Wikipedia:Reference desk/Archives/Mathematics/2007 November 7

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November 7

Sums of successive powers

Why is

$1\,+\,r\,+\,r^{2}\,+\,r^{3}\,+\,\cdots \;=\;{\frac {1}{1-r}},$

true? Imaninjapirate^{talk to me} 00:51, 7 November 2007 (UTC)

Multiply both sides by $1-r$ and see! It is derived from something called a Taylor expansion. Unfortunately, it only converges for $|r|<1$. SamuelRiv 00:58, 7 November 2007 (UTC)

Try this:

${\begin{aligned}S&=1+r+r^{2}+r^{3}+\cdots \\rS&=\quad \ \ r+r^{2}+r^{3}+r^{4}+\cdots \\S-rS&=1\\S&={\frac {1}{1-r}}.\end{aligned}}$

This should make the conclusion plausible. A proper formal proof requires careful attention to omitted details. --KSmrq^T 06:13, 7 November 2007 (UTC)

Or, if you are not comfortable with manipulating infinite series, start with the finite case:

$1\,+\,r\,+\,r^{2}\,+\,r^{3}\,+\,\cdots +\,r^{n-1}\;=\;{\frac {1-r^{n}}{1-r}},$

which is simply an algebraic identity. Then assume |r|<1 and show that you can make the right hand side as close as you like to ${\frac {1}{1-r}}$ if n is large enough. Gandalf61 10:25, 7 November 2007 (UTC)

Irksome Quotes

I wonder, is there any quote more grating for a true mathematician to read than the passage from Kings in the bible that details a round shape: "It was round in shape, the diameter from rim to rim being ten cubits...and it took a line thirty cubits long to go around it" Sappysap 02:03, 7 November 2007 (UTC)

Yes, "I'm no good at math" and "I hate math". A close second is "Do you ever watch that 'Numbers' show?" Donald Hosek 02:34, 7 November 2007 (UTC)

I'd put those marginally below "Why should I care about math?", and about the same level as "So, you just work with numbers, do you?" Confusing Manifestation 05:52, 7 November 2007 (UTC)

My all-time favorite is, "There is research in mathematics?", with varying degrees of astonishment.

Regardless, I wonder why people insist on perpetuating the false claim that "Bible says $\pi =3$". Never mind the common explanation, that 30 cubits is the circumference of the inner circle and 10 is the diameter of the outer circle (which is a bit of a stretch of the text, but supported by other findings). There is a simpler explanation (which may or may not be true) - the circumference is 30 and the diameter is 9.55, rounded to the nearest integer, 10 (or 30.34 and 9.66, if you prefer it). -- Meni Rosenfeld (talk) 09:16, 7 November 2007 (UTC)

Or that 10π=30 to the specified one significant digit. And to answer the original question, "I'm never going to use this." Followed closely by, "How to I know how much to tip?" — Lomn 14:18, 7 November 2007 (UTC)

One explanation is that at that junction of time and space the universe had a locally high curvature. If that doesn't work for you, consider that only "true mathematicians" would assume that a shape called "round" must be modelled mathematically as being perfectly circular. Thus, the grating suffered is self-induced and self-inflicted.  --Lambiam 09:46, 7 November 2007 (UTC)

"Will this be on the quiz?" is one that I especially loathe. I once had a student ask me what the largest number was, but that was more frightening than it was irksome. "Plug and chug" is another thing I hate; it sounds like the type of thing that college students of a weak moral fiber would engage in on weekends. –King Bee ^{(τ • γ)} 19:12, 7 November 2007 (UTC)

An acquaintance of mine (who lectured anthropology) had a standard answer to your first example: 'Will this be on the exam?' 'It will now.' I think I might hate 'plug and chug' more if I knew what it meant. And surely the largest number is the cardinality of the set of all sets? Algebraist 20:27, 7 November 2007 (UTC)

Just in case you were serious about your last comment - in ZFC and similar axiomatizations, the collection of all sets is not a set. It is a proper class, and as such, has no cardinality. I don't know if the set of all sets has a cardinality in those other set theories that have it. -- Meni Rosenfeld (talk) 21:38, 7 November 2007 (UTC)

As for "plug and chug", it means having a set of memorized formulas or algorithms and substituting in the values from the question to get to the answer. It is indeed, very much the sort of thing that students of weak moral character engage in on weekends (assuming that homework is due on Monday). Donald Hosek 00:27, 8 November 2007 (UTC)

I'm sure I have forgotten 110% of what I learned while getting my degree, but I always hated hearing "I'll bet you have no trouble balancing your checkbook!" --LarryMac | Talk 21:25, 7 November 2007 (UTC)

My ex-wife is a bookkeeper who can't do arithmetic (that's what machines are for). —Tamfang 04:38, 10 November 2007 (UTC)

Magnetopneumodynamic Drive

I asked a follow-up quesion about this on the Science Desk on 6 November 2007 and I thought I'd spread my bets by putting a reference here as well. I'm seeking a way of improving on the thruster design for an LTA ionocraft. (Sorry, I don't know how to link to the question from here.) If anyone can offer guidance, I'm mainly concerned with improving the airflow through the ionising system to get the greatest possible thrust from the power supply. Retarius | Talk 02:12, 7 November 2007 (UTC)

The question wasn't answered because we're still not sure what you're talking about. Do you have any links or can you give a better description? SamuelRiv 14:32, 7 November 2007 (UTC)

Okay, now I know how to do some links! (Wasn't as hard as I expected.) My question is at Reference Desk/Science November 6: Question 5.4. The relevant article here is Ionocraft and the best external link is Blaze Labs Research.

SteveBaker has provided a response on the Science Desk which I wouldn't fault within the context of the underlying assumptions. With thanks for the time he took, I'm really looking to explore whether the basic design of the thruster can be changed to increase airflow pressure. (I know it's useless for practical purposes as it stands.) For example, would a curvilinear form for the electrodes and placing thrusters of diminishing diameter in series turbocharge the gizmo? Would placing baffles between the concentric sets of electrodes channel the ion cloud more efficiently?

My plan is to put the thing on a test bed and tinker with it to see what thrust I can coax from it. I have made some model aerostats previously (I savvy the buoyancy/aerodynamics involved) and I wouldn't bother to put one together unless I had a powerful enough drive to make it feasible. Retarius | Talk 03:33, 8 November 2007 (UTC)

GCD

Hi. Here's another problem related to gcd. Prove that (a,b,c)=((a,b),c). I tried to take (a,b,c)=g and (a,b)=d. Now g|a,g|b,g|c,d|a,d|b,g|d. What now? Help.--Shahab 17:04, 7 November 2007 (UTC)

Denote also $e=(d,c)=((a,b),c)$. You need to show that $e=g$, for which it is sufficient to show that $e\leq g$ and $g\leq e$. Remember, the standard way to show that the gcd of some numbers is at least some other quantity, is to show that this quantity is a common divisor of the numbers. -- Meni Rosenfeld (talk) 17:48, 7 November 2007 (UTC)

Thanks for the reply. I am extremely grateful. I solved the problem by showing e|g and g|e. Can you help me with two more: (1)Prove that no integers x,y satisfy x+y=100 and (x,y)=3. The only thing that I can use is the definition of gcd. It gives me A,B so that Ax+By=3.

(2)Prove that if an integer is of the form 6k+5 then it is necessarily of the form 3k-1 but not conversely.

What now?--Shahab 18:11, 7 November 2007 (UTC)

For (1), remember that if (x,y)=3, then there exist integers a,b such that x=3a and y=3b. You should be able to figure it out from there.

For (2), right the second expression as 3m-1 which should help make it clearer what you're looking for. You should be able to find a relationship between k and m which makes the implication clearly true and provides an easy path to a counterexample to show the converse is false. Donald Hosek 18:22, 7 November 2007 (UTC)

Thanks for the hint. I proved it now: (1) 3(a+b)=100 implies 3 divides 100 which is false and for (2) 2k+2=m. An odd m proves the converse. Cheers --Shahab 18:54, 7 November 2007 (UTC)

It is almost trivial if you use the following defining characteristic for gcd:

For all natural d ≥ 1, d|(a₁,...,a_n) ⇔ (d|a₁ ∧ ... ∧ d|a_n).

 --Lambiam 21:47, 7 November 2007 (UTC)

Poll question

Hello all, it's been a while since I've been to the ol' refdesk. I hope you will all indulge me as you have in the past.

The question before you is as follows: Suppose you had to invent notation for the collection of all k-element subsets of an n-element set. What notation would you use? Thanks! –King Bee ^{(τ • γ)} 17:25, 7 November 2007 (UTC)

I don't see how the n is relevant - the collection depends on the precise set. If I wanted to refer to the collection of k-element subsets of A, I would simply write. $\{B\subseteq A:|B|=k\}$. If I was forced to invent a notation, I would go for $P_{k}(A)\;\!$. -- Meni Rosenfeld (talk) 17:38, 7 November 2007 (UTC)

Actually that notation already exists and means something else -- though ordinarily k would be infinite. $P_{\kappa }(A)$ is the collection of all subsets of A having cardinality strictly less than κ. Sometimes [A]^k is used for the desired notion. --Trovatore 05:28, 8 November 2007 (UTC)

I would say n is relevant; see Hypergraph and Combinatorial design. Thanks for answering the question, though! I'll post the other suggestions I've gotten from mathematicians in the department here in a bit. –King Bee ^{(τ • γ)} 17:41, 7 November 2007 (UTC)

My graph theory lecturer, following Béla Bollobás's Modern Graph Theory, used A^(k) for the k-element subsets of A. He also had a tendency to write [n] for {1, 2, ... n}, so I suppose [n]^(k) would be the k-element subsets of the standard n-element set. By the way, does your break from the refdesk indicate that all your qualifying exams are passed? If so, congratulations! Algebraist 20:21, 7 November 2007 (UTC)

I prefer the notation ${A \choose k}$, similarly how you one can use $2^{A}$ for the power set of $A$. —Preceding unsigned comment added by B jonas (talk • contribs) 20:36, 7 November 2007 (UTC)

Dixon and Mortimer's Permutation Groups text (p34) also uses Ω^{k}. This is comparing k-transitivity with k-homogeneity and so it wishes to emphasize the similarity with Ω^k, the set of ordered k-tuples from Ω. JackSchmidt 20:42, 7 November 2007 (UTC)

Algebraist - yup! All that nonsense is completed finally. I'm putting the final touches on a paper that I'm almost ready to submit. I have probably a year and a half to two years to do some research before the final defense, so I'm pretty pumped about that. In fact, I was the only one out of 8 people who passed this most recent incarnation of the Analysis exam. Scary stuff.

JackSchmidt - Indeed. I've seen that Erdos and Rado (1956) also use similar notation, so it's a little more standard. I'm partial to the binomial coefficient-looking thing suggested by Jonas myself. –King Bee ^{(τ • γ)} 20:52, 7 November 2007 (UTC)

why do you need division if you have multiplication?

The first is just the opposite of the second, so why do you need such superfluous thing? —Preceding unsigned comment added by 83.57.66.241 (talk) 18:51, 7 November 2007 (UTC)

Perhaps you can clarify your question. If we have numbers x and y, we need some way to refer to the number that when multiplied by y gives x. The difference between $x/y\;\!$, $x\cdot y^{-1}$ and "the number that when multiplied by y gives x" is merely of convenience. -- Meni Rosenfeld (talk) 19:14, 7 November 2007 (UTC)

what is so convinient? Having one symbol instead of two would be much convenient, don´t you think? —Preceding unsigned comment added by 83.57.66.241 (talk) 19:19, 7 November 2007 (UTC)

No. "Convenient" means that the effort of writing an expression should be as little as possible. You haven't clarified how would you prefer to refer to $x/y\;\!$. I assume you mean $x\cdot y^{-1}$. Note that not only is this expression longer, it also requires the additional symbol ${}^{-1}$ (similarly to risk's comment below). We can't afford to have the notation for something as fundamental and frequent as division any longer than necessary. -- Meni Rosenfeld (talk) 20:01, 7 November 2007 (UTC)

(after edit conflict) Why do we need the word cold, when it's just the opposite of hot? We could just refer to typical winter temperatures as "not hot", right? I'm trying very hard to get into the kind of mindset that could produce such a question, and I'm having some trouble. Why would the fact that some notion is the opposite of something else, make it superfluous? Try to do the simplest mathematics without explicitly using division, and you see that the notion is inescapable. True, you can write 10/4 as 10 * 0.25, but if you want to calculate how many apples each person in a group of four gets when ten apples are divided fairly, you'll somehow need to figure what to multiply ten by. You can only find that number by dividing 1 by 4, so you still need division. This page from Dr. Math makes some great points about what inverse operators actually are. risk 19:29, 7 November 2007 (UTC)

You can express 10/4 as 10 × 4⁻¹, so your last point isn't really valid. I totally agree with the first part though. (By the way, that's one of the things I hate about Esperanto, where the word for "cold" really is malvarma ("anti-hot").) —Keenan Pepper 19:58, 7 November 2007 (UTC)

Now, my mathematical confidence is beginning to desert me, but I think there are two ways of interpreting x⁻¹ here. Either as the inverse operation of the group of integers under multiplication, in which case it represents 1/x, or as a negative exponentiation x^-n, which is defined (I think) as x = 1/xⁿ. It does mean you can manage without the division symbol, but I'd hardly call it an improvement. risk 20:20, 7 November 2007 (UTC)

There is a "group of integers under multiplication" now? Someone should tell me these things!

Anyway, I still have no idea what the question is about, but I am inclined to think it is, indeed, about doing away with the division symbol. -- Meni Rosenfeld (talk) 21:18, 7 November 2007 (UTC)

Right, sorry. That would be the rationals (sans 0, even). Perhaps I should read more about group theory than the first paragraph of the wikipedia article before I start bringing it up in conversation. risk 21:33, 7 November 2007 (UTC)

To return to the original question, yes, division is a redundant operator, provided as a convenience to be a shorthand for xy^-1 (likewise subtraction is a shorthand for x+(-y)).

Convenience and conceptual understanding are the key reasons for its existence. Both play big roles in students' learning, but pointing out the equivalence of the two concepts is helpful for getting through certain key concepts in algebra, particularly getting past the deceptive My Dear Aunt Sally mnemonic for the order of operations and simplifying operations with rational expressions (including, at the most basic level, fractions). Likewise, learning that subtraction is equivalent to adding the opposite enables students to better deal with operations on negative reals.

risk, there really is only one interpretation of x⁻¹. Note that your first example is a specific case of the second example with n=1. Note that in rings, when we talk about division, we define division in the expected way: x÷y = xy^-1. Donald Hosek 00:23, 8 November 2007 (UTC)

I agree with risk's description of $x^{-1}$. Sure, both interpretations end up being the same thing, but I think they differ in the point where you are able to define them. That is, if you have a group, first of all you need some notation to denote the operator of taking inverses. Once that's done, you can move on to define powers, and in particular, negative powers. From this it will follow that a power of -1 is exactly taking the inverse. In light of this, the notation ${}^{-1}$ is adopted to denote the inverse to begin with. But these are still two conceptually different things. -- Meni Rosenfeld (talk) 08:28, 8 November 2007 (UTC)

If you are prepared so solve equations you do not need division, nor subtraction, nor square roots. The solution to a+x=b is a subtraction and the solution to ax=b is a division and the solution to xx=a is a square root. Just do algebra on the equations. Bo Jacoby 12:05, 8 November 2007 (UTC).

There is a lot of redundancy in mathematics. Take the trig functions, for example. For convenience, sine, cosine, tangent, cosecant, secant and cotangent are defined and used, but only one of them is necessary. If "s" is sine, say, then cosine is root(1-s^2), tangent is s/(root(1-s^2)), cosecant is 1/s, and so on. Using only sine would be possible, but ridiculously long-winded.→86.132.238.228 17:57, 8 November 2007 (UTC)

Almost... "$\cos x={\sqrt {1-\sin ^{2}x}}$" is only correct up to sign. What you can do is $\cos x=\sin(x+{\tfrac {\pi }{2}})$. -- Meni Rosenfeld (talk) 18:25, 8 November 2007 (UTC)