b. pi as a decimal number is an approximation that is only as long as you calculate it to be. Moreover, decimals do not contain non-numeric information. That's just your own interpretation. Imagine a number system where pi is 1. Clearly no names in 1. Now imagine a number system where pi is expressed as π, like... the number system that we use. Clearly no names in π.
But Pi is not equal to our approximations of Pi. That's backwards. Pi owes nothing to our lame efforts to approximate it. When someone asserts that Pi may be an infinite sequence, the truth of the proposition doesn't depend on someone computing an infinite sequence of Pi's digits.
> Moreover, decimals do not contain non-numeric information.
Are you aware that your typing is promptly translated into numbers for transmission through the Internet? That A = 65 (in the old ASCII encoding), B = 66, etc.? So yes, decimals do contain non-numeric information if we choose to interpret them that way. And we do.
Of course it is -- although choice of base is quire arbitrary. Pi is as much a decimal number as it is a binary number.
> Every decimal representation of pi is an approximation.
Our inability to represent Pi doesn't constrain Pi, it only constrains us. The fact that we cannot fully represent Pi has no effect on Pi or its identity. And Pi has the same identity in any base -- the ratio of a circle's circumference to its diameter.
> Pi is not an infinite sequence
Pi is an infinite sequence.
http://en.wikipedia.org/wiki/Pi#Infinite_series
Quote: "The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49] Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques."
> Pi is not an infinite sequence
Pi is an infinite sequence.
I was talking about an infinite number of digits, but even this new definition is wrong. Pi is one single value. There are many ways to compute this value. Some of them include infinite series. Note the wording: "The calculation of π was revolutionized by the development of infinite series techniques". Other techniques were used before, and they are still valid. The only advantage of infinite series is that they are easier to calculate, and the approximations converge faster when you compute them on real (finite) hardware.
That's your problem, not mine. You keep trying to say that Pi is defined by our approximations of its value.
> Pi is one single value.
So, tell me, which part of "Pi is an infinite sequence" are you actually disagreeing with?
> There are many ways to compute this value.
You persist in confusing Pi with our efforts to approximate its value.
> Other techniques were used before, and they are still valid.
False. 22/7 is not valid. 355/113 is not valid. This is valid:
https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png
The reason? Pi is a mathematical idea that happens to have a numerical value, but the idea transcends the value. The value is a coincidence, which is why choosing to express it as 1 to the base Pi changes nothing, and why arguing about the size of its approximations changes nothing.
> The only advantage of infinite series is that they are easier to calculate ...
No, they are much more difficult to calculate, but they convey more meaning. Infinite series are why Pi isn't just a number, any more than e is.
Right, those are approximations. Those are not what we are talking about.
> This is valid: https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png
I agree that it is valid. It is one of many ways of calculating the value of pi. Here is another one: http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-... Also there are iterative algorithms. http://en.wikipedia.org/wiki/Pi#Computer_era_and_iterative_a...
> No, they are much more difficult to calculate, but they convey more meaning.
Again, I didn't mean approximations like 22/7. I was comparing them to e.g. the geometric method which took hundreds of years to extend to a few hundred digits. The geometric method conveys the same "meaning" because it is an exact description of pi, just like the infinite series and the iterative algorithms.
> Pi isn't just a number, any more than e is.
They are both just numbers. e is not infinite, in fact it is less than 3.
No, they are ideas. The base of natural logarithms isn't an arbitrary number, it has special properties. It's the same with Pi -- they're ideas that happen to be expressible as numbers. But their numerical value is less important than their identity as ideas.
> e is not infinite, in fact it is less than 3.
Straw man. No one claimed otherwise. But e appears to have an infinite digital sequence, i.e. is "normal" in the mathematical sense.
But, as with Pi, the fact that e is likely "normal" is much less important than the idea it represents.
That would be because Pi is special because there are infinite series, and integrals, and mathematical identities, and limit expressions, that describe it in ways that give it a special meaning.
> So the number 2 is just as "infinite" as pi.
You're confusing the existence of a summation with its outcome. Obviously the sum of 2^-n for n between 0 and infinity (inclusive) is equal to 2, but that doesn't make 2 an infinite digital sequence, or in any other sense "infinite".