Wikipedia:Reference desk/Archives/Mathematics/2007 April 16

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April 16

Pi in other bases

Wikipedia states "The mathematical constant π is a transcendental (and therefore irrational) real number, approximately equal to 3.14159". Can anyone:

1. Confirm if Pi (the ratio of a circle's circumference to its diameter in Euclidean geometry) calculated in a non-base 10 number system (base 16 for example) is transcendental (and therefore irrational) ?
2. Tell me how to calculate Pi in a non-base 10 number system, preferably base 16, but if Pi in base 16 is transcendental then any base in which it is not transcendental (ie provide proof) ?

(no this is not homework, just curious :-)

Glover 06:16, 16 April 2007 (UTC)

For 2), bases are all calculated as followed: ... base^2 base^1 base^0 base^-1 base^-2 etc, so for base 2, you get ...2^3 2^2 2^1 2^0 2^-1 2^-2 2^-3 etc, which yields the well known 8 4 2 1 0.5 0.25 0.125 etc. So for hex, you would get 16^2 16^1 16^0 16^-1 16-2, etc, which yields 256 16 1 1/16 1/256 etc. Thus when you have 123₁₆, you do (1 * 256) + (2 * 16) + (3 * 1) = 291₁₀.

To convert 0.14159 etc, you would see how many times 16^-1 fits in 0.14159. Subtract that number times 16^-1 from 0.14159 and get a new value. Then see how many times 16^-2 fits in the new value, and repeat from there. So for other bases, you would replace 16 with whatever base. I know this is very confusing, and if you don't get it, I'd be more than happy to further try to explain it. --Wirbelwindヴィルヴェルヴィント (talk) 07:47, 16 April 2007 (UTC)

To get the hexadecimal expansion of a number between 0 and 1, an easier way is to repeatedly multiply by 16, taking each step the integral part as the next hex digit and continuing with the fractional part, thus:

16 × 0.1415926535... =  2.2654824574... →  2

16 × 0.2654824574... =  4.2477193189... →  4

16 × 0.2477193189... =  3.9635091037... →  3

16 × 0.9635091037... = 15.4161456606... → 15 = F

etc. So the hexadecimal representation of pi goes like 3.243F... . --Lambiam^Talk 11:04, 16 April 2007 (UTC)

Being transcendental is a property of numbers, not of representations of numbers. (In particular, every non-base 10 expansion will be non-periodic.)--80.136.150.8 08:20, 16 April 2007 (UTC)

Here's a simple Python program that computes digits of pi (and other mathematical constants) in arbitrary bases. Fredrik Johansson 10:22, 16 April 2007 (UTC)

For fast methods for calculating the digits of pi in a hexadecimal expansion and other bases, see Computing π#Digit extraction methods.  --Lambiam^Talk 10:48, 16 April 2007 (UTC)

Curious is good. Let's try to give answers worthy of the questions.

1. To answer the first question, we need to understand what it means for a number to be irrational and transcendental. You seem to be thinking it is something like ¹⁄₃, which can be written simply in some notations but has an infinite decimal expansion in base 10. Nope. A number is irrational ("not a ratio") if it cannot be expressed as a ratio of two integers. It is usually easy to show that a number is rational, but harder to show that a number cannot be rational. An early, famous example of an irrational number is the square root of two, the length of the diagonal of a unit square. Luckily, we have a fairly simple way to prove that most integers have irrational square roots. A transcendental number is always irrational, but no square root of an integer can be transcendental. Consider the polynomial x²−5; its two zeros are ±√5. Thus the square root of five can be called an algebraic number, meaning it can be defined as the zero of an polynomial with integer coefficients. Note that all integers and rational numbers are algebraic, since bx−a defines ^a⁄_b. The numbers that are not algebraic are transcendental. Not surprisingly, it is often much more challenging to prove a number is transcendental! Mathematicians suspected π was transcendental before they could prove it, and the easiest proof I know of involves calculus. If all you want is a non-repeating expansion in any base, irrational will suffice. A more intriguing property is being a normal number; but here we have not been able to prove anything about π, though all the evidence suggests it is normal.
2. It is a stroke of good fortune that you ask for π in base 16, because in this very special case we have the remarkable Bailey-Borwein-Plouffe formula. For a typical base, we would have to compute earlier "digits" along the way to computing a given "digit", but this formula allows us to directly produce a desired hexadecimal (base 16) "digit". For other bases, an easy method to try is the Gauss-Legendre algorithm, which requires addition, multiplication, doubling, halving, and square root. This can be done using any desired base for the arithmetic. Or try one of the many other known computations.

Now here's a recent twist relating your two questions: A 2003 article in Scientific American suggests the BBP formula for base 16 may hold the key to proving the normality of π. --KSmrq^T 11:46, 16 April 2007 (UTC)

200% of 200

How much is 200% of 200? —Preceding unsigned comment added by Prncss poo (talk • contribs)

100% of x is the same as x. 200% is twice 100%. See also Percentage.  --Lambiam^Talk 14:58, 16 April 2007 (UTC)

Agreed, but note that 200% more than 200, or a 200% increase over 200, is actually 300% (that's 200% plus the original 100%) of 200. This sometimes causes confusion. Even worse, people sometimes incorrectly say "a 200% reduction" when they really mean a 50% or 66.67% reduction. StuRat 15:24, 16 April 2007 (UTC)

That bugs the heck out of me. Black Carrot 23:16, 16 April 2007 (UTC)

Me too. StuRat 02:13, 17 April 2007 (UTC)

That's not what the OP was asking, however.

10% = 0.1

20% = 0.2

50% = 0.5

100% = 1.0

200% = 2.0

200% of 200 is 2.0 * 200 = 400

400 is the answer.

He who is anonymous. —The preceding unsigned comment was added by 202.168.50.40 (talk) 23:58, 16 April 2007 (UTC).

Yes, and we answered what was asked, without giving away the answer, in case this is homework. StuRat 02:13, 17 April 2007 (UTC)

50% = 0.5 and thus 200% = 2.0 is hardly a difficult mathematical concept. And if this is homework, then it is a very very very very easy homework. —The preceding unsigned comment was added by 202.168.50.40 (talk) 23:58, 16 April 2007 (UTC).

Hey now, try to remain polite. What's easy for you might be difficult for someone else. I think that most problems involving applications of Sylow's theorem and any exercise that requires the Rayleigh-Ritz theorem are very easy; but I don't put people down because they feel otherwise. –King Bee ^{(τ • γ)} 16:28, 17 April 2007 (UTC)

Divergent series

In the (excellent) article on 1 − 2 + 3 − 4 + · · · we see that

$1-2x+3x^{2}-4x^{3}...={\frac {1}{(1+x)^{2}}}=(1-x+x^{2}-x^{3}...)^{2}$

and so we throw caution to the winds, make our analysis professor turn pale, and set x equal to 1 to conclude that

$1-2+3-4...=(1-1+1-1...)^{2}$

which is indeed correct, since 1 − 1 + 1 − 1 + · · · = ${\frac {1}{2}}$ and 1 − 2 + 3 − 4 + · · · = ${\frac {1}{4}}$. Giddy with success, we then set x equal to -1, and conclude that

$1+2+3+4...=(1+1+1+1...)^{2}$

but this is not correct, because 1 + 1 + 1 + 1 + · · · = $-{\frac {1}{2}}$ and 1 + 2 + 3 + 4 + · · · = $-{\frac {1}{12}}$. Actually, I am surprised that this hugely non-rigorous argument gives consistent results in even one of these cases - but, since it does, why does it then fail to give consistent results in the second case ? Gandalf61 15:24, 16 April 2007 (UTC)

Rescued from displaying my ignorance by seeing the double brackets in edit mode, and following the link. Well, I didn't know that 1 + 1 + 1 + 1 + · · · = $-{\frac {1}{2}}$, but I am listening. --NorwegianBlue ^talk 16:12, 16 April 2007 (UTC)

Is the series expansion valid for x = -1?

I see that this is apparently interpreting these sums by analytic continuations. For most normal applications these sums are infinite. Otherwise I cannot and will not comment. Root⁴(one) 17:13, 16 April 2007 (UTC)

In the first case, one is essentially assuming Abel summation, which cooperates perfectly with the Cauchy product of series. In the second case, Abel summation gives infinity, and one is falling back on methods I don't understand to get finite answers. Apparently these methods don't play nice with the Cauchy product. Melchoir 19:26, 16 April 2007 (UTC)

Thanks for the responses. The articles on divergent series have really opened my eyes - I thought this was a dull backwater of mathematics, but now I see there is some really interesting stuff in there. I suppose the fact that Hardy wrote a whole book on the subject should have been a clue ! Gandalf61 08:28, 18 April 2007 (UTC)

The above discussion seems to misunderstand the fundamental issue: by themselves, divergent series do not, and can not, converge, by definition. That's why they are "divergent". In particular, in the absence of extra context, 1 + 1 + 1 + 1 + · · · does not equal a finite number -- and this is not controversial, despite quotes that may make it sound controversial.

The only loophole is if one makes an interpretation of a divergent or conditionally convergent series in a very specific context. Anytime you hear experts making wild claims like 1 + 1 + 1 + 1 + · · · equals some constant, it's because they are taking some context for granted: e.g. as a limit of the Riemann zeta function, one can interpret 1 + 1 + 1 + 1 + · · · to mean something. One cannot interpret it that way in the absence of such a context.

Certainly that seems strange, and it did in fact take centuries of development for mathematicians to figure out all that strangeness. I wouldn't be surprised if it turned out that more scholarly papers and books have been published that address aspects of this than on any other mathematical research topic ever in history.

The very basic issues, such as were not really addressed above, could be considered a dull backwater, since they were settled by the 1800s, but that's just the basics. More sophisticated issues related to this area continue to be actively researched today. Dougmerritt 17:54, 21 April 2007 (UTC)

SE(3)

What kind of a group is SE(3) ? —The preceding unsigned comment was added by Deeptrivia (talk • contribs) 19:18, 16 April 2007 (UTC).

It is a Lie group; see this tutorial. The elements correspond to all positions in 3-space in all possible orientations (see slide 6).  --Lambiam^Talk 21:41, 16 April 2007 (UTC)

Localisation of a graded ring, how can be it be a graded ring as well?.

Hello,

"let $R$ be a ring (commutative, with unity) and let it be a graded ring. Let $p$ be a homogeneous ideal in that graded ring $R$. Let $R'$ be $R(R\backslash p)^{-1}$, so this is a localization.

Prove that it is a graded ring, and show that the zeroeth component $R'_{0}$ is a local ring."

This is the problem I have to solve. My book leaves it as an exercise, but it isn't that easy though. There is actually a problem, since some people seem to claim that 'normally', you're supposed to use the complement of only the homogeneous elements of the prime ideal $p$. So that makes this problem particulary hard, because people seem to question the problem itself! Can anyone help me? Thank you very much, Evilbu 20:20, 16 April 2007 (UTC)

Which book are you reading?--80.136.137.26 09:17, 17 April 2007 (UTC)

It's not really a book actually, it's a syllabus... But I find the answer in the mean time : it was WRONG. The multiplicative set that we need to use is NOT the complement of the prime ideal, but the complement of set of homogeneous elements in the prime ideal. The degree was the degree in the numerator minus the degree in the denominator. Evilbu 13:07, 17 April 2007 (UTC)

But you stated in the original question that $p$ should be homogenous, so everything is fine... 80.130.184.227 14:15, 21 April 2007 (UTC)