Would finding a definate Pi value easier if...

[zygote]

Cheesehead wrote

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

Actually, that's not my claim at all. I am saying that 1.1010010000001... has a simpler pattern in
base 10 than in other bases. Take base 7 for example. According to my calculations it is equal to
1.046433444442... when expressed in base 7. Although I haven't defined "simple", I see no
justification for saying that the base 7 representation is just as simple as the base 10.

The key point, however, is that the computational speed of calculating this number depends on
the base. Calculating a trillion digits of this number in base 10 is so easy that it can be done with
a pencil and paper in a few minutes. It has a "1" in the following powers of 10 and a "0"
elsewhere: 1, -1, -3, -6, -13, -38, -159, -880, -5921, -46242, -409123, -4037924, -43954725,
-522956326, -6749977127.

In contrast, the base 7 calculation is extremely challenging. I see no better method than first
calculating the base 10 value and then applying a base conversion algorithm to get to base 7.
Doing this with a trillion digit precision is far beyond a pencil and paper calculation.

This means that 1.1010010000001... is a counterexample to the claim that the speed of
calculating a transcendental number is independent of base. That's the question that Xtreme2k
raised for pi. Experience says that a change of base probably won't speed up the calculation, but
mathematically the question is open. Some transcendal numbers can be computed more easily in
one base than another. Maybe pi is one of them.

[Paulie]

Is it being easier to compute in base 10 vs. base 7 a issue for the base, or for the "calculator" in question being used to base 10? :)

I remember reading that IBM used to make electronic multiplying calculating punches and some calculators (as they called electromechanical computers in the time) in base 10 (wasn't the 1401 series base 10?) There must have been a reason modern computers are all base to (probably because of the nature of transistors, eh? :) )

[cperciva]

It should be noted that Pi has BBP-like formulas only in base 2, and Pi^2 only in bases 2 and 3; so Pi isn't entirely base-agnostic.

So even if Pi cannot be computed any faster in one particular base, the computation can be verified faster in those bases.

[cheesehead]

Originally Posted by Paulie

(wasn't the 1401 series base 10?)

Though I never ran a program on a 1401, I read the manual and IIRC it did operate in base 10.

I know for certain that the IBM 1620 (which, BTW, was IBM's first 100%-transistorized computer) operated in base 10.

There must have been a reason modern computers are all base to (probably because of the nature of transistors, eh? :) )

Well, base-2 is more efficient than base-10 in any computer using binary transistor logic.

In the 1620, each addressible memory location held a single decimal digit and a flag bit. (Actually, each memory location had six bits, though the bits were not individually accessible by machine instructions. All instructions operated on the six-bit content as a whole. Four bits were for the binary-coded-decimal (BCD) digit, one was the flag bit, and one was the parity bit.) Number fields were variable-length. In a number field, the low-order (highest-address) digit had the flag bit on to indicate end-of-number. The address by which one referenced the field was the address of the high-order (lowest-address) digit. A negative number was signified by having the flag bit on in the high-order digit. (So, each number field had to be at least two digits long because the high-order digit's flag bit could not be the end-of-number indicator.)

Internally (in the circuitry, below the level of machine instructions) the 1620 actually performed binary arithmetic on the four-bit BCD digits, then further processed the results so as to make the result correctly represented as BCD digits. This was, of course, inefficient for computational purposes, but it was thought that it was easier for beginners in computer courses to deal with a base-10 computer. IBM targeted its marketing of the 1620 to colleges and universities.

[cheesehead]

Originally Posted by cperciva

It should be noted that Pi has BBP-like formulas only in base 2, and Pi^2 only in bases 2 and 3; so Pi isn't entirely base-agnostic.

But it's not _pi_ that lacks base-agnosticity --- it's the BBP-like formulas that are not base-agnostic.

Suppose someone discovered another class of formulas that were especially efficient in bases 109 and 191. That discovery wouldn't affect the base-agnosticity of the number pi itself; it would only affect the base-agnosticity of computation using those new 109-191 formulas.

So even if Pi cannot be computed any faster in one particular base, the computation can be verified faster in those bases.

Again, it would be the computational method's speed that is affected by choice of base, not pi itself.

[cheesehead]

Originally Posted by zygote

I am saying that 1.1010010000001... has a simpler pattern in base 10 than in other bases.

Again, thank you for persevering. Maybe now I'll get it right.

Okay, 1.1010010000001... has a simpler pattern in base 10 than in other bases, but that is because of the predominance of the integer 10 in its infinite series representation: 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!))).

The definition of your number is strongly tied to the integer 10. Can it be transformed into another reasonably-"simple" definition that does not feature the integer 10 or power thereof? I'd guess not.

(BTW, I'm suspicious of the transcendence of 1.1010010000001... Can anyone point me to a proof?)

The formula Pi = 4 times the sum over i from i = 0 to infinity of ((-1)^i/(2*i+1)) has every odd integer appear in it, so does not lead to a representation in which some integer pattern predominates.

There are other formulas for pi, such as 4*(4 ArcTan[1/5] - ArcTan[1/239]), which might lead to representations in which particular integers, such as 4, 5, and 239, predominate. However, when the featured integers are relatively prime, as in this case, their patterns tend to overlap and obscure one another in any base representation.

Take base 7 for example. According to my calculations it is equal to 1.046433444442... when expressed in base 7. Although I haven't defined "simple", I see no
justification for saying that the base 7 representation is just as simple as the base 10.

Unless someone comes up with a simple infinite-series representation of your number that features the integer 7 predominantly, I'd agree.

The key point, however, is that the computational speed of calculating this number depends on the base. Calculating a trillion digits of this number in base 10 is so easy that it can be done with a pencil and paper in a few minutes. It has a "1" in the following powers of 10 and a "0"
elsewhere: 1, -1, -3, -6, -13, -38, -159, -880, -5921, -46242, -409123, -4037924, -43954725,
-522956326, -6749977127.

... because of the simple way that the integer 10 appears in the definition of your number, without featuring any other integer to create interfering patterns.

In contrast, the base 7 calculation is extremely challenging. I see no better method than first calculating the base 10 value and then applying a base conversion algorithm to get to base 7. Doing this with a trillion digit precision is far beyond a pencil and paper calculation.

This means that 1.1010010000001... is a counterexample to the claim that the speed of calculating a transcendental number is independent of base.

Okay, within the class of transcendental numbers, a few can be defined such that one integer, or some such simple pattern, predominates in a way that makes it easy to construct the number's representation in some base with "pencil-and-paper" methods. But that doesn't apply to transcendentals in general.

That's the question that Xtreme2k raised for pi. Experience says that a change of base probably won't speed up the calculation, but mathematically the question is open. Some transcendal numbers can be computed more easily in
one base than another. Maybe pi is one of them.

I still think the answer is "no". However, I don't know whether it's proven that it's impossible to come up with an infinite-series formula for pi that features one or a few integers in such a way as to produce a "simple" pattern in pi's representation in some integral base. Maybe if I knew more about how to prove that 1.1010010000001... is transcendental...

[cheesehead]

Originally Posted by zygote

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.

Were you thinking of Liouville's Constant 0.110001000000000000000001... = sum over n from n = 1 to infinity of 10^-n! ? (see http://mathworld.wolfram.com/LiouvillesConstant.html)

[cperciva]

Originally Posted by cheesehead

But it's not _pi_ that lacks base-agnosticity --- it's the BBP-like formulas that are not base-agnostic.

Not really. Other numbers have BBP-like formulas in other bases; the property of "has BBP-like formulas only in base 2" is special (but not unique) to pi.

[cheesehead]

Originally Posted by cperciva

Not really. Other numbers have BBP-like formulas in other bases; the property of "has BBP-like formulas only in base 2" is special (but not unique) to pi.

There are many, many formulas for pi other than the BBP-like ones, featuring many integers other than 2 or any power of 2 (e.g., 4, 5, and 239 in one I quoted earlier).

Therefore, if there are no BBP-like formulas for pi in bases other than 2 or 16 (the famous BBP formula I'm thinking of produces individual hexadecimal digits of pi) it is the BBP-like formulas which are restricted in base, not pi itself.

[cperciva]

Originally Posted by cheesehead

There are many, many formulas for pi other than the BBP-like ones, featuring many integers other than 2 or any power of 2 (e.g., 4, 5, and 239 in one I quoted earlier).

BBP-like formulas have the special property that they allow computation of specific digits in polylogarithmic space and almost-linear time. This relates to the original question I addressed: Pi can be considered to be "simpler" in base 2, because it is possible to compute individual digits.

Still, whether you consider this a special property of pi, or a special property of "computing individual digits of a number", is up to you.

[cheesehead]

Originally Posted by cperciva

BBP-like formulas have the special property that they allow computation of specific digits in polylogarithmic space and almost-linear time. This relates to the original question I addressed: Pi can be considered to be "simpler" in base 2, because it is possible to compute individual digits.

How about that! I was just coming back here to amend my preceding posting with what I'll put here instead:

While chewing the cud and mulling over the question of base-agnosticism, it occurred to me that I was being too stubborn -- that if it had been proven that (a) BBP-like formulas, with which I have not kept up recently, existed for pi in fewer bases than for other numbers, and (b) BBP-like formulas have enough generality, then it could indeed be proper to categorize numbers in accordance with their BBP-like-formula base-existences.

cperciva, I now think you're right, if the above conditions pertain (which I suspect, but don't know for sure).

Still, whether you consider this a special property of pi, or a special property of "computing individual digits of a number", is up to you.

I'll have to learn more about BBP-like formulas ... and (*sigh*) that will probably take me longer now than it would have when I was a young math-whiz. (Why couldn't B, B, and P have gotten together sooner?!?) ... Pardon my ruminations.