What Are Some Interesting Properties of Pi?

[Good4you]

So if Pi is an irrational number, and therefore has an infinite line of numbers after the decimal point; my intuition tells me it would take an infinite amount of time to determine its exact value.

a) Do calculators and computers somehow know Pi's exact value or is it just an estimate?
b) Are mathematicians out there that concern themselves with finding ever more accurate definitions of pi? Does at some point it become moot, and no practical application requires such accuracy?

 

[Office_Shredder]

Any calculator or computer you use will only have an estimate of pi, not an exact value

As far as how accurate estimates of pi are, computing digits of pi is often used to demonstrate the power of a new supercomputer; these estimates go into the billions of digits. It's not the value of pi that's interesting at that point though; it's the computer's ability to run the algorithm that computes so many digits

 

[DaleSwanson]

We currently know over two trillion digits of pi.

To give some perspective on how much the digits of pi matter here is an example.

The Persian mathematician and astronomer, Ghyath ad-din Jamshid Kashani (1380–1429), correctly computed 2\pi to 9 sexagesimal digits. This figure is equivalent to 16 decimal digits as 2\pi = 6.2831853071795865 which equates to pi = 3.14159265358979325.

The Windows calculator uses:
pi = 3.1415926535897932384626433832795
If we calculate the size of Earth's orbit around the Sun using these two values it gives:
Radius = 150,000,000,000 meters
942,477,796,076.93797153879301498385 meters for Window calculator value of pi
942,477,796,076.937975 meters for the 15th century value of pi.
The error using the 15th century value is about 3.5 micrometers.

Using pi = 3.14159 to calculate Earth's circumference gives an error of about 34 meters.

http://en.wikipedia.org/wiki/Numerical_approximations_of_π

 

[g_edgar]

1/3 is a rational number, but it also has infinitely many digits after the decimal point. Do you say it "takes an infinite amount of time to determine its exact value"?

 

[disregardthat]

g_edgar, I don't think that's a fair rebuttal. pi is always made intelligible by means of a limit. Speaking of pi is the same as speaking of the limit of a recursively defined sequence (an algorithm), but 1/3 is not. The algorithm is never 'completed', that would be nonsense. Therefore it does not make sense to speak of the totality of the decimal expansion, we can only speak of the algorithm which produce an infinitely long decimal expansion.

So we don't know the exact value of pi, because it does not make sense to talk about it. This applies to all numbers made intelligible purely by the means of limits.

 

[HallsofIvy]

On the contrary- we do know the value of \pi as well as we know the value of any number! The fact that we don't have any convenient way of writing it in terms of numerals has nothing to do with "knowing the exact value of pi".

 

[disregardthat]

On the contrary- we do know the value of \pi as well as we know the value of any number! The fact that we don't have any convenient way of writing it in terms of numerals has nothing to do with "knowing the exact value of pi".

You are comparing the decimal expansion of pi to an 'enormous' decimal expansion. We have no convenient way of writing the numerals to a sufficiently large decimal expansion either, but we can (in principle) know the totality of it. We cannot (in principle) know the totality of the decimal expansion of pi, it doesn't make sense to speak of it.

pi's decimal expansion is defined in terms of a recursive algorithm (in one way or another), so what do you actually mean by the 'exact value of pi'? Is this not a constructed way of talking about 'the limit' of some recursively defined sequence?

 

[Tac-Tics]

So if Pi is an irrational number, and therefore has an infinite line of numbers after the decimal point; my intuition tells me it would take an infinite amount of time to determine its exact value.

Do you know the exact value of the number 1? I mean, do you really, reeeeally know the value?

To be technical, you're talking about writing down the digits of pi, yeah? Yes, it would take an infinitely long time to write down the digits.

As another poster said, the same is true of 1/3 and many rational numbers.

The difference is, of course, that all rational number eventually reach a fixed repeating pattern. We might say 1/3 = 0.(3), where the numbers between the ( ) are said to repeat. So 1/7 = 0.(142857).

This allows us to "write down" the value of all rational numbers in a finite way. Of course, we haven't actually written down all the digits, but we've given the reader enough information to figure out the rest of the digits on their own, to whatever precision they want.

But pi is irrational. This notation doesn't work. But that doesn't mean we can't figure out another technique that helps us achieve the same thing.

We might use a formula instead of a fixed number. Often, these take the forms of series. Here's a whole page of ways you can "write down" pi in a finite way:

http://en.wikipedia.org/wiki/List_of_formulae_involving_π#Efficient_infinite_series

Infinity doesn't really bother mathematicians much. Certain kinds of infinity are easy to work with. Others aren't. For example, there are an infinite number of integers, but that fact is so mundane and well-understood, it sounds funny to say it that way. A very important place for things NOT to be infinite is when you have to prove something or do something. If you can prove something in an infinite number of steps, you haven't proved anything. If you can do something, but it would take you infinitely long (writing down all the digits of pi), you can't actually every finish it. (You can't do it).

a) Do calculators and computers somehow know Pi's exact value or is it just an estimate?

Something I wondered when I was a kid.

Storing information in memory (be it a hand held calculator or PC) is the same as writing it down on paper. You can only write so much down. The more you write, the more paper (or memory) it takes.

Computers are discrete machines. They effectively only work with integers. "Real numbers" on a calculator are actually approximations. We use the term "floating point" number instead of "real number" to highlight this fact.

b) Are mathematicians out there that concern themselves with finding ever more accurate definitions of pi? Does at some point it become moot, and no practical application requires such accuracy?

Mathematicians, I think, generally don't care about the digits of pi. It's not really useful for what anyone does. Twenty digits is way more than anyone would actually ever need.

Pi is still interesting in other ways. It interacts with a whole lot of seemingly unrelated mathematics. However, knowing the digits rarely helps with finding these connections.

 

[Hurkyl]

Since nobody answered this question:

b) Are mathematicians out there that concern themselves with finding ever more accurate definitions of pi?

An exact definition of pi is already known. Even the ancient Greeks had one, if you forgive them for outright assuming the notion of "length" is reasonable.


Sometimes people aren't interested in pi -- they are instead interested in things like:

a 20 digit floating-point decimal whose value in the real numbers is close to pi​

or

A computer program that could, on input n, compute a rational number p/q such that |p/q - \pi| < 1/n. (given enough time and memory to calculate)​

These things aren't all that hard to do either. And if simplistic methods to answering these questions is inefficient for your tastes, people have devised pretty fast algorithms too.