Simple proportion one that is a one to one function

[dynamic998]

is a simple proportion one that is a one to one function of some sort. I don't understand this term.

 

[Integral]

To say A is poportional to B is to say that there is some constant k such that A=kB. It makes no other statement or assumptions on the behavior of A and B.

With that said I am going to bump this into the Math Forum

 

[r637h]

I don't want to create a new thread. This thread may be a start for my question:

I'm trying to determine whether there's a relationship between Phi=Golden Mean, and the natural log base=e. If there's a relationship, (other than by a simple constant - not what I'm looking for) it is almost certainly irrational.

I'm running into a brick wall; perhaps because there is no relationship. Can anyone help?

Gratefully,

 

[HallsofIvy]

Of course, there's a relationship (there's a "relationship" between any two numbers!) but that's exactly what you say you are not looking for. It's pretty easy to calculate that phi= (1+ sqrt(5))/2 and don't think you'll find any simple relationship between an algebraic number like that and a transcendental number like e.

 

[plus]

Originally posted by HallsofIvy
Of course, there's a relationship (there's a "relationship" between any two numbers!) but that's exactly what you say you are not looking for. It's pretty easy to calculate that phi= (1+ sqrt(5))/2 and don't think you'll find any simple relationship between an algebraic number like that and a transcendental number like e.

Some people might have once said that about Pi!

Although this number does not seem to be as important as Pi.

 

[HallsofIvy]

pi is transcendental like e.

You are not going to find a "simple" relationship between an algebraic number and a transcendental number.

Of course, it is possible to make up some formula that changes phi into e. Given any two numbers, it's possible to make up a formula that will change one into the other. That's clearly not what r637h meant.

 

[r637h]

I would like to avoid number theory (very uncomfortable):

Phi is an algebraic number, granted, and satifies the quadratic, x^2-x-1=0. (Although that contains the irrational square root of 5.)

But e is transcendental, and cannot satisy a quadratic or any other algebraic expression, as far as I know.

But Phi is also an irrational number, which can satisfy a Taylor Series or be expressed as a continued fraction.

I thought a number had to be either irrational or algebraic. Is that the flaw in my thinking?

Or is Phi "unique", in that it is both? Surely not.

Anyway, Pi and e can easily be related, and although the relationship may be complex, can be demonstrated in several series.

Am I running around in circles? Is the reasoning non-sequetur?

Help!

 

[HallsofIvy]

No, a number is either "rational" or "irrational".

Both rational and irrational numbers can be "algebraic".

An algebraic number is any number that can be found as a root of polynomial equation with integer coefficients. (If "n" is the lowest possible degree of such a polynomial, the roots are "algebraic of order n".)

Rational numbers are precisely those numbers that are algebraic of order 1: x= a/b if and only if x satifies bx- a= 0, a polynomial equation of order 1 with integer coefficients.

Square root of 2, on the other hand, is not rational but is algebraic of order 2.

Phi, since it satisfies the equation x^2- x- 1= 0, is also algebraic of order 2.

Transcendental numbers are those numbers that are not algebraic of any order. Pi and e are the best known of those, but, technically speaking, "almost all" numbers are transcendental.

 

[r637h]

To Halls of Ivy:

Thanks. I was confusing "rational" with "algebraic."

Right: Order 2.

Any thoughts on "constructing" a relationship of Phi with e.(other than a simple constant)?

If Pi with e, then why not Phi with e?

Again, Thanks.

 

[plus]

Originally posted by HallsofIvy
pi is transcendental like e.

Am I meant to be aware of some result which states that there exists no 'simple' relationship between any transcendental number and a non transcental number.

Clearly e^1 = e,

Which gives a simple relationship between e and 1.

I think that we require a working definition of 'simple'.

 

[r637h]

As it turns out, relating phi to e is simpler than I thought:

phi=2cos(pi/5), etc. and the identity:
e^(i.pi)+1=0, (special case of Euler's Formula, with x=pi)

From there, a simple matter of substitution.

Now, the answer may be a complex number (I haven't figured it out, yet), but at least the relationship is expressed.

Now comes a tougher job: Trying to relate fundamental (or natural) # like i,pi,e,phi, to physical # such as Planck's constant, c, gravity acceleration, Avogadro's Number, etc.

Any thoughts?