Wikipedia:Reference desk/Archives/Mathematics/2011 February 6

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

February 6

Un-deducting percentages

I have a net amount of $150.98, and need to find out the gross amount that 4.95% and 1.73% were deducted from to reach the net of $150.98. Can you help — Preceding unsigned comment added by Nannyno1 (talk • contribs) 06:25, 6 February 2011 (UTC)

This looks like homework. We will not do people's homework for them. If you try to solve the problem yourself and get stuck, feel free to show your work up to that point, and we'll try to point you in the the right direction. In this case, your first thing to do would be to figure out whether the two percentages were deducted sequentially (first take away 4.95% of the gross, then 1.73% of what remains) or at once (take away (4.95+1.73)% of the gross in a single operation). –Henning Makholm (talk) 08:51, 6 February 2011 (UTC)

Click here for a basic explanation. The second example is particularly relevant. Have a go at following that, and see if you can work out how to solve your problem. You are very welcome to ask for further help when you've had a go, perhaps explaining where you are stuck or what you've tried so far. 86.164.58.119 (talk) 11:42, 6 February 2011 (UTC)

If you know algebra, these sorts of problems are somewhat easy to work out. If you start with the original value (call it x), subtract off 4.95% and 1.73% (figure out how to write these in terms of x), the result is $150.98. Write that down in equation form and solve for x. -- 174.24.195.38 (talk) 17:58, 6 February 2011 (UTC)

... but if algebra is not to your liking, you can still easily solve the problem by noting what operations you would do on the original "unknown x" to arrive at $150.98, then you can start with $150.98 and just apply the inverse operations in reverse order to find the "unknown x". Division is the inverse of multiplication. Dbfirs 22:51, 6 February 2011 (UTC)

But that is just doing the algebra without calling it that. And being able to see that, say, $x-7\times (x/100)$ is the same as $0.93\times x$ is essentially algebra. –Henning Makholm (talk) 09:08, 7 February 2011 (UTC)

Yes, of course, but some people are turned off by algebra, and it it not necessary to set up an equation to solve this type of problem. In the UK, the "School Mathematics Project" used to introduce algebraic solution of equations through inverse operations. From my experiments (OR) I observed that students grasped this method more quickly, but were subsequently slower to learn the "rules of algebra". Dbfirs 10:36, 7 February 2011 (UTC)

Oddly, I've found that many of those who are capable of inverse operations, but panic at algebra, are quite capable of algebra if you use symbols that are not letters. It's a very specific block that I assume must come from our surrounding culture somehow. Little pictograms are easily handled, but throw in an x and they don't understand :/ 212.183.128.67 (talk) 11:59, 7 February 2011 (UTC)

Yes, I've found that, too. An empty square is often used for an "unknown". I don't use algebra myself for reverse percentages, perhaps because I learnt to calculate them before I had ever heard the word "Algebra". Dbfirs 01:17, 8 February 2011 (UTC)

I don't think there's any argument that for a question like the OP's, explicitly writing down an equation with a named variable is unnecessary overhead. -- Meni Rosenfeld (talk) 07:32, 8 February 2011 (UTC)

Integers

Are integers defined by our experience of reality, or are they something more innate? I.e. Is there a definition which appeals to something more fundamental than that they are the numbers which indivisible 'stuff' comes in? If we lived in a universe with a fractional dimensionality, would our experience of which numbers are integers be different? Would indivisible 'stuff' come in different quantities? —Preceding unsigned comment added by 129.67.37.227 (talk) 13:13, 6 February 2011 (UTC)

Our articles on the Natural Numbers and the Integers have some explanations for this. I would say that in short, yes, there is something fundamental about the natural numbers (and the integers). If something is 'indivisible', then it really is counted by the natural numbers (although, I have to admit, I'm not sure about this concept of living in 'fractional dimensions'). Invrnc (talk) 14:05, 6 February 2011 (UTC)

I can't answer the question, but I'll just note that there are many mathematical concepts which on the surface seem only applicable to integers but in fact can be generalized, such as fractional Hausdorff dimension and fractional calculus. -- Meni Rosenfeld (talk) 14:53, 6 February 2011 (UTC)

It's a partial generalization. You can have fractional dimensions in certain senses (self-similarity or the behavior of the measure of balls of a certain radius), but you can't have one-and-a-half mutually orthogonal directions.

Things that you do one at a time are really, inherently different from things that vary continuously. I remember as a kid I didn't like this and it took me a long time to come to terms with, but it's really true.

On the other hand, things that you do one at a time are not limited to things you can index by the integers. Set theorists routinely use transfinite recursion to do uncountably many things, one at a time. --Trovatore (talk) 19:26, 7 February 2011 (UTC)

I don't know if this is "more innate" than the "numbers which indivisible 'stuff' comes in", but to construct the integers, one needs only the notion of the empty set, and the notion of set inclusion. This construction of natural numbers has the same general idea. Presumably the empty set and set inclusion exist in any universe, but whether you agree may depend on your philosophy of mathematics.SemanticMantis (talk) 03:23, 7 February 2011 (UTC)

“God gave us the integers; the rest is the work of Man.” —Leopold Kronecker. Sławomir Biały (talk) 14:02, 7 February 2011 (UTC)

I like that quote too, but once you've accepted the existence of the empty set and set inclusion, the existence of real numbers becomes a necessary truth as well ;-)SemanticMantis (talk) 18:36, 7 February 2011 (UTC)

Kronecker's God was too small. (Or, more precisely, Kronecker had too small a conception of God.) --Trovatore (talk) 18:52, 7 February 2011 (UTC)

Differential equations are probably more innate than integers. Your dog probably can't count past 5 so, but if you throw a stick, s/he can predict where it will land and run towards that spot. 71.141.88.54 (talk) 07:52, 8 February 2011 (UTC)

mathematics

50oranges were in a bag,25 is bad.what is the probability of picking a bad oranges at random.what is also the probability of choosen a good one without replacement. —Preceding unsigned comment added by 41.206.12.1 (talk) 14:48, 6 February 2011 (UTC)

If I take your first question literally, the answer is trivial: 50%. Your second question is nonsense, because you haven't specified how many oranges you remove without replacement. However, suppose you remove n oranges. Then the probability of all of them being bad is:

${\frac {\binom {25}{n}}{\binom {50}{n}}}$

That is, the number of ways of choosing from just the 25 bad oranges, over the total number of ways of choosing from 50 oranges. So the probability of any one of them being good is 1 minus that:

$1-{\frac {\binom {25}{n}}{\binom {50}{n}}}$

See Binomial Coefficient if the notation is unfamiliar to you. In short, ${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}$ is the number of ways of choosing k things out of a collection of n things. HTH. --COVIZAPIBETEFOKY (talk) 16:24, 6 February 2011 (UTC)

Calculus

What's the integral of (f(x))^2? --75.15.161.185 (talk) 16:33, 6 February 2011 (UTC)

That depends on what f(x) is. There isn't a general formula. Perhaps you really want the integral of $f(x)^{2}f'(x)$. To do that, use Integration by substitution (u-substitution), which is the analogue of the chain rule for integration. In that case, you should get $(1/3)(f(x))^{3}$. Staecker (talk) 18:03, 6 February 2011 (UTC)

Input-to-state stability

What is input-to-state stability? I discovered this article while new-page patrolling; the language is confusing, and it has no references or links that can help me understand the topic. It looks mathematical, and Google results for the phrase are mostly from university math department websites, but anything beyond that is more than I can understand. Nyttend (talk) 20:28, 6 February 2011 (UTC)

Hmm, I just discovered that the article is a copyvio of one of the top Google results. However, since pretty much that entire page was copied to here, it doesn't help me understand at all. Nyttend (talk) 20:30, 6 February 2011 (UTC)

If you still want to know, it looks like a topic in dynamical systems theory. It resembles Lyapunov stability which I see has a redlink to it. There is a book by Hirsch, Smale, and Devaney (ISBN 0123497035) that should be a very good introduction to the general subject. I'm only familiar with the older edition which had a somewhat different approach, but I liked that one a lot, and the Amazon reviews say that the new one is more accessible. 71.141.88.54 (talk) 12:13, 7 February 2011 (UTC)

I don't understand the terminology in that handout, but I think it basically describes a system like a rocking chair, which is at equilibrium at a certain displacement (call this 0), and if you push it 3 inches with your finger and then let go, it will start rocking back and forth, but the amplitude of the rocking will be bounded. That contrasts with an umbrella balanced on its tip, so if you poke it, it falls over (unbounded displacement). 71.141.88.54 (talk) 14:06, 7 February 2011 (UTC)

Compact manifold without boundary not contractible

Could anyone please link me to a proof that a compact manifold without boundary is not contractible? I would prefer if it was not based on things like cohomology but rather closer to first principles, though fairly basic or fundamental theorems in differential geometry such as the inverse function theorem/preimage theorem/Sard's theorem are okay. I am teaching myself material from a set of online notes alongside a number of other courses, but unfortunately the proofs are omitted, and while i would like to derive the proof myself, I'm afraid I don't really have the time currently. Any help greatly appreciated, Estrenostre (talk) 23:12, 6 February 2011 (UTC)

Sorry, I don't know a proof from just calculus, but this follows quite easily from the homotopy-invariance of the n-th singular homology group over Z₂. If I were pressed to not use any topology, then I might try to construct a nonconstant harmonic function on a given contractible manifold. I have some ideas how this can be done, but nothing that seems particularly easy to me. Sławomir Biały (talk) 13:50, 7 February 2011 (UTC)

The way it's stated, the claim is false: A point is a zero-dimensional compact manifold without boundary, and it's contractible by definition. To make a working argument you somehow need to use the positive dimensionality; which leads us to homology (or maybe homotopy) because it's an obvious way of getting at positive dimensional information. I have a feeling that every proof we might come up with in some way comes down to homology. Ozob (talk) 12:05, 8 February 2011 (UTC)

Could you not use the fact that every open cover of a compact space has a finite subcover? Assuming the space to be Hausdorff should give a contradiction some how. If you assume a homotopy does exist then you could prove that the homotopy wasn't continuous, and that's a contradiction. I know it's very hand-wavy. It's just an idea of what I might try to do. — Fly by Night (talk) 15:41, 8 February 2011 (UTC)