Is There a Significance to the Imaginary Number in the Series for Pi/4?

[arydberg]

I find this interesting.
You can approximate pi/4 with the Gregory and Leibniz series pi /4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 ... (1)
btw it takes a lot of terms to get a reasonable approximation for pi. The formuli is pi / 4 = [ ( -1 ) ^ ( k + 1 ) ] / ( 2 * k -1)

But there is another simpler equation.

pi / 4 = [ ( i ) ^ ( k - 1 ) ] / ( k )

where k = 1 2 3 4 5 ... if we assume that simpler is better we get the terms 1/1 i/2 -1/3 -i/4 1/5 ... or the same answer as (1) for pi / 4 in the real term and another number in the imaginary term .

To sum the terms of k = 1 to 10,000 using Matlab . I get 3.1413926 + 1.386094 * i .

So my question is, is there any significance to the number 1.386094?

(Yes i know it is simply the sum of the terms 1/2 - 1/4 + 1/6 - 1/8...) but I am looking for something more.

 

[andrewkirk]

It's suspiciously close to $\frac{e}{2}$, yet not quite close enough - it misses by just over 2%, which is well above the error for that number of terms.

I suggest looking into the derivation of that complex formula for pi. That will probably suggest some meaning for the imaginary part.

 

[fresh_42]

So my question is, is there any significance to the number 1.386094?

This is $4 \cdot \ln{|1-i |} = 4 \cdot \ln \sqrt{2} = \ln 4$.

$$\sum_{k=1}^{\infty}\frac{(i)^{k-1}}{k} = -i \cdot \sum_{k=1}^{\infty}\frac{(i)^{k}}{k} = i \ln(1-i)
= i \ln (\sqrt{2} \cdot e^{-i \frac{\pi}{4}}) = i (\ln \sqrt{2} + (-i \frac{\pi}{4})) = \frac{\pi}{4} + i \ln \sqrt{2} $$

with one branch of the logarithm and thus $\pi = (4 \sum_{k=1}^{\infty}\frac{(i)^{k-1}}{k}) - i \ln 4$.

 

[HallsofIvy]

Technically, the set of all real numbers is a subset of the complex numbers- every real number, including $\pi$, is a complex number.

 

[andrewkirk]

Technically, the set of all real numbers is a subset of the complex numbers

I suppose it depends on how technical we want to be. If we get a bit more technical still - moving on into the most common set theoretic construction of the numbers - it becomes not a subset again, because the real numbers is a set of Dedekind Cuts and the Complex Numbers is a set of ordered pairs of Dedekind Cuts. The real numbers are isomorphic to the subset of the complex numbers whose second coordinate is zero, and in Complex Analysis we usually identify the two, but set-theoretically they are different..

 

[jbriggs444]

I suppose it depends on how technical we want to be. If we get a bit more technical still - moving on into the most common set theoretic construction of the numbers - it becomes not a subset again, because the real numbers is a set of Dedekind Cuts and the Complex Numbers is a set of ordered pairs of Dedekind Cuts. The real numbers are isomorphic to the subset of the complex numbers whose second coordinate is zero, and in Complex Analysis we usually identify the two, but set-theoretically they are different..

The question of whether "$\pi$" is or is not a "complex number" ought not depend on whether the set theoretical construction that we choose to demonstrate the consistency of the axioms for "complex numbers" uses Cauchy sequences or Dedekind cuts, uses a copy of the natural numbers starting at zero or at one, defines the rational numbers using ordered pairs with the numerator first or with the denominator first or any of a host of ultimately equivalent approaches.

In mathematics as in every other field of human endeavor, context counts. In a context where one is contemplating any particular model of the complex numbers then "$\pi$" denotes the complex number that fills the role of $\pi$ in that model. So yes, $\pi$ is a complex number regardless of technicalities.

 

[andrewkirk]

I absolutely agree that context counts jb. In fact that was my point.
In the original context 'complex' was being used to distinguish from real, so that 'complex' in that context meant 'cannot be fully characterised as a real number'. In that context pi is not a complex number.
If we shift the context to the subset of the complex numbers that is isomorphic to the reals then we can say that the image of pi in that subset is a complex number. But that is a different context from that of the original question.
The point I was making was that, with one shift of context, we change the answer to 'is pi a complex number?' from 'No' to 'Yes', and then with an additional shift of context we change it back to 'No'.

 

[micromass]

I absolutely agree that context counts jb. In fact that was my point.
In the original context 'complex' was being used to distinguish from real, so that 'complex' in that context meant 'cannot be fully characterised as a real number'. In that context pi is not a complex number.
If we shift the context to the subset of the complex numbers that is isomorphic to the reals then we can say that the image of pi in that subset is a complex number. But that is a different context from that of the original question.
The point I was making was that, with one shift of context, we change the answer to 'is pi a complex number?' from 'No' to 'Yes', and then with an additional shift of context we change it back to 'No'.

Give me one reputable source that says $\pi$ is not a complex number.

 

[andrewkirk]

Give me one reputable source that says $\pi$ is not a complex number.

In my experience, reputable sources don't waste time on trivial issues of taxonomy like this.

 

[Mark44]

Give me one reputable source that says $\pi$ is not a complex number.

In my experience, reputable sources don't waste time on trivial issues of taxonomy like this.

This doesn't seem worth arguing over to me. One might consider $\pi$ to be a complex number whose imaginary part is 0. Someone else might consider it to not be complex for precisely this reason.

Can we end this dispute, please?