For an infinitely small circle with an area greater than zero, is pi still defined?
Your question has a logical contradiction, and if it doesn’t, is certainly not answerable in the domain of real numbers.
Good luck with that.
For a circle with an infinite area, is pi still defined?
Assuming that “infinitely small” means “arbitrarily close to zero” - sure. It’s pi. The ratio of diameter to circumference is always a constant value, so the limit as diameter-> zero is easy to find
If it has an infinite area, it’s not a circle. Same if it has an area of zero.
It sounds to me like you’re trying to ask “what is the limit of the ratio of a circle’s diameter to its circumference as the area approaches zero?” In that case, the limit is pi.
In addition, this has nothing to do with the pi being transcendental. The ratio of the perimeter of a square to one of its sides is 4 no matter how big or small. 4 is an integer. The ratio of the perimeter to the diagonal is 2*sqrt(2) an irrational number but not transcendental.
The phrasing of these questions makes it sound as if you have the idea that each circle has its own value of pi. I think the mainstream view is that pi is a constant.
Is that the same if the area approaches infinity?
Yes. Pi is Pi, by definition; it happens to be the ratio of a circle’s diameter to its circumference if that circle is drawn on a flat plane, no matter how large or small, as long as the circle has some nonzero and finite area; if you draw a circle on a sphere or some other non-flat surface, the ratio of its diameter to its circumference will fail to be Pi. Pi doesn’t change, you just drew something which isn’t a circle on a flat plane.
What does any of this have to do with pi being transcendental
?
Absolutely nothing.
Say it again!
I’m not sure that I understand your question. By definition the circumference of a circle is given by C = [symbol]π[/symbol]D and so the ratio is C/D = [symbol]π[/symbol]. Are you asking what the limit of a constant is as some other value approaches infinity?
Pretty much answered in #7 – nothing.
Might want to be clear on what sort of limit you are trying to compute. Limits have a particular definition so how you ask the question is pretty important. This is true of most things in math. You might think you are asking one question but a mathematician may parse it differently.
Also have no clue what this has to do with pi being a transcendental number.
It may help to learn just what is a transcendental number: “In mathematics, a transcendental number is a real number or complex number that is not an algebraic number—that is, not a root (i.e., solution) of a nonzero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e.”
Pi (π) is transcendental and irrational but it is a RATIO of constants, only one of which can be an integer. I doubt that pi floats around much when its circle changes size from infinite (uncountable) to infinitesimal (negligible), not even if warped into extra dimensions. Well, in biblical text, a well’s diameter may be 3x its diameter, but that’s a miracle. 
And no, Illinois didn’t pass a law that pi=3.0.
Pi, Huh!
For any circle in flat space with non-zero finite size, the ratio of the circumference to the diameter is the same, and is equal to a single value somewhere in between 3.1415926535897932384 and 3.1415926535897932385. We can then take the limit of this as the diameter approaches infinity, or zero, or anything else, because a constant is the easiest thing in the world to take a limit of. And if you don’t understand what I’m talking about with limits, then you need to take a calculus course before you’re even qualified to ask the questions you’re asking.
Oh, and don’t get too hung up on that qualification of “in a flat space”. The real world isn’t a flat space, and that’s completely irrelevant to this question, because the real world doesn’t contain any perfect circles, either. The only perfect circles are the abstractions we talk about in math, and we can set them in whatever kind of space we want, and the space most often wanted (because it’s the simplest one) is a perfectly flat space.
I get more hung up on the idea that Pi is the ratio of two dimensions of a circle. I like the second sentence of Wikipedia’s article on Pi:
“Originally defined as the ratio of a circle’s circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.”
For one thing, circles don’t have Pi as the exact ratio of their circumference to diameter if there is any mass in the neighborhood, which Chronos may have been alluding to. This might appear a fastidious nitpick, but hey, we’re talking about a number that has been calculated to “many trillions of digits” (same article).
IMHO the definition in terms of the ratio of two dimensions sounds almost like an experimental result.
It was Indiana, and the actual story is still dumb.
Not quite. Chronos was referring to two different things (1) the fact that mathematical abstraction doesn’t rely on the real world and (2) to the extent we know anything, it’s that the real world doesn’t conform to Euclidean geometry (the sort of geometry we teach the kids). Hyperbolic geometry fits better, but at relatively small scales, things act much like they do in classic geometry.
A circle is a circle. The phrase ‘mass in the neighborhood’ does not compute - there isn’t such a thing as ‘mass’ in geometry.
That leads to the question of how much the real world can be accurately modeled by classic geometry, but that’s an entirely different discussion.