# Math Help - A good place to end pi

1. Originally Posted by StevenBrown
That point is well taken, but my point is that rounding at a sequence of zeroes or the highest digit of the number system gives you the highest degree of accuracy for the number of digits specified. You can argue against that, but it remains my point.
Well it's just a matter of stating things carefully, since usually we are concerned with absolute error rather than the type of error you're referring to.

2. Originally Posted by undefined
Well it's just a matter of stating things carefully, since usually we are concerned with absolute error rather than the type of error you're referring to.
Can you elucidate the difference between those two types of error?

What I'm saying is that when 1.4999999 is rounded to 1.5, there is a great deal of hidden precision in just two digits.

3. $\pi$ provides a good example of what I mean with regard to economy of rounding. 3.14 is often the value used, because the 159 that follows is not seen as altering the value of the preceding 4 that much. However, I prefer to use the value 3.1416, and that seems a good, practical place to end it, because the 6 is a close rounding of the 59265 that occurs at that decimal place. Now, suppose the value of pi were 3.1449.... In that case, 3.14 would be a less useful approximation, and most likely, 3.145 would be more commonly used.

And that brings me full circle back to the title of this thread, "A good place to end pi." For most practical purposes, 3.1416 seems a good place to end it, providing more than enough precision. Of course, close-tolerance mechanical, engineering, or scientific work may require more precise values of $\pi$.

4. Originally Posted by StevenBrown
You're saying that $\pi$ is an approximation, and I take that to mean it is an approximation to the "real" value of $\pi$ that applies to the geometry of space. Is it possible that real value is rational? And, by implication, whatever that real value is, $\pi$ is equal to it only out to a certain number of decimal places. Beyond that decimal place, additional digits are only the result of a mathematical exercise and have no correlation to reality. Approximately where, in your opinion, is the decimal place that $\pi$ deviates from the real value?

I know the difference between math and physics. What I meant was that in a real-world application, $\pi$ to 761 decimal places, even the real value, if we knew how to calculate it, is so close to the "final" value, that any additional precision would be insignificant compared to quantum fluctuations of spacetime.
Do not use $\pi$ for these two different purposes $\pi$ is a mathematical constant and transcendental, no approximation. We use approximations (but the approximation is not $\pi$ or another " $\pi$")

CB

5. Originally Posted by StevenBrown
Can you elucidate the difference between those two types of error?

What I'm saying is that when 1.4999999 is rounded to 1.5, there is a great deal of hidden precision in just two digits.
Suppose we have a real number $\displaystyle a$ which we round $\displaystyle n$ digits after the decimal point, obtaining an approximation $\displaystyle b$. Then you are expressing interest in finding small values of $|a-b| \cdot 10^n$. Contrast with the absolute error which is just $|a-b|$.

Edit: Another thing I thought of; I see the usefulness of writing 1.5 in place of 1.4999999 as it allows us to write the approximation 1.4999999 $\approx$ 1.500000 with fewer characters (to save our poor fingers from all the typing/writing they do), but consider that in working with significant figures and scientific notation we will often want to specify the precision, so we might end up writing 1.500000 anyway.

6. Originally Posted by undefined
Suppose we have a real number $\displaystyle a$ which we round $\displaystyle n$ digits after the decimal point, obtaining an approximation $\displaystyle b$. Then you are expressing interest in finding small values of $|a-b| \cdot 10^n$. Contrast with the absolute error which is just $|a-b|$.
I see. That is a good mathematical elucidation.

7. Originally Posted by StevenBrown
I see. That is a good mathematical elucidation.
Oh I edited my previous post just now. I didn't realise you'd replied, otherwise I would have made a new post. So, I'm just writing this post to direct your attention to the edit in the previous post (#20).

8. Originally Posted by CaptainBlack
Do not use $\pi$ for these two different purposes $\pi$ is a mathematical constant and transcendental, no approximation. We use approximations (but the approximation is not $\pi$ or another " $\pi$")

CB
If I understand you correctly, all that mathematicians have been able to obtain is an approximation of the true value of $\pi$. Is that because we cannot use current methods to calculate $\pi$ out to an infinite number of decimal places, or is it because the methods yield results that cannot be replied upon to give correct digits all the way to infinity? In other words, when I read that $\pi$ has been calculated to a few hundred million decimal places, are those hundred million digits assumed to be correct, as far as it goes? Or is it that an error may creep in long before the calculation is stopped?

9. Originally Posted by undefined
Oh I edited my previous post just now. I didn't realise you'd replied, otherwise I would have made a new post. So, I'm just writing this post to direct your attention to the edit in the previous post (#20).
Yes, it may be desirable to specify the precision by writing or typing all the zeros. But it would be useful for a universal constant to have more precision than implied by the number of digits, as long as the actual precision is common knowledge.

10. Originally Posted by StevenBrown
If I understand you correctly, all that mathematicians have been able to obtain is an approximation of the true value of $\pi$.
No! We can calculate $\pi$ to whatever decimal or any other precision we like, but because $\pi$ is transcendental no finite rational representation will represent it exactly.

Strange as it may appear being just morals we are restricted to finite representations. But the above does not mean that we have no finite representations that represents $\pi$ exactly. A transcendental number is an equivalence class of sequences of rational numbers and any element of an equivalence class may be used for that number. We have a Turning machine which given an integer $N$ will return the $N$-th decimal digit of $\pi$. This Turing machine characterises $\pi$ exactly, it is (exactly) $\pi$ in most senses that make sense.

(note I am avoiding restricting myself to spigot algorithms, so though the known spigot algorithm for $\pi$ is in base 16 that is not relevant here, but it to would do as an exact representation of $\pi$)

CB

11. Originally Posted by StevenBrown
If I understand you correctly, all that mathematicians have been able to obtain is an approximation of the true value of $\pi$. Is that because we cannot use current methods to calculate $\pi$ out to an infinite number of decimal places, or is it because the methods yield results that cannot be replied upon to give correct digits all the way to infinity? In other words, when I read that $\pi$ has been calculated to a few hundred million decimal places, are those hundred million digits assumed to be correct, as far as it goes? Or is it that an error may creep in long before the calculation is stopped?
The digits are correct if the algorithms have been implemented correctly

CB

12. Regarding the whole physics/math/approximation/granularity of the universe discussion: there is a well-known joking quote, "An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care." I mention it because I believe both directions of approximation have been mentioned in this thread.

As for practicality of having so many digits of pi... it seems there is at least one use: quoting from a Wikipedia article, "These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers." Granted, I don't feel this quote is entirely reliable in terms of how it's written and its belonging to a freely editable encyclopedia, but I'm guessing it's true that calculating digits of pi is a reasonable test for the performance of supercomputers.

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