Mathematical Constants
By Jeff Lindsay on Monday, January 21,
2002 - 12:11 am:
In physics there are a small-ish number of physical constants
(such as the speed of light, the charge on the electron, the
gravitational constant and so on). One aim of the theoretical
physicist is to explain where the constants come from or, at the
very least, to reduce the number of fundamental constants to a
minimum as they appear (to humans) to be arbitrary.
My query comes in several parts. Firstly are there a finite set
of fundamental mathematical constants from which all other
constants can be calculated?
Certain constants can be defined in terms (infinite) series of
rational numbers. Could all constants be defined in this
way?
Certain constants can be defined in terms of the primes. Can all
constants be defined in this way?
Is it possible to compute the value of e in terms of p (or vice versa)?
Thanks
Jeff
By Gavin Adams on Monday, January 21, 2002
- 12:43 am:
1) There's one constant that everything can be calculated
from: 1.
2) Yes, all constants can be defined that way because any
constant can be expressed as a continued fraction. Whether or not
there's a pattern in the continued fraction i don't know - and i
guess i trivialized your question by stating this. As far as
patterns converging to a constant, i don't know and a proof of
this wouldn't be very intuitive...
3) 3-2 = 1 where 3 and 2 are primes. Every constant can be
constructed in terms of the number 1. (again i guess i
trivialized your question)
4)
|
[(-12/p2) |
¥
å
n=1
|
(1/n2)cos(9/(np+ |
Ö
|
n2p2-9
|
))]-3=e |
|
-Gosper
Crazy formula, but it relates p and e
By Dan Goodman on Monday, January 21,
2002 - 12:49 am:
The answer to the question of whether or not there is a
finite set of numbers which all others can be calculated from is probably no,
it depends what you mean by "calculated from". If we mean you're only allowed
to add, multiply, subtract and divide then no. Actually, you can go further.
If you only allow a finite set of numbers to start with, a finite number of
operations (like add, multiply, and so forth) and you require that you can get
to any number with a finite number of operations, then you can't get all the
numbers. In fact, in this setup you can hardly get any of the numbers that
there are. Do you know about the ideas of countability and uncountability in
sets? If so, I can explain why the above is true.
Next question: all numbers can be defined in terms of infinite series of
rational numbers. In fact, this is one way of defining what we mean by a
"real number".
Not sure what you mean by "can be defined in terms of the primes".
Final question: can you compute e from p? Well, it depends what
operations you are allowing. If you're allowing only a finite number of
multiplications, divisions, additions, subtractions, and that sort of thing,
then the answer is probably no (although I think it is unknown). The usual
way of defining this is by saying that two numbers x and y are
algebraically independent if there is no polynomial f(u,v) with coefficients
rational numbers such that f(x,y)=0.
By Dan Goodman on Monday, January 21,
2002 - 12:56 am:
In addition, in response to Gavin's post, continued
fractions are one way of finding a rational series converging to any number,
but there's an easier way. Just let pn = ë2n xû
(ëyû is the smallest integer less than or equal to y) and
qn = 2n. Then pn/qn is a rational and as n gets large pn/qn
converges to x.
The equation in (4) relating e and p is a bit of a cheat, because it
uses trigonometrical functions and infinite series. For example,
-2cos(p)cos(i)=e+1/e and you can probably work out something better.