Wikipedia:Reference desk/Archives/Mathematics/2007 November 30

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November 30

100 Coin Flips

Given infinite time, is it possible for there to be heads every time in 100 coin tosses?74.139.233.170 (talk) 02:30, 7 December 2007 (UTC)

I take the question to mean: In an infinite sequence of tosses of a fair coin, does a string of 100 consecutive heads exist with nonzero probability? Yes. —Tamfang (talk) 00:46, 30 December 2007 (UTC)

Correcting forecasts

Are there any other means of correcting forecasts by taking into account how they matched the actual in the past? I am aware of Theil's optimal linear correction, also called Theil's OLC. But are there any others please? I seem to recall that Trigg had something perhaps. 80.0.109.29 (talk) 00:13, 30 November 2007 (UTC)

I'm nor familiar with Theil's OLC, but here is a relatively simple idea. Given a method for making forecasts, you can plot on the x-axis forecasts produced for various data sets as they were available in the past versus on the y-axis the actual outcomes. Mark where the difference is negative or positive. If the difference is due to normally distributed random effects, there should be about as many sign flips as non-flips, as x increases. If the number of the non-flips is significantly higher than expected under the "random error" hypothesis, a correction may be possible. Fit a function from a class of simple models to the plot (for example using linear regression), and apply that function to the forecast produced by the model. Success in the sense of an actual improvement is not guaranteed, but more likely if the significance of the flip discrepancy was high and the function shows a nice fit. For example, if Y_i = rⁱ, where i = 0, , 2, ..., and the forecast model extrapolates the best linear fit to the last three observations, you get the forecast X_i = (4r²+r−2)/(3r³) × Y_i, with a systematic linear bias.  --Lambiam 14:18, 30 November 2007 (UTC)

Thanks, that is similar to what I have found out about Theil's OLC. If the forecast was perfect, then if forecast and actual were plotted on a X-Y graph you would get all the forecasts falling on a 45 degree straight line with the origin at zero. Theils OLC apparantly finds the best-fitting line (by linear regression) that the forecasts actually fall on, and adjusts points on that line into the 45 degree, zero at origin one. I have been thinking that you could extend this by fitting a curved line if you had enough data points to make it worthwile. Or, if you had forecasts over a long time frame you could fit a plane in an X-Y-Time graph rather than just a line, which would take care of gradual changes in accuraccy. Do you think there are any other ways of doing it? (And I wonder if you'd get continuing improvements in accurracy if you applied the above over and over again?)80.2.193.243 18:07, 30 November 2007 (UTC)

There is an article on exponential smoothing, a systematic way of updating a forecast by analysing the ongoing error. The article is fairly introductory and makes no reference to Trigg's tracking signal, a specific technique aimed at improving the forecasts. A Google search gave plenty of references, however. 86.152.77.73 19:20, 30 November 2007 (UTC)

Exponential smoothing applied to a case like Y_i = rⁱ systematically underestimates (if r > 1) the next value, just like extrapolation of the best linear fit to the last three observations but more so, and may likewise be improved by a correction. The input to a method making the forecast could be some description like the style, condition, age, dimensions, and number of colours of a Persian carpet, together with descriptions and prices obtained for similar carpets in the past, and the model might then attempt to forecast how high the winning bid will be in an auction. The model can be arbitrarily complex, beyond anything that smoothing methods can cope with, give good results, and still possibly benefit from a correction eliminating bias.  --Lambiam 00:39, 1 December 2007 (UTC)

Number bases that are not natural numbers

Prepare your minds for a very strange and dumb question.

I am wondering if it is possible to count in a number base where the radix is not a natural number, or even a rational number. A couple days ago, I asked someone to state the value of π in the most precise way he could, and he said 10_π (10 in base π). This follows the logic that 10₂ = 2₁₀, 10₈ = 8₁₀, etc. At first I thought he was a genius because apparently he could exactly state the value of π using only two digits. But after thinking about it, I began to doubt the possibility of counting in such a base.

For example, say you wanted to count in base 2.5. According to the above logic, 10_2.5 = 2.5₁₀. But how do you count to 10_2.5? You could start with 1, then 2, then 10 (2.5 in base 10), then 11 (1 + 2.5 = 3.5 in base 10), then 12 (following this logic, 4.5 in base 10), then 20 (5₁₀). The Radix article on Wikipedia defines "radix" as "the number of various unique digits, including zero, that a positional numeral system uses to represent numbers." That makes it seem like there cannot be a base 2.5, for that would require 2.5 unique digits - which doesn't make sense. The counting system above uses 3 unique digits: 0, 1, and 2. Another weird thing is that the difference between 10_2.5 and 2_2.5 is half the difference between 2_2.5 and 1_2.5 or between 1_2.5 and 0_2.5. (2.5 - 2 = .5 while 2 - 1 = 1 and 1 - 0 = 1.) Why can't the small space be counted first instead of third? Can't 2.5 be written as .5 + 2? And what about decimals? How do you divide up the smaller space? We don't run into this problem in normal bases because the spaces between the numbers are the same. What I have come up with is a number base that does not make any sense.

In fact, forget that paragraph. Is it possible to count in a number base where the radix is not a natural number? And if not, why not? Other mathematical principles seem to apply to all numbers, including fractions, decimals, irrational numbers, negatives, imaginary numbers, etc.

Of course, any smartass could make up a numbering system and define 10 as having the exact value of π, and then work in the rest of the numbers around that. But that would just be stupid.

Also, what is 1 + 1?

71.227.1.59 (talk) 03:34, 30 November 2007 (UTC)

First of all, I realize this question has nothing to do with $\pi$'s definition. It's about its expression in some numerical format/base/system.

My first suggestion is a look at non-standard_positional_numeral_systems, which extends your inquiry in other directions.

My second suggestion (maybe it should be my first one!) is: consider a numerical base b as a system that expands b-adically a number n in the following sense:

$n=\textstyle \sum _{k=-\infty }^{\infty }a_{k}\cdot b^{k}$

...but here the problem arises: concretely, which values can $a_{k}$ take? In binary every $a_{k}$ is either one or zero.In the decimal system, $a_{k}$ ranges from 0 to 9. But what about the system your friend has announced? Pallida  Mors 05:42, 30 November 2007 (UTC)

Not all 'mathematical principles apply'. Consider base 3/2 for example. A repeating fraction 0.1111..._(3/2) value would be the sum of a geometric series with the first term equal $0.1_{(3/2)}={\tfrac {1}{3/2}}={\tfrac {2}{3}}$ and the common ratio also $0.1_{(3/2)}={\tfrac {2}{3}}.$ That is ${\tfrac {2}{3}}\cdot {\tfrac {1}{1-{\frac {2}{3}}}}={\tfrac {2}{3-2}}=2.$ On the other hand 1.0_(3/2) = 1 < 2, so we got:

0.1111..._(3/2) > 1.0_(3/2)

and even

0.1111..._(3/2) > 10.0_(3/2) = 3/2

which breaks the rule of comparing and subtracting numbers digit-by-digit. --CiaPan (talk) 07:39, 30 November 2007 (UTC)

First, a general observation - just because someone comes up with something clever doesn't mean he is a genius (especially since you don't know if he thought about it originally). Disregarding any potential problems with non-natural bases, this is not even that clever since it basically just says $\pi =\pi$.

Another general observation: Your idea that "Other mathematical principles apply to all numbers" suggests that you are not familiar with enough mathematical principles. Every mathematical structure has its own properties, and not all of them exhibit the same phenomena. Of course, it is always wonderful when we are able to take some concept we have originally thought about in the context of some structure, and extend it to a more general structure - but that is not always possible.

Now, I second the suggestion of reading Non-standard positional numeral systems; in particular, note the possibility of Negative bases and some noninteger bases, such as Golden ratio base. However, let's look at it more generally. For a given base b, we will define an (as opposed to the) expansion of a number x in base b to be a sequence $(a_{k})_{k\in \mathbb {Z} }$ of real numbers such that:

$x=\sum _{k=-\infty }^{\infty }a_{k}b^{k}$

Now what we have is the problem of uniqueness - we would like (almost) every real number to have exactly one expansion in base b. However, for that we clearly have to pose some restriction on the $a_{k}$. For some bases we know of good ways to do that. The most well-known method is for a natural $b\geq 2$, where we pose the restriction that $a_{k}$ are integers between 0 and $b-1$. For $b=-3$ we can require that $a_{k}\in \{-1,0,1\}$. For $b=\varphi$ we can require that $a_{k}\in \{0,1\}$ and that "11" doesn't appear anywhere.

However, it is not at all clear that for every base b we can come up with some simple restriction that guarantees uniqueness. It might be possible for every algebraic number, but I doubt it is possible for any transcendental number like π. This doesn't, however, invalidate the claim that 10 (i.e., the sequence $a_{1}=1$ and $a_{k}=0$ otherwise) is an expansion of π in base π. So your friend has made a correct, but not really informative, statement. -- Meni Rosenfeld (talk) 08:15, 30 November 2007 (UTC)

As for 1+1, see our articles 1 (number) and addition. -- Meni Rosenfeld (talk) 08:36, 30 November 2007 (UTC)

To get a canonical expansion 0.d₁d₂d₃... of a number x, 0 ≤ x < 1, in base β > 1, you can take for each next d_k the largest β-ary digit – being an integer in the half-open interval [0, β) – such that (0.d₁d₂d₃...d_k−1d_k)_β ≤ x. For values of x ≥ 1, take the expansion of x/β and shift the point one to the right. You get, for example, e = (2.2021201002111...)_π and π = (10.1010020200021...)_e.  --Lambiam 15:31, 30 November 2007 (UTC)

Alright, I see. Somewhat strange, but it works. Just to complete your statement, for really great numbers, you find k such that

0 ≤ x/β^k < 1,

then expand and finally shift k places to the left, I guess. Pallida  Mors 21:54, 30 November 2007 (UTC)

Eulers function proof

Hello. To prove the following formula: $\varphi (p^{\alpha })$=$p^{\alpha }-p^{\alpha -1}$ my book states that "note that the numbers from 0 to $p^{\alpha }-1$ which are not prime to $p^{\alpha }$ are precisely those that are divisible by p and there are $p^{\alpha -1}$ of those." What I don't understand is as how do we get the figure $p^{\alpha -1}$ for such numbers. ( p is a prime number and $\varphi (n)$ is the number of nonnegative integers relatively prime to n & less then it). Please help.--Shahab (talk) 10:48, 30 November 2007 (UTC)

How many integers between 1 and 15 are multiples of 3 ? Answer: 5 because every 3rd integer is a multiple of 3 and 15/3=5.

How many integers between 1 and 25 are multiples of 5 ? Answer: 5 because 25/5=5.

How many integers between 1 and 5p are multiples of p ? Answer: 5 because 5p/p=5.

How many integers between 1 and kp are multiples of p? Answer: k because k=kp/p.

Now suppose kp=p^α. What is k ? Gandalf61 (talk) 11:54, 30 November 2007 (UTC)

Incidentally, your book seems to be using a non-standard definition of $\varphi (n)$, which is usually defined as "the number of positive integers less than or equal to n that are coprime to n". I guess the definitions are equivalent because 0 is a multiple of every integer, and so not coprime to any integer. Gandalf61 (talk) 12:19, 30 November 2007 (UTC)

Thanks.--Shahab 16:05, 1 December 2007 (UTC)

...because 0 is a multiple of every integer... I don't understand everything I see here, but isn't this a typo? -SandyJax 20:30, 4 December 2007 (UTC)

Not at all. It doesn't come up much in school, but it's true. An integer X is a multiple of an integer Y if there is some integer Z such that X = Y*Z. If X = 0, then with Z = 0 we have 0 = Y*0, which is true for any integer Y. There are times when "multiple" is used to mean "positive multiple", and then 0 doesn't count. Black Carrot (talk) 04:29, 5 December 2007 (UTC)