Does Pi terminate or never?

[drkitten]

A "normal number" is a number in whose decimal expansion the digits are uniformly distributed (i.e., they all occur with equal probability), as well as all series of two digits, of three digits, etc.; and not only in decimal expansion, but in expansion in an arbitrary (integer) base.

Nearly all numbers are normal.

I suppose technically that what Ziggy was saying was not that pi has not been proven "normal," but that pi has not been proven "disjunctive", which is a weaker property. But since pi is almost universally believed to be normal, and all normal numbers are disjunctive, the distinction is quite weak.

Oh, and since nearly all numbers are normal, nearly all numbers are disjunctive, too.... (Duh.)

 

[readme.txt]

I suppose technically that what Ziggy was saying was not that pi has not been proven "normal," but that pi has not been proven "disjunctive", which is a weaker property. But since pi is almost universally believed to be normal, and all normal numbers are disjunctive, the distinction is quite weak.

Oh, and since nearly all numbers are normal, nearly all numbers are disjunctive, too.... (Duh.)

That's exactly it. Normality (therefore "disjunctivity") on every irrational number is strongly suggested, but still unproven, thus what Ziggy said.

 

[stilicho]

The point is, Pi never repeats. It does not matter what sort of counting system you use. It never repeats. Ever. IF PI REPEATED IN BASE 10 NUMBERS IT WOULD ALSO END IF WE USED A DIFFERENT COUNTING SYSTEM. Pi is the length it takes to draw a circle divided by its diameter. The diameter can be an easily determined number 1, 2, 3... but use that straight line as a unit to represent the circumference and you will get a number that can never be accurately 100% represented because you will always be using what amounts to straight line sebments to represent a curve.

The fact that people will debate and argue mathematics is proof that we are not an intelligent species. That is what I am getting from this discussion thread.

I disagree. Some of us aren't professional mathematicians. (I'd gauge rather more than a few). Curiosity is actually one sign of possible intelligence.

But I can absolutely see the logic in what you're saying above. I trust you're saying that the ratio of the circumference to the radius of a circle has to be transcendental in all numeric systems if it transcendental in any of them. That makes logical sense to me since any number would have to have the same characteristics in all systems to have them in any one of them. There's nothing magical about our base ten system AFAIK.

Is this true of all number "types"? Irrational numbers? Imaginary numbers? Negative numbers? Whole numbers? If "type" isn't the right word then tell us what it is.

Are there different rules that apply to ratios?

 

[Ladewig]

I have a question about pi that is not worth its own thread, so I'll ask it here.

Why is knowing the first two trillion digits of pi considered so much more worthwhile than knowing the first two trillion digits of radical 2 or e ?






ETA: changed billions to trillions.

 

[readme.txt]

Why is knowing the first two trillion digits of pi considered so much more worthwhile than knowing the first two trillion digits of radical 2 or e ?

Well, there's some kind of unexplainable metaphysical adoration for Pi. It might be because of its "mysteriousness"... I dunno really.

There's also another explanation that has a lot to do with physics, since Pi is absolutely useful in angle determination : the more digits you know, the more accurate you will be, for a higher distance (i.e. in astrophysics and all related fields, Pi might be more useful than "e" or sqrt(2) ). I don't want to go further in my "explanation", since I'm more of a mathematician than a physicist.

 

[ddt]

But I can absolutely see the logic in what you're saying above. I trust you're saying that the ratio of the circumference to the radius of a circle has to be transcendental in all numeric systems if it transcendental in any of them. That makes logical sense to me since any number would have to have the same characteristics in all systems to have them in any one of them. There's nothing magical about our base ten system AFAIK.

Is this true of all number "types"? Irrational numbers? Imaginary numbers? Negative numbers? Whole numbers? If "type" isn't the right word then tell us what it is.

Are there different rules that apply to ratios?

What you call "numeric system" is actually only the representation of the number as a string of digits. The inherent properties of a number - if it's a whole number, or if it's rational, or algebraic, or transcendental - do not change whether you write out the number in base-10 or in base-2 or in base-37 or whatever positive integer base you might choose. Note, I restrict myself to positive integer bases, as they're the only ones commonly regarded.

How the representation looks like is simple.

Whole (=integer) numbers have always a representation without fraction, irrespective of the base.

Rational numbers have a representation with a finite fraction part, or a repeating fraction. There is at least one base in which a rational number has a finite fraction: a rational number is the division of two whole numbers, so just take the denominator as the base.

Irrational numbers have in every base a representation with an infinite, non-repeating fraction part.

Transcendental numbers are a subset of the irrational numbers, so they always have an infinite, non-repeating fraction part.

 

[Ziggurat]

Why is knowing the first two trillion digits of pi considered so much more worthwhile than knowing the first two trillion digits of radical 2 or e ?

Well, pi and e both have a special place in mathematics in a way that roots of integers just don't. Plus, circles are cooler than right triangles. And e has no simple corresponding geometric shape. So people think pi is just more interesting. On some level it's a subjective evaluation, but that's OK.

 

[stilicho]

What you call "numeric system" is actually only the representation of the number as a string of digits. The inherent properties of a number - if it's a whole number, or if it's rational, or algebraic, or transcendental - do not change whether you write out the number in base-10 or in base-2 or in base-37 or whatever positive integer base you might choose. Note, I restrict myself to positive integer bases, as they're the only ones commonly regarded.

Perfect. That's what I wanted to know. It's the property of the number that doesn't change.

What's this about non-positive and/or non-integer bases now?

 

[3point14]

Which makes for an excellent compression algorithm. Any movie, mp3, application or what have you can be reduced to 2 integers; starting position and length.

That first integer might be remarkably big on occasion.

 

[Ziggurat]

That first integer might be remarkably big on occasion.

In fact, that integer should be about the same size as the file, on average. So... not really compression at all, actually.

 

[Towlie]

In fact, that integer should be about the same size as the file, on average.

Why?

 

[3point14]

In fact, that integer should be about the same size as the file, on average. So... not really compression at all, actually.

The same? Or bigger?

The first integer is going to be somewhere between 1 and infinity. I'm not sure how you project an average for that. Is it possible? I suspect if you do the average is going to be a very, very, very big number.

Actually, how long is the string of numbers that represent, say, a movie? Surely that's got to be the starting point...?

I'm very bad at maths.

 

[dasmiller]

Why?

Let's do a quick estimate of how far down the string of digits your movie is likely to be.

Let's say you're encoding a tiny picture that requires 1000 digits. A particular random sequence of 1000 digits has 1 in 10^1000 chance of being identical to the one you want. Thus, if you had 10^1000 sequences of 1000 digits, you'd have a reasonably good chance of having the one you want.

As you travel down pi, every new digit is the beginning of a 1000 digit sequence, so you'll likely have to go on the order of 10^1000 digits down the line until you find the sequence you want. And how many base-10 digits does it take to specify a number on the order of 10^1000? Why, around 1000, of course.

 

[orange31]

re the attention to Pi,

as many useful functions involve cycles or repetitive functions, the unit circle is widely distributed throughout classical and quantum physics (ie, think how often you see sine, tangent, etc), therefore pi is all over the place.

Slightly different, but I always loved how Feynman in his undergraduate physics course lecture, once as an aside (I think the topic was vectors) told the students in his NYC accent- "anytime ya see somethin' with a bunch of square roots in it, ya got a^2 + b^2 = c^2 in dere somewhere....the Pythagorean theorem."

 

[Ladewig]

Well, there's some kind of unexplainable metaphysical adoration for Pi. It might be because of its "mysteriousness"... I dunno really.

I can buy that

There's also another explanation that has a lot to do with physics, since Pi is absolutely useful in angle determination : the more digits you know, the more accurate you will be, for a higher distance (i.e. in astrophysics and all related fields, Pi might be more useful than "e" or sqrt(2) ). I don't want to go further in my "explanation", since I'm more of a mathematician than a physicist.

I am neither a mathematician nor a physicist, but I cannot believe that any scientist in the world who is measuring things would ever need to use more than 200 digits of pi.

Earlier in this thread a claim was made that 39 digits was all that was needed to measure the circumference of the observable universe to a degree of gradation involving Plank lengths. A quick survey of the internet show estimations for the number of digits of pi for this feat range from 39 to 61.

 

[Ladewig]

re the attention to Pi,

as many useful functions involve cycles or repetitive functions, the unit circle is widely distributed throughout classical and quantum physics (ie, think how often you see sine, tangent, etc), therefore pi is all over the place.

I will agree that pi is all over the place, but that does not account for the publicity surrounding and the desire to know more than the first trillion digits.

 

[Complexity]

Will this never end?

 

[Ladewig]

Will this never end?

I thought we already established the irrationality of this thread.

 

[Complexity]

I thought we already established the irrationality of this thread.

Thank you for not suggesting it is transcendent as well.

 

[ddt]

Perfect. That's what I wanted to know. It's the property of the number that doesn't change.

Yep.

What's this about non-positive and/or non-integer bases now?

Oooh, I opened the door to that, didn't I? I haven't thought about it myself until there was a thread on this a month or so ago in this very section in which it came up. I'll give you a very fragmentary reply.

Let's first see how a base-N system works for a positive integer N > 1. There are digits with values 0, 1, 2, ... N-1. The number representation
abc.de
(where a, b, c, d and e are digits) denotes the number
a * N^2 + b * N + c + d * N^-1 + e * N^-2

The nice thing about this is that you get a unique representation for each number. Well, for nearly each number. There's the obvious duplicate
1.000... = 0.999...
and likewise, all numbers with a finite fraction part can also be represented with an endless fraction of repeating 9's (or in the general case: repeating (N-1)-digits). This is very systematic so it doesn't bother much. We call the finite fraction representation the "standard representation" and we're done. Everybody from grade 4 upwards understands this.

Now let's turn to other possible bases.

0 is out because you'd have no digits. Even if you'd allow a digit 0, you could only represent the number 0.

1 is out because you'd only have digit 0. Even if you allow digit 1, you could only represent integer numbers. There's a thing called "unary numbers", but they're nothing else than tally marks.

Likewise, each base between 0 and 1 suffers from the lack of available digits.

Non-integer bases > 1: as an example, let's take the golden ratio phi as base. Phi is the solution of the equation
phi^2 - phi - 1 = 0
and has the value phi = (1+sqrt(5))/2, approximately 1.6. So you'd have as digits 0 and 1. Now, the representation of a number is not anymore unique. As a simple example, the number phi^2 has two finite representations in base-phi: 100 and 11, as follows from the equation above. The wiki page on this has an algorithm to determine a "standard representation", but as you see it's much more complicated. There's also a separate wiki page explicitly about the golden ratio as base.

ETA: another drawback of a non-integer base is that you can't easily determine if a number is an integer or a rational number. Look at the above example of the golden ratio: phi is an irrational number, so 10 denotes an irrational number, and phi^2 is irrational too, so 100 denotes another irrational number. On the other hand, writing 2 in that base would require a fraction (and possibly an infinite fraction at that too).

Finally, with negative bases you'd first have to answer what the applicable digits would be. Would base (-4) have digits 0, -1, -2 and -3? Or somesuch? And then, negative numbers have the obnoxious property that their powers are alternating positive and negative. I'm getting headaches just thinking about it while I type this.