irrationality and infinity

[AntonioLao]

irrational numbers, when expressed in decimals do not repeat nor terminate, which extend toward infinity. One of the widely used transcendental number called [math]\pi[/math] is also an irrational number. It is transcendental because it can never be a solution to any algebraic equation. But it is defined as the ratio of the circumference over the diameter of a circle.

Although the circumference can be approximated by adding the sides of a regular polygon inscribed , the only way that the sum can ever be the same is for the length of each side equal to zero or the number of sides is exactly infinite. But what is the sum of infinite number of zeros? This is a question that even the theories of infinitesimal calculi could never be able to provide a satisfactory answer. Not even the theory of limit, which requires the existence of an [math]\epsilon[/math]. The answer lies within the connection of zero and infinity once they are joint together as a common point as that of a serpent has completely swallowed itself and vanish into thin air!

[SinJin]

This sounds similar to the fibonacci sequence. This sequence can basically support itself through infinity by constantly having one number that can be added to the next. In fact if the sequence excluded 0 and started with 1, then you could invoke the rule to always take the current number (1) and add it to the previous number (nothing in this case) to get the next number. Starting with 1 and invoking this simple rule, you can get a self-sustaining infinity. However, if you start with just 0 and trying invoke that same rule, nothing will arise. Similar to what you said, the sequence involves some unknown jump from 0 to 1, which leads to infinity.

As a side note, in relation to this discussion I find it a strange coincidence that the ratio of each sequential fibonnaci number is phi, or at least gets infinitely closer to phi, which is also the golden ratio, supposively found throughout nature and the universe. Don't know if it means anything...

[AntonioLao]

reference:Keith Devlin, 'The Language of Mathematics: Making the invisible Visible,' page 141, Freeman, NY, 1998.

The golden ratio has a particularly intriguing representation as a fraction that continues forever, namely,

[math]1+\frac{1}{\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}} }[/math]

[Guille]

I always think about 0, infinity, irrational numbers...I have concluded that everything depends.

If you look at zero as a number, then it must be that anything times/plus/minus/divided by zero will give you a number. 1*0=1, 2*0=2....

But if you look at zero as a concept representing nothingness or absence, then, anything times/plus/minus/divide or any other mathematical work you can do to a number, will give you zero when this one is involved.

In the first of the cases, 0*0=0 like 1*1=1. But in the second case, 0*0=20 (not twenty, but 2 zeros (here zero is like a letter or constant). Although it makes no sense as concept because having two absence is equal to haivng no absence.

This again shows that mathematics is NOT a completely logical language. Although it is the nearest.

Yesterday I was searching on the web for information about the fine structure constant and someone had founded an equation with pi and phi leading to the fine structure of EM:


I think that all irrational important numbers have a connection. Actually, I think that all constants have connections. But there is something that science doesn't and can't explain yet, and it is WHY the constants have those numbers. why is pi always 3.1415926564... You might say that it is because circumference/diameter is always that, but this doesn't explain it. Why is it this number? This is one of the hardests questions in physics and mathematics, and I think it has never been answered, or has it?

[AntonioLao]

why is pi always 3.1415926564... You might say that it is because circumference/diameter is always that, but this doesn't explain it.

In Einstein's general theory of relativity, the above statement is false.

[Guille]

In Einstein's general theory of relativity, the above statement is false.

Do you mean the statement of pi beign 3.1415...? do you mean the statement that pi is circumference/diameter? Or do you mean the statement that saying the previous two, doesn't explain why pi is that?

if any, why?

[AntonioLao]

the circumference is not given by just the product of diameter and pi but a relativistic correction must be inserted into the calculations.

[Guille]

the circumference is not given by just the product of diameter and pi but a relativistic correction must be inserted into the calculations.

Why? What is this correction? I have used this formula many times and it has always given correct answers.

[AntonioLao]

What is this correction?

correction due to the spacetime curvature tensor. If the metric tensor has non zero off-diagonal element then the the quadratic form contains products of the components. For example

[math] d^2=x^2+y^2+z^2[/math]

becomes

[math]d^2=x^2+y^2+z^2+2xy+2xz+2yz[/math]

[Guille]

correction due to the spacetime curvature tensor. If the metric tensor has non zero off-diagonal element then the the quadratic form contains products of the components. For example

[math] d^2=x^2+y^2+z^2[/math]

becomes

[math]d^2=x^2+y^2+z^2+2xy+2xz+2yz[/math]

I see now.