Simple proportion one that is a one to one function

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In summary, a simple proportion is one that is a one-to-one function between two variables, typically represented as A=kB. This does not make any other assumptions about the behavior of the two variables. The conversation then moves on to discussing whether or not there is a relationship between the Golden Mean (Phi) and the natural log base (e). While there is technically a "relationship" between any two numbers, the person asking the question is looking for a more complex relationship than a simple constant. It is then explained that Phi is an algebraic number of order 2, while e is a transcendental number. The conversation then turns to discussing various relationships between Phi and e, with the conclusion being that while there is a relationship, it
  • #1
dynamic998
is a simple proportion one that is a one to one function of some sort. I don't understand this term.
 
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  • #2
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
 
  • #3
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,
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 
  • #7
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!
 
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  • #8
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.
 
  • #9
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.
 
  • #10
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'.
 
  • #11
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?
 

What is a simple proportion?

A simple proportion is a mathematical relationship between two quantities that states that the ratio of the first quantity to the second quantity is equal to the ratio of two other quantities. This relationship can be expressed as a:b=c:d, where a and c are the first pair of quantities and b and d are the second pair of quantities.

What is a one to one function?

A one to one function is a mathematical function in which each element in the domain is paired with only one element in the range. This means that for every input, there is only one corresponding output. In other words, the function has a unique output for every input.

How do you determine if a function is a one to one function?

To determine if a function is a one to one function, you can use the horizontal line test. If a horizontal line can intersect the graph of the function at only one point, then the function is a one to one function. Another way is to check if the function passes the vertical line test, meaning that a vertical line can only intersect the graph at one point.

What is the importance of a one to one function?

A one to one function is important because it ensures that each input has a unique output. This is useful in many real-world applications, such as in finance and economics, where there should be a one-to-one relationship between different variables.

How can simple proportions be used in real life?

Simple proportions can be used in many real-life situations, such as cooking, where a recipe may call for a certain proportion of ingredients. They can also be used in financial calculations, such as calculating interest rates or exchange rates. In addition, simple proportions can be used in scaling or resizing objects, such as maps or blueprints.

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