Wikipedia:Reference desk/Archives/Mathematics/2010 August 7

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August 7

Solve this equation

FIND THE VALUE OF K IF:9^K+12^K=16^K —Preceding unsigned comment added by 122.170.28.77 (talk) 03:57, 7 August 2010 (UTC)

A simple bit of iteration. When K=1, the left side is greater. When K=2 the right side is greater. so the answer is somehere between the two. Set lower_limit=1, set upper limit=2, try (lower_limit + upper_limit)/2 and if the result has the left side larger than the right replace the lower_limit by the trial value otherwise replace the upper_limit by the trial value. Repeat until the two sides are equal. -- SGBailey (talk) 05:39, 7 August 2010 (UTC)

So K ≈ 1.67272093446, but is there an analytic solution? -- 119.31.121.93 (talk) 06:04, 7 August 2010 (UTC)

There is no need to use iteration. In fact, observe the following easy solution. First write the equation as:

$(3^{k})^{2}+4^{k}(3^{k})-4^{2k}=0.$

Put $x=3^{k}$ and note that the above equation can be rewritten as:

$x^{2}+(4^{k})x-4^{2k}=0.$

This is a quadratic equation in x and hence can be solved using the quadratic formula; we obtain:

$x={\frac {-4^{k}+{\sqrt {4^{2k}+4^{2k+1}}}}{2}}.$

(We discard the other solution since $3^{k}$ is positive for all real k.) Hence:

$x=4^{k}({\frac {-1+{\sqrt {5}}}{2}}).$

Recalling that $x=3^{k}$ we obtain that:

$({\frac {3}{4}})^{k}={\frac {-1+{\sqrt {5}}}{2}}.$

Taking logarithms to the base ${\frac {3}{4}}$ yields the result:

$k=\log _{\frac {3}{4}}({\frac {-1+{\sqrt {5}}}{2}}).$

PST 07:10, 7 August 2010 (UTC)

(edit conflict). If 9^K+12^K=16^K then (9/16)^K+(12/16)^K=1. Then (3/4)^2K+(3/4)^K=1. Set x=(3/4)^K. Then K=log(x)/log(3/4) and x²+x=1. Then K=log((√5−1)/2))/log(3/4)=(log(√5+1)−log(2))/(2log(2)−log(3)) is an analytic solution. Bo Jacoby (talk) 07:32, 7 August 2010 (UTC).

So it's log_4/3φ. Wonderful!-- 119.31.121.78 (talk) 00:51, 8 August 2010 (UTC)

vectors

Why vector division is not defined? —Preceding unsigned comment added by Mahendrasingh4u (talk • contribs) 05:30, 7 August 2010 (UTC)

See Quaternion#Algebraic_properties. Bo Jacoby (talk) 07:07, 7 August 2010 (UTC).

First you would have to define "vector multiplication". Then you would deem "vector division" to be the inverse of this operation. But to even speak of "inverses", you need to define a multiplicative identity of this "vector multiplication" operation. What multiplication operation do you propose that has such a multiplicative identity? (The usual inner product does not work since its output is not a vector but a scalar. The usual cross product does not work either since while it does define a product operation on a set of vectors, it does not have a (non-trivial) multiplicative identity.) PST 07:20, 7 August 2010 (UTC)

Does the cross product have a trivial identity? The cross product is always perpendicular to the vectors being multiplied and no vector is perpendicular to itself (other than the zero vector, I suppose, depending on the details of your definition). --Tango (talk) 03:32, 8 August 2010 (UTC)

I probably should have been more clear. The cross product certainly does not have an identity when defined on positive dimensional Eucldiean space. It is conceivable that by deleting some "trivial vectors" one can fix this problem, or put differently, that the problem can be fixed by restricting the cross product to some proper subset of Euclidean space. When I said "non-trivial" identity, I meant that the only vector space in which this problem is totally fixed is the zero-dimensional vector space. But that is, of course, a trivial example. PST 06:38, 8 August 2010 (UTC)

Given orthogonal vectors A and C, it is easy invert the cross product to describe the family of vectors B such that A × B = C, but I don't recall ever needing to do that. This is clearly not a general form of division, as the two given vector must be specifically related and the result is not a single value. -- 119.31.121.88 (talk) 00:03, 12 August 2010 (UTC)

What is C^r topology and C^r distance in the space of differentiable maps C^r(M,N)

in which the domain (M) and codomain (N) are both manifolds. Thanks for your answer.SongJie@NTU (talk) 14:23, 7 August 2010 (UTC)

You are refering the the Whitney C^k-topologies, named after Hassler Whitney. There isn't an article on Wikipedia about the Whitney C^k-topologies. They use the differences of partial derivatives to define the distances between mappings. Planetmath has an article, see here— Fly by Night (talk) 18:43, 7 August 2010 (UTC)

Thanks! that's helpful.SongJie@NTU (talk) 02:52, 8 August 2010 (UTC)

Probability question

I understand this is an incredibly simple problem compared to the questions I have seen posted here, but here goes. I have a bet each week on soccer results. I usually look for odds of around 10/1. I am currently on a losing streak of 50 weeks. I figured I have a 90% chance of losing (ignoring the bookies 1-2% advantage) on any one bet, therefore the odds of this losing streak is 0.9^50 (sorry haven't worked out how to do superscripts) or about 1 in 200. I explained this to a friend who said my logic is flawed. Who is right? Si1965 (talk) 23:11, 7 August 2010 (UTC)

Why do you think you have a 10% chance of winning any one bet? The odds are set to make money for the bookie, so your chances are probably less than 10%. Maybe a lot less. My chance of getting a date with Keira Knightly is certainly less than 1/1000 (more like zero), and a bookie might give 10-to-1 odds against it, but probably would not give 100-to-1 much less 1000-to-1. 75.62.4.94 (talk) 03:35, 8 August 2010 (UTC)

The OP already said he was ignoring the bookie's advantage which, as he says, is usually only a few percent. --Tango (talk) 03:42, 8 August 2010 (UTC)

Well, it's not actually 90%, since it's odds of 10 to 1, not a probability of 1 in 10. See odds. So actually your chance of losing is 10/11 or 91% (the difference is more significant for more likely events, eg. odds of 2/1 against isn't 50%, it's 33%). Otherwise, your logic is fine. Did your friend say what the flaw he saw was? --Tango (talk) 03:37, 8 August 2010 (UTC)

That said, it is worth pointing out that even a small change in the probability you start with can have a big change in the end result. If we use the correct figure of 91% and then add on 2% for the bookie's advantage (using your numbers, I'm not sure exactly what it usually is) to get 93% then we see that 0.93^50=0.027, which is about 1 in 40. So just a 3% difference has taken us from 1 in 200 to 1 in 40. --Tango (talk) 03:42, 8 August 2010 (UTC)

Why do you consistently choose to bet on the longshots? Is it because, even though you expect the average results to be the same, you'd rather occasionally win big than regularly win small? See Favourite-longshot bias. You and other longshot favoring bettors may be shifting the odds so that while your bet may pay 10:1, the team's chances of winning may be much less than 1/11. This was discussed recently in WP:Reference desk/Archives/Miscellaneous/2010 August 2#Betting on the ponies -- 119.31.121.72 (talk) 05:44, 8 August 2010 (UTC)

Note also that although this is the correct way of calculating the probability of this losing streak, i.e. losing in all of these 50 weeks, that's not the same as the probability of ever experiencing a losing streak of 50 weeks or more if you've been betting this way for considerably longer. Calculating that is quite a bit more tricky... --Qwfp (talk) 06:50, 8 August 2010 (UTC)

A bookie's odds are based on measuring (or estimating) collective public perception, not actual odds. For simplicity, suppose A is playing B. The bookie's goal is to get the amount of money bet on A winning to be equal to the expected payout if B wins and vice versa. In other words, there goal is the ensure that losers pay the winners (after taking a cut for the house, which could either be explicitly expressed, or implicit by not having the total odds sum to unity). Getting the accounts to balance on each side is based on judging how likely people are to bet either side given a specific rate of return, and not on how likely an outcome actually is to occur. Hence it is about perception and willingness to bid, rather than true odds. Collective public wisdom is actually pretty good on average, so the offered odds are often similar to the true odds, but it is subject to biases. One of these is the favourite-longshot bias, which says that people tend to overestimate the odds of longshots winning. Because of this, we can generally expect people to bid on a longshot more often than they should statistically. Bookies know this. In order to get the books to balance, they need to reduce the expected payout for the longshot in order to compensate and make their expected payouts balance. To make a long story short, because people tend to overbid longshots, when a bookie gives 10:1 odds, the true odds might be something more like 25:1. Given true odds of 25:1, we would expect a 50 week losing streak to occur roughly 1 time in 8. Hence, it would be somewhat unlikely, but not a surprising event. Dragons flight (talk) 07:50, 8 August 2010 (UTC)

Thanks to you all for your responses, very interesting and helpful. Si1965 (talk) 09:22, 8 August 2010 (UTC)

Strange notation

The article titled binary logarithm contains this:

For integer domain and range, the binary logarithm can be computed rounding up or down. These two forms of integer binary logarithm are related by this formula:

$\lfloor \log _{2}(n)\rfloor =\lceil \log _{2}(n>>1)\rceil +1,{\text{ where }}n\geq 1.$

Can someone translate this into standard language and notation? Michael Hardy (talk) 23:33, 7 August 2010 (UTC)

Are you questioning the ⌊⌋, the ⌈⌉, or the >>? Note that n>>1 = ⌊n/2⌋ (and n>>k = ⌊n/2^k⌋), and that for these positive n, logical shift and arithmetic shift are equivalent. -- 119.31.121.78 (talk) 01:24, 8 August 2010 (UTC)

The formula is also incorrect, as shown by evaluating with n=7. I believe that it should read:

$\lfloor \log _{2}(n)\rfloor =\lceil \log _{2}(n+1)\rceil -1,{\text{ where }}n\geq 1.$

and that it was broken by this edit. -- 112.142.28.242 (talk) 03:47, 8 August 2010 (UTC)

And, of course, all that it is saying is that the rounded up logarithm is one more than the rounded down logarithm, except for exact powers of two where they are equal. -- 112.142.28.242 (talk) 03:51, 8 August 2010 (UTC)

I've fixed it. I suspect that the editor who broke it was going for the correct formula:

$\lfloor \log _{2}(n)\rfloor =\lfloor \log _{2}(n>>1)\rfloor +1,{\text{ where }}n\geq 2.$

which is related to the code snippet that follows in the article. -- 119.31.126.83 (talk) 15:09, 8 August 2010 (UTC)

I was not familiar with the "logical shift" denoted by ">>".

I've changed the word "where" to "if". One writes, for example,

"p + q

where p is the probability of success", or

"$(x_{1}+\cdots +x_{n})/n\,$

where n is the sample size", etc. The word "where" then defines the notation. That's the right way to use "where". Michael Hardy (talk) 16:54, 8 August 2010 (UTC)

The >> notation comes from the C programming language. It's not really appropriate to use it in a general-mathematics context. --Trovatore (talk) 17:46, 8 August 2010 (UTC)

I think it's probably okay in that context if a line explaining it is put in. Whatever is done one would need an explanation. Dmcq (talk) 18:53, 8 August 2010 (UTC)

Surely $\left\lfloor {\frac {n}{2}}\right\rfloor$ is both simpler and clearer ? Gandalf61 (talk) 20:10, 8 August 2010 (UTC)

Depends on wether you consider a binary logarithm a mathematics or computer science concept. Much of the material of this article comers from "Hackers Delight", which is all about low level programming. In that context >> is the correct operation to use as it is a much faster operation than other ways of performing division.--Salix (talk): 21:39, 8 August 2010 (UTC)

It's still kind of language-specific — C and languages that copied it from C. I suppose it's possible that it's been absorbed into CS generally, but I haven't heard about it. --Trovatore (talk) 21:48, 8 August 2010 (UTC)

It is not an issue here because >> is no longer used in the article other than in a C code snippet. In my (admittedly dated) experience, we used >> freely in both pseudocode and general formula where appropriate in numerical analysis classes. In such contexts ⌊n/2⌋, while mathematically equivalent, is misleading as it suggests a non-integer division followed by a floor. -- 119.31.121.87 (talk) 23:38, 8 August 2010 (UTC)