Wikipedia:Reference desk/Archives/Mathematics/2007 October 1

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[edit] October 1

[edit] Math

I would like to know what are the instruments of addition? (A list of instruments) —Preceding unsigned comment added by 71.252.34.246 (talk) 01:39, 1 October 2007 (UTC)

Clarinets of summation? Strad 02:18, 1 October 2007 (UTC)
Organs of increase? -- JackofOz 03:40, 1 October 2007 (UTC)

Maybe he means the cymbals.

+,OR,||,...--Mostargue 04:50, 1 October 2007 (UTC)

Perhaps he means devices used to perform addition calculations, like an abacus, adding machine, or calculator. StuRat 14:51, 1 October 2007 (UTC)

[edit] What's the biggest set in use?

Okay, we have the finite sets, then the integers, then the reals, the the complex, but what after that?--Mostargue 05:03, 1 October 2007 (UTC).

The set of irrational complex numbers. 24.250.139.137 06:25, 1 October 2007 (UTC)ForgotMyLogin

The previous response is nonsense. See cardinal number --- there's a whole theory about this sort of thing (and there is no biggest set). If you want an explicit example of something larger, I should first note that the cardinality (think "size") of the complex numbers is the same as that of the reals. For an actual example of something larger, see power set; the power set of the reals is larger than the set of reals. (Also, I should note that the reals have the same cardinality as the power set of the integers.) As for biggest set "in use," that's presumably subjective and I have no idea, but the references I gave should get you started.Kfgauss —Preceding signed but undated comment was added at 06:31, 1 October 2007 (UTC)

It's difficult to answer your question, because the mathematical objects you name don't really fit into a sequence. Most of the sets you describe are sets of numbers, which fit into a hierarchy:

  1. The natural numbers
  2. The integers
  3. The rational numbers
  4. The real numbers
  5. The complex numbers

Each step in this hierarchy is obtained from the last by making the set closed under certain arithmetic operations:

  1. The integers are obtained from the natural numbers by closing under subtraction
  2. The rational numbers are obtained from the integers by closing under division
  3. The real numbers are obtained from the rational numbers by closing under the process of taking limits
  4. The complex numbers are obtained from the reals by closing under square roots.

In some sense, the process of adding more numbers is complete at this point. In fact, the fundamental theorem of algebra states that any polynomial equation over the complex numbers has a solution. On the other hand, various generalization of the complex numbers have been defined, including the quaternions and Clifford algebras. Matrices can also be viewed as a generalization of the concept of number to multiple dimensions.

If you're more curious about sets than numbers, the first step is indeed finite set, followed by infinite set (such as the integers or rationals), followed by uncountable set (such as the real or complex numbers). See also cardinality and the other links provided by Kfgauss. It is important to realize that not every set in mathematics is a set of numbers: as has already been mentioned, the simplest way to make larger sets is to consider sets of sets (e.g. power sets). Jim 06:47, 1 October 2007 (UTC)

If you think my first answer is nonsense then prepare for this one: The largest set is defined recursively as the superset of itself. So just think of any set, take the superset of it and you will get the largest set if you keep applying this definition an infinity of times. 24.250.139.137 08:17, 1 October 2007 (UTC)ForgotMyLogin
Befor anyone can understand your response, I think you have to explain what you mean by "take the superset". You are clearly not using superset in the standard sense of the term, because (a) a superset of A is any set that contains A as a subset - how do you distinguish the superset ? (b) with infinite sets, a set and one of its (proper) supersets can have the same cardinality - for example, the set of even integers has the same cardinality as the set of integers - so how do you ensure that your operation of "take the superset" produces a superset that has a greater cardinality ? Gandalf61 08:55, 1 October 2007 (UTC)
Perhaps anon has meant power set, or the union of a set with its power set? Even if so, the proposed method is flawed, since you can continue after taking the union of ω applications, and anon hasn't specified any other interpretation of "applying an infinity of times". Of course, in ZFC set theory there is no largest set; at best, there is the collection of all sets, which is the largest class but is not a set.-- Meni Rosenfeld (talk) 09:24, 1 October 2007 (UTC)

Hi. Yes I am talking about cardinality. A finite set has a smaller cardinality than an infinite set, right? The only thing I had to think twice about was the jump from reals to complex. The set of complex numbers, is it a higher cardinality of the set of real numbers? --Mostargue 18:50, 3 October 2007 (UTC)

No, the cardinality is the same. Think of a complex number in terms of its real and imaginary parts, and interleave the decimal digits of those to get the decimal expansion of a real number. Well, roughly speaking anyway. You might have to do a little work to make sure the 999... problem doesn't cause any ambiguity.
This is the result that made Cantor write "Je le vois, mais je ne le crois pas". --Trovatore 19:03, 3 October 2007 (UTC)

I think I get it, but I'm not quite sure. Is there any formal proof that complex numbers and real numbers have the same cardinality?--Mostargue 21:49, 3 October 2007 (UTC)

Yes, there is. Trovatore has indicated one method; the 'little work' referred to is not particularly difficult, and can be made easier by use of the Schroder-Bernstein theorem. Algebraist 16:44, 4 October 2007 (UTC)

[edit] Coupling from the Past

OK, so random Wiki-browsing led me from Markov Chains (which I understand) to MC Monté Carlo (which I kinda get) to Coupling from the Past, which I get, a little. I get enough to wonder about one thing.

In this paper, they rightly make the point that the first place where all the different chains coalesce might not be able to take a certain state, even if that state has a presence in the stationary distribution (e.g. if that state is only "fed" directly from a previous state, where all the chains would have had to have coalesced one time interval ealier). But they then say that if, rather than looking at the first time T after 0 when the chains coalesce, you look at the first time M such that STARTING at -M the chains coalesce at 0, THEN you get an accurate representation of the stationary distribution.

Is this because, when you're running "from the past", adding the new random seeds (rt) to the start of the chain means that the later chain is just "messed up" somehow, whereas adding the seeds at the end of the chain doesn't have this "messing up" effect?

So yes. This is all pretty muddled. I don't usually come across maths I don't understand! I hope that, despite this subject not having a wiki article, someone might be able to help me out here ^^

Rawling4851 15:47, 1 October 2007 (UTC)

Your link doesn't seem to work for me, and I don't quite get what you mean by "messing up", but have you checked out the original Propp & Wilson paper available here?
The basic trick in the proof that the output of the algorithm is from the stationary distribution of the chain, is that once you have coalesced the n-step maps, you can conceptually extend the chain further backward, as if you had run the chain forward for a large fixed number of steps. Then you can let the number of steps approach infinity, and you can do this without requiring any extra work in the algorithm.
On the other hand, if you run multiple copies of the chain forward until they coalesce, the same extension trick does not work - you would have to actually run the chains forward for a large, fixed number of steps in order to approach the stationary distribution. 84.239.133.38 19:16, 1 October 2007 (UTC)
Ah, I Think I'm a little out of my depth. Better keep taking my Probability units, eh? Cheers Rawling4851 07:57, 2 October 2007 (UTC)
Hey, if you are interested, it's worth your while to take another look. It is a cool algorithm and the proof is surprisingly simple. 84.239.133.38 17:22, 2 October 2007 (UTC)

[edit] Rydeberg formula question

According to Rydberg formula, this is the equation:

\frac{1}{\lambda} = R_{\mathrm{H}} \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)

Now, I was asked to solve for "n2". So this is what I did:

\cfrac{1/\lambda}{R_{\mathrm{H}}} = \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)

\cfrac{1/\lambda}{R_{\mathrm{H}}}-\frac{1}{n_1^2} = -\frac{1}{n_2^2}

-\cfrac{1/\lambda}{R_{\mathrm{H}}}+\frac{1}{n_1^2} = \frac{1}{n_2^2}

-\cfrac{R_{\mathrm{H}}}{1/\lambda}+\frac{n_1^2}{1} = n_2^2

\sqrt{-\cfrac{R_{\mathrm{H}}}{1/\lambda}+n_1^2} = n_2

But then I'm still wrong. What did I do wrong?? --Jeevies 18:47, 1 October 2007 (UTC)

-\cfrac{1/\lambda}{R_{\mathrm{H}}}+\frac{1}{n_1^2} = \frac{1}{n_2^2}
-\cfrac{R_{\mathrm{H}}}{1/\lambda}+\frac{n_1^2}{1} = n_2^2
The incorrect step is between these two lines. It looks like you tried to take the inverse of both sides, but the inverse of A + B is not 1/A + 1/B. —Keenan Pepper 18:52, 1 October 2007 (UTC)

Then what is? --Jeevies 19:10, 1 October 2007 (UTC)

The inverse of A + B is 1 / (A + B). That probably doesn't help solve the problem though. --LarryMac | Talk 19:21, 1 October 2007 (UTC)
I hate to tell you this, and I don't know how to say it without sounding patronizing, but by the time you are assigned such problems as solving the Rydberg formula for hydrogen for one of the two numbers, you ought to be well beyond the stage in which a student makes such errors. It seems to me that you'd be well advised to invest some time in brushing up on elementary algebra.  --Lambiam 22:40, 1 October 2007 (UTC)

>_< [[Chilling Effect|Chilling.] It took me a while to get that whole thing to "parse" right! Anyways, I looked through a formula sheet, but I'm still confused. How do I do this? Please? --Jeevies 23:06, 3 October 2007 (UTC)

[edit] Help teaching Scientific Notation?

I am currently doing field experience in a Middle School in New York State, and one of the 7th grade students is having trouble with scientific notation. It seems the heart of the problem is that he has not learned the faster method of multiplying by ten, since he uses the classic

   1262
   x 10
   -------

to solve what should be a simple mental calculation. Unfortunately, I am at a loss as to how to help this student. Does anyone have any ideas on how I can teach the method of adding zero/moving the decimal, other than standard math classroom methods? Some concrete examples would be very helpful as well, since I can't think of any. US Sports examples would be especially helpful, as I am know next to nothing about sports.

209.51.73.60 21:58, 1 October 2007 (UTC)

Can't you tell the student that in a multiplication you are allowed to shift final digits 0 from one number to the other: 7 × 800 = 70 × 80 = 700 × 8?  --Lambiam 22:24, 1 October 2007 (UTC)
This kind of learning gap can be usually be addressed by means of visualization. Building on the example provided by Lambiam, you might suggest that 70 × 80 means "exactly the same thing" as 700 × 8 because the rules of multiplication allow the zeros to "jump" back and forth.
Eventually, you will have to introduce that 700 × 8.0 and 7000 × .8 and 70000 × .08 also mean "exactly the same thing" but that the "jump" rules are a little different when dealing with decimals, because you have to remember about the "dot". Then you just give the student a few iterations of applying the "jump" rules on their own.
Some math teachers wince at this kind of method, because it is rather crude, imprecise, inelegant and informal (for example, this assumes Base10), but many times a student's eyes will light up because they will be able to picture the symbolic manipulations more easily, and it will be less "abstract" to them. Sometimes this is far more helpful than trying to talk about the numbers in the context of sports scores or applied mathematics, because the student is not struggling with practical application of the knowledge, but with the fundamental customs of symbolic knowledge representation as have been handed down to us by fiat. dr.ef.tymac 02:08, 2 October 2007 (UTC)
I humbly suggest consulting the experienced teachers at The Math Forum.
Like all personalized teaching, it's a matter of probing the student's existing mental model and helping the student construct a more effective revised and expanded model. You've made a start, and it looks like several concepts are missing.
Better understanding of multiplication will be crucial. At most a dozen exercises of multiplying by powers of 10, plus a tiny amount of discussion, should solidify the idea of shifting decimal points. Ideally, the student understands the theory of position notation, but I'm guessing not.
Next we need to bring in the commutative law, so that we can go from (a×10b)×(c×10d) to (a×c)×(10b×10d). We fervently hope this is already solidly familiar, and merely reinforce its existence and applicability.
The next step is to discuss adding exponents; this is an important concept that must not be taken as "obvious". Thus we gain the ability to reduce 10b×10d to 10b+d. Be sure to cover negative exponents.
Finally we come to the hardest part. Having introduced the freedom to slide decimal points around by adjusting exponents, we now must confront the need, in addition and subtraction, to align decimal points. Our theoretical basis is the distributive law. That is, given (a×10b)+(c×10b), where the exponent in both summands is b, we can write (a+c)×10b. Without great emphasis and a solid mental model a student will be irresistibly tempted to skip the alignment step, with disastrous consequences.
Most likely we will have already discussed a normal position for the decimal point already. Since our calculations will often leave the decimal point somewhere else, we must finally remember to "normalize" the result.
At each stage we may wish to have a few exercises/examples to display a pattern, then discussion of theory to point out the pattern and its importance and use, then adequate exercise to build competence and confidence.
I haven't offered specific exercises to use (again, try The Math Forum); I do hope the framework is helpful. --KSmrqT 05:09, 2 October 2007 (UTC)
Response to KSmrq
Like all personalized teaching, it's a matter of probing the student's existing mental model and helping the student construct a more effective revised and expanded model
Ideally, the student understands the theory of position notation, but I'm guessing not.
I humbly suggest consulting the experienced teachers at the Math Forum.
All of these are excellent points, and in fact reinforce my original emphasis on (the possibility) of using fanciful visualizations as an option. Because it tends to be the last method considered by mathematicians, and the first method to make sense to young people.
Concrete and simplified example:
To clarify what I am talking about, here's a very simple example. The question is:
 "When dealing with inequalities, why do you flip the inequality sign when   
 multiplying by a negative number?"
The following links all show different mathforum.org responses to this "stumbling block" for students, with a summary description of the answer methodology enclosed in parenthesis:
  • [1] ;; (algebraic description )
  • [2] ;; (verbal description [based on textbook definition] )
  • [3] ;; (visualized description [number line] )
  • [4] ;; (visualized description [balanced scales with "anti-gravity"] )
  • [5] ;; (visualized description [number line and balanced scales])
All of the answers here are legitimate, but note the last answer. The respondent clearly indicates there are different ways to think about this, by saying:
   The answer depends on whether you need to convince yourself or a 
   mathematician. Here's how I like to convince kids ...
This answer is quite unique, because it is the only one out of the bunch that acknowledges there are different ways to "convince" someone of this basic principle of arithmetic. Each kind of "convincing" employs a different methodology. Some students simply are not "convinced" by algebraic descriptions or an explanation of the distributive law, even though these are the "modalities of choice" for a trained mathematician, even though these are (ostensibly) the principles that you are intending to convey to your students.
All of this just re-emphasizes the point already made: it's a matter of probing the student's existing mental model -- and sometimes that mental model does not respond well to the symbolic and theoretic apparatus of arithmetic and algebra. dr.ef.tymac 15:34, 2 October 2007 (UTC)