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Sorry for the retarded question, but isn't...
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You are currently reading a thread in /sci/ - Science & Math

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Sorry for the retarded question, but isn't pi:

4/1 - 4/3 + 4/5 - 4/7 + 4/9 - ... = Pi

Similar to:

1/2 - 1/4 - 1/8 - 1/16 - ... = 0

Pi, like 0 will have a number you can write out completely, we just don't know how to work out the exact circumference of a circle so we use approximations.

Am I right in thinking this, /sci/?
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>>7802104
So, how is high school going? Still trolling le /b/?
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>>7802110
So it was a retarded question? I'm sorry, anon, but can you please tell me why it's retarded?
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>>7802110
You're the one that sounds like you're in high school.
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Pi is irrational and thus it can't be expressed as a sum of rational numbers.
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>>7802146
>literally never took calc 2
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>>7802104
If by "write out completely" you mean "write every digit" the answer is no.
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>>7802146
1/2 - 1/4 - 1/8 - 1/16 - ... is irrational, but after an infinite number of steps you can write it as 0.
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pi_over_4.png
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>>7802104
Not really, pic related. For the first sum it oscillates about a certain value (pi/4) with each term bringing you closer to it, the second sum is strictly decreasing and tends to 0.
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Yes, π is an example of Zeno's "the Tortoise and Achilles" paradox.

π is a number with a finite number of digits, but all we have is an approximation of π, we can closer and closer but we'll never reach the end without an "infinite" number of steps.
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>>7802245
Anon, are you bullshitting? Because the idea that π has a finite number of digits pleases my autism.
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>>7802104
0 can be written as a finite sum of rationals.
Pi can only be written as an infinite sum of rationals.
That's the difference.
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>>7802245
Nope, you are an approximation.
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
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>>7803805
After an infinite number of steps it becomes rational, just like:

1/2 - 1/4 - 1/8 - 1/16 - ... becomes rational after an infinite number of steps, which we know is 0.

You can write all the digits of pi out, we just don't know what they are so we use an approximation, we can get closer and closer to pi, but like in Zeno's "the Tortoise and Achilles" paradox we'll never get to the end.
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>>7802264
Yes, these people just don't seem to understand infinite series.
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>>7802161
>after an infinite number of steps
>after
>an infinite number
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>>7804136
He's clearly not saying infinity is a number, anon.

Infinite by definition means limitless or endless, so just think of it as, "after a limitless or endless number of steps."
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>>7804162
>after an endless amount of steps
I think he's pointing out the contradictive nature of the sentence itself
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>>7804173
But infinities in maths can have an end result.

Like the "1/2 - 1/4 - 1/8 - 1/16 - ... = 0" example he gives, it's basically saying after an infinite number of steps it equals 0, which is correct.

They question is, like "1/2 - 1/4 - 1/8 - 1/16 - ...", does "4/1 - 4/3 + 4/5 - 4/7 + 4/9 - ..." have an end result you can write out completely?
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you only need to know pi to 30 decimal places to calculate the circumference of the universe down to within the width of an atom
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>>7804162
>by definition
You've never defined infinity formally, you pleb. No true mathematical definition uses hand-wavy bullshit like "growing forever" or "the idea of being boundless"
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>>7804367
then why is it possible to have a better resolution than a planck length? we can use arbitrary precision
that's kind of weird
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>>7802104
This isn't a stupid question. Actually, there's a lot about the categorization of number in here. Let me give you a couple terms that might help you out.

Decimal notation expresses numbers as the sum of a finite polynomial of 10 (i.e. the number 9403.8 is 9*x^3 + 4*x^2 + 0*x^1 + 3*x^0 + 8*x^-1, where x = 10). It's a shorthand way of writing out an integer number. Otherwise, numbers would just be done by counting, and that would take a while for larger numbers.

When folks try "write out" pi as 3.1415..., they are representing it as a sort of infinite sum already. Problem: you can't represent irrational (non-fraction) numbers finitely in decimal, nor is there any repeating pattern in irrational numbers that iterates forever. (Because if you could, then you could write it out as a fraction).

But as you suggested, there's another infinite sum which can be finitely represented: [math]4\cdot\sum_{k=0}^N (-1)^k \cdot \frac{1}{2*k + 1} [/math]. As N gets larger, it converges to the number pi. So we say that pi is computable. Yes, there are uncomputable numbers. A lot of them.

Are you thinking this right? Roughly, yeah. You're talking about finite representation. I hope this helped give you a primer to explore wikipedia on your own dangit.

P.S. I skipped a bit where I explain that pi can't be the root (i.e. zero) of a polynomial; hence it's transcendental as opposed to algebraic.
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>>7804312
[math]\ learn_{2 \to limits}\ you=faggot[/math]
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>>7804312

Yes, convergence is a thing, but pi does not converge to a rational number.
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>>7802161
But that number is not irrational. That's the point. You said "1/2 - 1/4 - 1/8 - 1/16 - ... is irrational," but it's not.
Pi is. That's the difference.
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>>7804124
No you're wrong though. That number is already rational. It doesn't "become rational" after an infinite number of steps. It meets the definition for being rational.
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>>7802245
You're just redefining things and spouting things that aren't true because they sound cool.
>Pi has a finite number of digits.

You have no reason to say this, nothing to back it up, nothing. You may as well be spouting that black is white because white is prettier than black.
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>>7804312
Sure. It converges to pi, which is irrational.

Whether something is convergent and whether it is rational are two separate definitions and you're conflating the two. They aren't related.
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The rationals are not complete, or something like that. I had to do some reading on this earlier because I was confused, and even emailed my math professor buddy. Then I started drinking, so forgive my improper use of language.

If anyone in this thread is actually confused, look into a Cauchy sequence. Such a sequence in the rationals does not converge within the rationals since they are not complete. That's where irrationals come from.

Not all infinite sums of rationals are therefore rational.
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