It's just weird how elliptic integrals arise from the arc length of an ellipse $s = \int ds = \int \sqrt{dx^2+dy^2} = \int \sqrt{(dx/d\theta)^2+(dy/d\theta)^2}d\theta$ but also when you compute the arc length in polar coordinates $s = \int ds = \int \sqrt{dr^2+r^2 d \theta^2} = \int \sqrt{(dr/d\theta)^2+r^2}d\theta$ and then randomly throw away $(dr/d\theta)^2$ to get the angular arc length $u = \int r d \theta$ right? I mean this weird angular arc is the argument of elliptic functions

its been the first time i see this chatroom at the bottom of math' chatrooms page

Why is pii considered a constant term in experssions? Surely pii goes on for infinity so is it really right to say its constant if we don't know its true value

@WDUK What does that even mean?

What do you think it should vary wrt.?

well it varies depending on how precise you want to be

so its not exactly constant

What?

what part are you confused by ?

Everything. You're not making sense...

i don't see how

a constant term is something like +4 in an expression

The definition of $\pi$ is independent of how many decimal places you write down.

but pii is never truely defined as we don't know its full value

Sure we do.

so you know the final digit in pii?

Pick your favorite infinite series.

@WDUK Not relevant.

of course its relevant

how is a never ending number a fixed number

it never ends

@WDUK What does this even mean?

depending on different precision that you choose in a calculation makes it more variable than constant

FWIW, it's "pi" not "pii".

@WDUK First of all, not all real numbers (aka constants) have a "definite" decimal expansion. Fractions do: the decimal expansions are always periodic. But eg $\sqrt{2}$ doesn't: does that make it non-constant? There exists constants the decimal expansions of which we cannot predict: they are still and always will be constant. The point is one can approximate them to arbitrarily well accuracy by taking longer and longer decimal expansions - that's all what is important.

Secondly, there is no "final digit of $\pi$" because that's nonsense. An infinite decimal expansion has no final digit. What's the final digit of $1/7$?

thats why i asked the question cos we don't know the final digit so we don't know its final value

i wasn't asking it literally

So we don't know the 'final value" of $1/7$?

not precisely we don't

because it just goes on forever

I think your understanding of "final value" is different from mine then.

most likely

By that logic we also don't know the final value of $1/3$.

It also goes on forever.

it's 3

for me a final value for pi is like asking the last digit in an infinite series of numbers - there isn't one so its never has a final value

you could only get infinitely more accurate at best

or precise if you prefer that word

i am getting ruthlessly fucked by my combinatorics homework

@WDUK Just work in base $\pi$, then $\pi$ has one digit.

Problem solved.

what do you mean base ?

@WDUK For sure. But there is no such thing in mathematics as "final value", which I think is the root of our confusion and debates. Nobody says that terminology.

well i mean 2+2 has a final value of 4 surely

Why not 3.9999... ? That's equal to 4.

even numbers with repeating series of decimals could arguable have an ending because we know the numbers repeat but doesn't pi have decimals that never repeat?

@WDUK Yes, but there do exist algorithms to determine, given any $n$, the $n$-th term in the decimal expansion of $\pi$.

Curveball: if you can represent a number by a series $\sum_n a_n$ then $\lim_{n \rightarrow \infty} a_n = 0$

yeah but when we use pi in equations its never expressed to what precision so in that regard is it really constant unless you define how precise in the equation

Sorry, that's vague and we're drifting away from mathematics.

well that was the original point of my question

The original point of the question was it's nonmathematical?

what subject have we drifted into =/

The kid in this video forgot to check that limit :p : youtube.com/watch?v=hBW4S9xcTOk

Because 'final value' seems to be a vague philosophical term, as far as I can see.

And here is the link to what I was talking about.

the final value would be the last number in a value with nothing after it which pi won't have obviously

@WDUK So 3.9999... has no final value?

no it doesn't you would have to define how many decimal places to make it a constant in an equation

It's equal to 4 though.

well i don't fully understand why it equals 4 but okay..

Which you said before has a final value (aka 4). So it both has a final value and doesn't at the same time?

can you explain how that is equal to 4?

ouch my brain

Your definition of 'final value' is not quite well-defined is the point. That's how real numbers work: a rational number is "explicit" (whatever that means), but in general a real number isn't. In fact as you move towards foundations of mathematics more, you'll see real numbers are defined by limit of a sequence of rational numbers (e.g., the limiting value of it's iterated decimal expansions). This does not mean they don't "have a specific value" or "they aren't constant".

It's the first obstacle one comes up with when going from rational to irrational numbers. It's a bit confusing, but learn to live with it.

When do they teach stuff like that? university?

Any high school math course worth it's salt should teach it, in my humble opinion.

@BalarkaSen I've never heard of that being high school material

It's intro analysis material

0.9999... being 1 is analysis intro material? Huh?

Oh, no

I thought you were talking about reals = equiv. classes of Cauchy sequences of rationals

Ah, that. No, 'course not.

Not in full generality of course. one would invariably end up mentioning a variant of that if one wants to teach real numbers in high school.

when do they introduce imaginary numbers?

In high school, certainly.

Agreed

@arctictern what's the standard bi-invariant inner product on $\mathfrak{so}(3)$?