Is the sinus function empirically proven?

[Orian]

I have googled intensively the origin of the number Pi, and have found many functions that if done infinitely (series) will approximate it.
I have also found proofs that A=pi*r*r, and though I understand the integral functions used in them, they all end up using a sinus function in the end, and the sinus function comes from the calculations related to the number pi.
alternatively, I understand something wrong and haven't read enough.

Can Pi be calculated without functions related to infinitisimal math? Is the number pi found only by more and more accurate measurements?

 

[endorfin]

... Is the number pi found only by more and more accurate measurements?

I think, the answer MUST be YES, because if you wish found whole pi, you will need infinite lists of paper to display the whole number pi, which is impossible. But maybe there is a trick available (if we will use some non-usual numbers), but I have no further information and I am not sure.

 

[Orian]

reply to reply

1/3 also has infinite numbers, so your argument is invalid
edit: as in 0.33333333333333333333333333333333333333333333333333.
edit2: the function is defined, the number is found from observation?
we know that the a^2 is the area of a square and it's surface is 4a.
for a circle it's pai*r^2 and 2pai*r.

Can that ratio, pai, be only found with more and more thorough calculus?

 

[stapel]

1/3 also has infinite numbers...

Do you perhaps mean that 1/3 (having two "numbers") can be presented as an infinite decimal expansion? If so, the point here is that, while 1/3 can be written as an infinite decimal expansion, said expansion is "repeating", indicating that the expansion is of a rational number.

The number \(\displaystyle \, \pi,\, \) on the other hand, is not only not rational, it is transcendental. As such, it has no repetitions in its expansion. The number can be written "exactly" as "\(\displaystyle \pi\)". It cannot be written out, in decimal form, "completely", because the expansion never terminates or repeats.

Can someone more pro answer?

It would help if we had a "more pro" question. As it currently stands, I'm not quite sure what you're even asking. What do you mean, exactly, by "is the sinus function empirically proven"? Even assuming that you mean "sine" function, it's just a function; it's defined, not "proven". So what are you talking about?

Please be specific. Thank you.

 

[pka]

1/3 also has infinite numbers, so your argument is invalid
Can someone more pro answer?

Are you familiar with Tanakh ,the also-called Old Testament, Book of Kings?
In there is a description of a circular pool or bath. It is said to have been 'three times around as the distance was across'. That is an amazing observation. The circumference of a circle is \(\displaystyle \pi\) times its diameter. And what is \(\displaystyle \pi:~3.14159265358979\cdots\). That is about three. That was just one on many approximations used for \(\displaystyle \pi\) used by ancient builders. By the time we had need of more precise measurements we knew about the series representations of functions. In particular the \(\displaystyle \arctan(x)~\&~\sin(x)\).

Because \(\displaystyle \pi\) is an irrational number, we never have an exact numerical value for it.

 

[Orian]

Do you perhaps mean that 1/3 (having two "numbers") can be presented as an infinite decimal expansion? If so, the point here is that, while 1/3 can be written as an infinite decimal expansion, said expansion is "repeating", indicating that the expansion is of a rational number.

The number \(\displaystyle \, \pi,\, \) on the other hand, is not only not rational, it is transcendental. As such, it has no repetitions in its expansion. The number can be written "exactly" as "\(\displaystyle \pi\)". It cannot be written out, in decimal form, "completely", because the expansion never terminates or repeats.

Please be specific. Thank you.

the question is how does one know that calculate what "\(\displaystyle \pi\)" is? Is it derived simple by making more and more accurate calculus?

 

[Orian]

Are you familiar with Tanakh ,the also-called Old Testament, Book of Kings?
In there is a description of a circular pool or bath. It is said to have been 'three times around as the distance was across'. That is an amazing observation. The circumference of a circle is \(\displaystyle \pi\) times its diameter. And what is \(\displaystyle \pi:~3.14159265358979\cdots\). That is about three. That was just one on many approximations used for \(\displaystyle \pi\) used by ancient builders. By the time we had need of more precise measurements we knew about the series representations of functions. In particular the \(\displaystyle \arctan(x)~\&~\sin(x)\).

Because \(\displaystyle \pi\) is an irrational number, we never have an exact numerical value for it.

2 "\(\displaystyle \pi\)"\(\displaystyle \pi\)" r = (2r)*"\(\displaystyle \pi\)". so yes.

I know you extract the "\(\displaystyle \pi\)" from this function, but this function is coming from "\(\displaystyle \pi\)", so this bugs me a bit.

 

[stapel]

the question is how does one know that calculate what "\(\displaystyle \pi\)" is?

The value of \(\displaystyle \, \pi\, \) is \(\displaystyle \, \pi,\, \) just like the value of "2" is 2. Do you mean to ask about the calculation of the decimally-expanded form of \(\displaystyle \, \pi?\,\) If so, then yes, one extends the number of decimal places in the expansion by doing further computations (which is what I'm guessing you mean by "more and more accurate calculus").

2 "\(\displaystyle \pi\)"\(\displaystyle \pi\)" r = (2r)*"\(\displaystyle \pi\)". so yes.

I know you extract the "\(\displaystyle \pi\)" from this function, but this function is coming from "\(\displaystyle \pi\)", so this bugs me a bit.

What, specifically, is "bugging" you?

 

[pka]

2 "\(\displaystyle \pi\)"\(\displaystyle \pi\)" r = (2r)*"\(\displaystyle \pi\)". so yes.
I know you extract the "\(\displaystyle \pi\)" from this function, but this function is coming from "\(\displaystyle \pi\)", so this bugs me a bit.

What, specifically, is "bugging" you?

I think what is 'bugging' Orian is simply that he/she has no idea what any of this is really about.
We can say what the exact value of \(\displaystyle 2\) is: \(\displaystyle 1\cup\{1\}\).
But the is no similar way to say what an exact value of \(\displaystyle \sqrt2\) is.
We know that \(\displaystyle \sqrt2=\text{GLB}\{x>0 : x^2>2\}\) and \(\displaystyle \pi = 4\sum\limits_{(k = 0)}^\infty {\frac{{{{( - 1)}^k}}}{{(2k + 1)}}} \)

 

[Orian]

we can close this thread

I think what is 'bugging' Orian is simply that he/she has no idea what any of this is really about.
We can say what the exact value of \(\displaystyle 2\) is: \(\displaystyle 1\cup\{1\}\).
But the is no similar way to say what an exact value of \(\displaystyle \sqrt2\) is.
We know that \(\displaystyle \sqrt2=\text{GLB}\{x>0 : x^2>2\}\) and \(\displaystyle \pi = 4\sum\limits_{(k = 0)}^\infty {\frac{{{{( - 1)}^k}}}{{(2k + 1)}}} \)

yea I've seen that function (though without the sigma, saw it as a series) - + - + .... gonna look up the proof for it tomorrow.

also, u've answered my question. it wasn't a good question just wanted to check if I understand something correctly.