − b, a ⋅ b and a / b are represented by strings as “sum of (description of a) and (description of b)”. obviously, every algebraic number is representable, so our field of representable numbers contains the algebrically closed field of the algebraic numbers. but then, our field also contains euler’s number e and archimedes’ constant π, so it’s strictly larger. this opens the question: how does it’s algebraic closure looks like? not too surprisingly, it turns out to be already algebraically closed: every element in its closure can be represented by “root of polynomial with coefficients (description of coefficients)”, as all coefficients are representable. hence, our field, being countable, is strictly larger than the smallest algebraically closed subfield of the complex numbers, but still countable. and it contains lots of transcendental numbers. isn’t that cool?
posts for september 27, 2007.
yesterday i fell in love. again. with a song, called into cold light, the first track on unholy‘s third album rapture. it catches me. absorbs me. makes me feel like floating, like sitting on a boat, moving on the water with high speed, like flying through space, stars passing by, infinitely, monotone, yet never boring, full of power, psychedelic. somehow reminds me a bit of dream theater‘s constant motion, in particular it’s video, as it looks to me as if everything is in motion, no way to stop, no way to rest, no way to look back. reminds me a bit of life. constantly moving forward in time, not able to stop for a moment, no possibility to rest. making decisions in every moment, some of them good, most of them not. what happened cannot be undone. the music’s intenseness increasing every second, then, guitars hitting in, increasing the pace. hell breaking loose. drop-dead gorgeous. stunning. haunting. so intense, so beautiful. then, fade to black. the stream flows away, leaving me in blackness. again.