Proof that pi is rational

>>8670171
proving that the sum of two rationals is rational doesn't prove the sum of three, four or infinite rationals is rational

you can get that the sum of a finite, arbitrary number of rationals is a rational by using induction, but that's it. the sum of an infinite number of rationals is any real number

>>8670181
>proving that the sum of two rationals is rational doesn't prove the sum of three, four or infinite rationals is rational
It proves that the sum of a finite amount of rationals is rational. You are right about the infinite case though. The infinite sum of rationals isn't necessarily rational

>>8670171
your "induction" fails because it shows that approximations for pi using a finite number of terms is rational, which it is.

the reason why it fails is kinda how like f(x+y)=f(x)+f(y) has pathological solutions

>>8670171
Proof that any [math]\sqrt(2)[/math] is rational:
We can write [math]\sqrt(2)[/math] as:
[eqn]1 + \frac{4}{10} + \frac{1}{10^2} + \frac{4}{10^3} + \frac{2}{10^4} + \frac{1}{10^5}...[/eqn]
By the lemma, since it is a sum of rational numbers, [math]\sqrt(2)[/math] is a rational number. We can generalize for any irrational numbers.

>>8670213
yeah thanks for quoting the first question and then saying exactly what the second question says, you fucking genius

>>8670232
How about being a cunt somewhere else, huh?

>admittedly slowly
I lol'd

If you have a alternating series that is not absolutely convergent and it's terms tend to 0, you can get that series to be equal to any real number just by changing the order of terms.

>rationals are dense in R
>therefore every real number is rational xD
saged

>>8670171
If pi is assigned a value of one unit, then radius is always irrational. All numbers are theoretical since no exact unit of measure exists in Reality.