NMC National Pi Day 3-14

[Charles E.]

Just wanted to wish everyone on the Cafe a happy Pi Day!

http://www.piday.org/

[BrianWilliam]

I love pie! Oh....

[Jim Bevan]

My daughter teaches college-level math, I could ask her, but it's more fun to bring this up here.

There's something I don't get about pi: yeah, it's the circumference divided by the diameter, but how do they measure it so accurately that it always comes up with an infinite number of digits? Like, I draw a circle, it's 22 cm, the diameter is 7 cm -- I get lots of digits, but not an infinite number of 'em, so I must not be measuring accurately enough, right?

[FLATROCK HILL]

Right.

Keep in mind though that you are discussing two ways to represent Pi... 22/7 and 3.14159265358979323 etc...a fraction and a decimal.

Just like 1/3 is truly one third, when you try to represent that number in decimal form...

.3333333333333333333333333333333333333333333333333 333333333333333333333333333333333

can go on for a long, long time.

[Jim Bevan]

Any idea what instruments they used to take the measurements when they first started getting hundreds of digits? What did they have that was so accurate back then?

And do they remeasure these days, or do they use the same measurements but just use more powerful computers that can generate more digits?

[FLATROCK HILL]

No idea. And to be honest with you, I don't know that it can be proven that the decimal places go on infinitely. I mean just because they've computed the number beyond a million decimal places, do they really know?

[fatt-dad]

couple of days ago, the fatt-dad-mobile turned to 314,159 and I thought of Pi-day about to come. . .

f-d

[FLATROCK HILL]

I mean just because they've computed the number beyond a million decimal places, do they really know?

couple of days ago, the fatt-dad-mobile turned to 314,159 and I thought of Pi-day about to come. . .

f-d

I just realized that they've now computed Pi to beyond a trillion digits. That, to me is almost as amazing as a car making it to 314,159 miles.

[CarlM]

Any idea what instruments they used to take the measurements when they first started getting hundreds of digits? What did they have that was so accurate back then?

The value of pi to the large number of decimal places is not arrived at by measuring circles and diameters. Pi comes up in a lot of other places in mathematics. The digits are calculated using summations of infinite series based on certain integral calculus formulas. Check the links below or google calculating the digits of pi for more.

https://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml

https://cs.uwaterloo.ca/~alopez-o/ma...xt/node12.html

I mean just because they've computed the number beyond a million decimal places, do they really know?

A discussion of the mathematical proof pi is irrational is at the link below

https://en.wikipedia.org/wiki/Proof_..._is_irrational

[David L]

Right.

Keep in mind though that you are discussing two ways to represent Pi... 22/7 and 3.14159265358979323 etc...a fraction and a decimal.

Just like 1/3 is truly one third, when you try to represent that number in decimal form...

.3333333333333333333333333333333333333333333333333 333333333333333333333333333333333

can go on for a long, long time.

Pi is NOT 22/7, that is just an approximation. Pi is an irrational number which means that it cannot be written as a ratio (fraction). 1/3 is a repeating decimal, which is a completely different thing.

[FLATROCK HILL]

Pi is NOT 22/7, that is just an approximation. Pi is an irrational number which means that it cannot be written as a ratio (fraction). 1/3 is a repeating decimal, which is a completely different thing.

This is true and I thought obvious. Except that although 22/7 is an approximation, it is commonly used as a fraction to represent Pi. Neither way...fraction nor decimal is a perfect representation.
My mistake for using a poor analogy.

Mr, Bevan's scenario stated that the circumference of his circle was 22 units and the diameter was 7 units (cm). He asked if he was measuring accurately enough. No...he was not. 22 divided by 7 results in a number close to Pi, but not Pi.

[mandroid]

And 3.14.1879..it's Albert Einstein's Birthday.. Coincidence?

He played the violin (began reluctantly, from age of 5), so read it's classical scores.
reportedly, More enthusiastically when he was 13..






...

[JeffD]

I was looking for the Peter Infeld strings with the pi on the label, but I could only find violin strings.

[JeffD]

No idea. And to be honest with you, I don't know that it can be proven that the decimal places go on infinitely. I mean just because they've computed the number beyond a million decimal places, do they really know?

I just assigned the proof of this to a few of my students as a challenge (I tutor math).

[fatt-dad]

pi can be related to a fraction. It's the fractional equivalent to C/r, where C is the circumference and r is the radius.

It's algebra!

Also, pi to six significant digits is 3.14159. So, Einstein's birthdate of 3.14.1879 is not quite right after the 3.14.

Yes, I'm a pedant!

f-d

[Mandoplumb]

Any idea what instruments they used to take the measurements when they first started getting hundreds of digits? What did they have that was so accurate back then?
And do they remeasure these days, or do they use the same measurements but just use more powerful computers that can generate more digits?

It's one of those things that just won't work out mathematically like the B string on a guitar. The diameter times pi equals the circumference except it won't quite so you keep adding to it to get closer and closer. Wonder if anyone justs splits the difference like on a B string.

[sblock]

And 3.14.1879..it's Albert Einstein's Birthday.. Coincidence?

He played the violin (began reluctantly, from age of 5), so read it's classical scores.
reportedly, More enthusiastically when he was 13..

...

But the value of Pi is not approximately 3.141879... (the Einstein date) -- Pi is approximately 3.141592653...

So, there's no "coincidence" to be confused about!!

[JeffD]

Any idea what instruments they used to take the measurements when they first started getting hundreds of digits? What did they have that was so accurate back then?

And do they remeasure these days, or do they use the same measurements but just use more powerful computers that can generate more digits?

You don't need to measure actually drawn figures (which of course are going to be inaccurate to some degree). The circle can be precisely defined mathematically, so that every point on the circle can be determined exactly. Then when trying to determine the distance around when the length across is exactly one, the resulting number is not a simple ratio of exact numbers.

Like so many mathematical ideas, it is easy to describe but hard to make obvious. Like our music world too I suppose, where we can know something is true, show others that it is true, but the explanations, do not always create the "obviousness" that is in our own heads.

[Jim Bevan]

Like so many mathematical ideas, it is easy to describe but hard to make obvious. Like our music world too I suppose, where we can know something is true, show others that it is true, but the explanations, do not always create the "obviousness" that is in our own heads.

Reminds me of when my afore-mentioned daughter did a high-school science project illustrating how the cycle of fifths (real fifths) should actually be a helix, and thus how equal temperament works. Without understanding what she was getting at, the teacher gave her a 99 because she obviously knew what she was talking about. (I told her that she didn't deserve the grade 'cuz she hadn't explained it well enough to make it obvious to the teacher.)

I fly a lot, and one Christmas I asked for "geek T-shirts", 'cuz I figured that geeks would appear safe to airport security types. I received a T-shirt with Pi on it, with the numerals getting infinitely smaller.

[Charles E.]

I am reminded of a Jerry Clower bit where one of the Ledbetters comes back from college and to impress Jerry says " well uncle Jerry, did you know Pi-R squared?" Jerry replied " what did they learn you up there? Everybody knows pie are round, corn bread are square"

[CarlM]

I just realized that they've now computed Pi to beyond a trillion digits.

To put that into perspective, if a person were to read those digits from a screen or page, reading ten digits every second it would take over 3000 years just to read the number.

Calculating digits of pi is one of the speed tests run on new supercomputers.

[JeffD]

To put that into perspective, if a person were to read those digits from a screen or page, reading ten digits every second it would take over 3000 years just to read the number..

Which is ridiculous because NASA can keep the space station going with pi to only 16 digits or so.

[Petrus]

For years I had pi memorized to about 30 digits, able to recite it without looking it up but only if I went at a steady speed and started at the beginning. If I stopped anywhere in the middle of that string, or tried to start several digits in, I couldn't (and still can't) do it. I conclude from this that my memorization was along the lines of memorizing a song or piece of instrumental music, where most of us have to start at the start (or at a particular phrase, previously memorized) or else risk getting completely lost. Though they are digits, they apparently occupy the same place in the memory as words or notes.

Nowadays I barely know pi to maybe ten or twenty places, and most of the time the approximation 3.14159265 is sufficient.

I think the transcendence of pi has been proven, though I certainly don't understand it that well. Several proofs have been offered, such as the Lindemann-Weierstrass theorem used to prove (or disprove) the transcendence of those type of numbers in general.

Suppose α is a nonzero algebraic number; then {α} is a linearly independent set over the rationals, and therefore by the first formulation of the theorem {eα} is an algebraically independent set; or in other words eα is transcendental. In particular, e1 = e is transcendental. (A more elementary proof that e is transcendental is outlined in the article on transcendental numbers.)

Alternatively, by the second formulation of the theorem, if α is a nonzero algebraic number, then {0, α} is a set of distinct algebraic numbers, and so the set {e0, eα} = {1, eα} is linearly independent over the algebraic numbers and in particular eα cannot be algebraic and so it is transcendental.

The proof that π is transcendental is by contradiction. If π were algebraic, πi would be algebraic as well, and then by the Lindemann–Weierstrass theorem eπi = −1 (see Euler's identity) would be transcendental, a contradiction.

https://en.wikipedia.org/wiki/Lindem...strass_theorem

Another thing to realize is that the relationship of a circle's circumference to its diameter or radius is basically a human-made concern, in so far as it is useful in many ways. But there is no particular reason in nature why this ratio should be exact as opposed to irrational or whatnot. Some people have argued that pi is not as useful a number as tau:

http://tauday.com/