Would finding a definate Pi value easier if...

[zygote]

Cheesehead wrote

"Pi is irrational and transcendental in all integer number bases. Generally, ease of computing its value in a particular base depends only on the computer's ease of calculation in that base, not on how Pi's value is expressed in that base."

Experience says that you are right, but I seriously doubt that this has been proven. There are lots of transcendental numbers that are trivial to write down if you pick the right base. An example is

1.1010010000001...

where the number of zeros between each 1 increases by n!. This number is simple to write down in base 10, but would have a much more complicated expression if converted to other bases.

Same thing with pi. There's no evidence that it has a simple expression in any rational base, but I am unaware of a proof. In fact, I would be very surprised if this has been proven.

[cperciva]

Originally Posted by zygote

Experience says that you are right, but I seriously doubt that this has been proven. There are lots of transcendental numbers that are trivial to write down if you pick the right base. An example is

1.1010010000001...

where the number of zeros between each 1 increases by n!. This number is simple to write down in base 10, but would have a much more complicated expression if converted to other bases.

Same thing with pi. There's no evidence that it has a simple expression in any rational base, but I am unaware of a proof. In fact, I would be very surprised if this has been proven.

What counts as "simple"? It has been proven that Pi doesn't have an expansion similar to the example you give above -- bounds have been given on how closely Pi can be approximated by rationals, and the value you give doesn't fall within those bounds.

[zygote]

Cerciva wrote

What counts as "simple"?

This could be interesting. In mathematics "simple" is hardly ever simple.

My first instinct is to say that a simple expression for pi would be one that resulted in a more efficient algorithm to approximate pi. It would be meaningful, for example, if the recent trillion digit calculation could have been done in half the time using a different base.[/quote]

[cheesehead]

Originally Posted by zygote

I seriously doubt that this has been proven.

Which part of what you quoted from me do you doubt has been proven?

That Pi is not only irrational, but also transcendental has been proven since 1882. Do a Google search on "pi transcendental proof".

Same thing with pi. There's no evidence that it has a simple expression in any rational base, but I am unaware of a proof. In fact, I would be very surprised if this has been proven.

The very definition of "transcendental" (from Webster's Third New International: "3 a : incapable of being the root of an algebraic equation with rational integral coefficients") implies that Pi cannot be exactly expressed in any rational base.

While Webster's specifies "integral" (Egad -- "rational integral" is rather redundant), and I specified "integral" in some of my preceding statements, distinguishing "integral" from "rational" is mathematically trivial and not really necessary to include to distinguish it from "rational" -- Any rational number can be multiplied by an appropriate integer to form an integer, so an equation with rational coefficients, or a number expressed in a rational base, can be converted to integral coefficients or base by integer arithmetic.

[cheesehead]

(Egad again. I've got to stop capitalizing "pi" when it's not the first word of a sentence.)

[Deamiter]

old habits from Mathematica perhaps?

[zygote]

Cheesehead wrote

Which part of what you quoted from me do you doubt has been proven?

Please forgive me if my point was unclear, but the subject of this thread is not the transcendent nature of pi, which is not in dispute, but whether a switch in base would make approximating pi easier. You made the assertion that "Generally, ease of computing its value in a particular base depends only on the computer's ease of calculation in that base, not on how Pi's value is expressed in that base." It's this statement that is very much in dispute. I was simply pointing out that this statement is demonstrably false for certain transcendental numbers, including my example of 1.1010010000001...

I doubt that anyone seriously believes that there is some base in which the value of pi turns out to have some miraculously simple pattern. However, practical experience that the base makes no difference is not a proof that it makes no difference. Xtreme2k asked whether the base makes a difference. As far as I know, this remains an open question. Do you know differently?

[xtreme2k]

Originally Posted by cheesehead

Nope. It's a fundamental property of all irrational numbers that they cannot be exactly expressed in a finite number of digits in a rational number base.

From my opinion isnt it irrational only because it is based 10? Just because you cannot express it 'rationnally' in base 10 does it make it irrational in all base as well?

Say you get 10 feet of string and divide it by 3. What do you get? You get 3.33333333333333~ feet. How ever there is a more simple way of putting it, 3'4", 3 feet and 4 inches... - 3.4[base12] How simple, How elegent.

As you see, using this method to put a floating point value in an alternative base, you can get a much more accurate value with much less digits... increasing the base gives you MUCH more chances of hitting an exact figure easier... (because you are not limited to 10 divisions before going to the next 'decimal') Given a big enough base (and accurate enough value) wouldn't this open up for more room for the calcuation of pi? Wouldnt it be easier to 'hit' the exact pi value easier?

Now the above might have been a bit too simplistic but that is just an example. That was what made me think about this pi thing...

[zygote]

Xtreme2k wrote

Say you get 10 feet of string and divide it by 3. What do you get? You get 3.33333333333333~ feet. How ever there is a more simple way of putting it, 3'4", 3 feet and 4 inches... - 3.4[base12] How simple, How elegent.

I'm afraid that this is strictly a property of rational numbers. A rational number in any rational base will exhibit either a repeating or terminating fractional value. You can always get a terminating fraction by switching to an appropriate base. Very convenient, but it won't work with pi. Pi is irrational and doesn't follow this behavior.

That doesn't completely eliminate the possibility that pi will exhibit some simple pattern in a different base. We don't know that a pattern won't exist, only that any such pattern won't be either a terminating or infinitely repeating set of digits. Take the number I gave earlier, 1.1010010000001... It shows about as simple a pattern as possible, but it is clearly irrational. It doesn't terminate, and there is no repeating sequence of digits, since there is an ever increasing number of zeros before you get to the next "1". The number is also transcendental, but proving that is dependent on a theorem which says (roughly) that algebraic numbers can't be approximated too closely by rational fractions.

If a base is ever found in which pi shows a simple pattern, it will be big news. The available evidence indicates that the digits in pi behave very similarly to a random number, regardless of base. That means that no one expects to find a useful pattern, but the possibility cannot be excluded.

[cheesehead]

Originally Posted by zygote

Please forgive me if my point was unclear, but

You unclear? No, no, it was I who was woolly-headed (so to speak).

Thank you for persevering.

the subject of this thread is not the transcendent nature of pi, which is not in dispute, but whether a switch in base would make approximating pi easier.

... to which the answer, as I've already written, is "No, a switch in base would not make approximating pi easier, because of the transcendent nature of pi." But you understandably want more details, AND you had a line of reasoning I did not catch earlier...

You made the assertion that "Generally, ease of computing its value in a particular base depends only on the computer's ease of calculation in that base, not on how Pi's value is expressed in that base." It's this statement that is very much in dispute. I was simply pointing out that this statement is demonstrably false for certain transcendental numbers, including my example of 1.1010010000001...

_Now_ I think I see what you're getting at. (Please correct me again if you don't think the following are fair restatements of your arguments.)

You're claiming that 1.1010010000001... has a simpler pattern than pi = 3.1415926535... I disagree.

Let's express both numbers as infinite series (which makes them number-base-independent, right?).

pi = 4 * (1/1 - 1/3 + 1/5 - 1/7 +- ...) is a pretty simple pattern. Pi = 4 times the sum over i from i = 0 to infinity of ((-1)^i/(2*i+1)).

Your number (which probably has someone's name attached) = 1 plus the sum over i from i = 0 to infinity of (1/10^-(i + the sum over j from j = 0 to i of (j!))).

Expressed that way, pi looks as simple as, or even a little simpler than, your number.

Is there a simpler infinite series expression for your number? If not, then it doesn't seem that your number actually has a "simpler" expression than pi does. And I think you'll find the same true for almost any transcendental number (I personally think e is a little "simpler" than pi). Any transcendental expressed as a sum of rational terms requires an infinite number of such terms -- so they're all about equally "simple".

I doubt that anyone seriously believes that there is some base in which the value of pi turns out to have some miraculously simple pattern.

This does turn on the meaning of "simple", doesn't it?

Is 4 * (1/1 - 1/3 + 1/5 - 1/7 +-...) a simple enough pattern for you? If not, why not?

(BTW, try defining "simpler" as "requiring fewer characters to express in natural language".)

However, practical experience that the base makes no difference is not a proof that it makes no difference.

The proof involves noting that expressing a number in some base is equivalent to expressing it as a sum of an infinite series. :)

One you approach it from that direction, it'll be clearer.

Xtreme2k asked whether the base makes a difference. As far as I know, this remains an open question. Do you know differently?

Yes, I do know differently. (But I'm not claiming I'm the only person who does! Far from it!! I learned all this from others who knew it before me.)

The base does NOT make a difference. This is not because of lack of counterexample -- it is because of the properties of transcendental numbers.

[cheesehead]

Originally Posted by xtreme2k

From my opinion isnt it irrational only because it is based 10?

No. "Irrational" means that it cannot be expressed as the ratio of two integers.

The number base in which integers are expressed is irrelevant, as long as it is a rational base.

Just because you cannot express it 'rationnally' in base 10 does it make it irrational in all base as well?

If a number is irrational, it is irrational regardless of the rational number base in which one expresses it. It is irrational because it is not the ratio of any two integers, not because of some choice of number base.

Say you get 10 feet of string and divide it by 3. What do you get? You get 3.33333333333333~ feet. How ever there is a more simple way of putting it, 3'4", 3 feet and 4 inches... - 3.4[base12] How simple, How elegent.

Or, just 10/3. Since it is the ratio of two integers, it is rational.

As you see, using this method to put a floating point value in an alternative base, you can get a much more accurate value with much less digits... increasing the base gives you MUCH more chances of hitting an exact figure easier

But doing that will never change a rational number to irrational. Either the number is the ratio of two integers, or it isn't -- regardless of number base.

... (because you are not limited to 10 divisions before going to the next 'decimal')

No, that has nothing to do with it. I already explained that earlier in this thread.

Given a big enough base (and accurate enough value) wouldn't this open up for more room for the calcuation of pi?

It will never, ever change pi to a rational number.

Wouldnt it be easier to 'hit' the exact pi value easier?

No. I already covered that earlier in this thread.

It's not a matter of "hitting" the exact value of pi. The exact value of pi has been known for a long time.

One way to express the exact value of pi is, as I wrote in my preceding posting, "Pi = 4 times the sum over i from i = 0 to infinity of ((-1)^i/(2*i+1))." There. That is one way to represent the exact value of pi.