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January 30

First digits of 10^n

How can I compute the first x digits of 10^n? N is not necessarily an integer. 149.169.132.90 (talk) 02:15, 30 January 2011 (UTC)[reply]

From first principles? Compute successive square roots of 10 to the desired precision (and add a few digits to absorb rounding errors in the next phase) until the value rounds to 1. Express the fractional part of n in binary, and multiply those of the square roots that correspond to 1 bits. (See exponentiation by squaring for theory of the latter step). If you want to be completely free of rounding errors, you can repeat the entire computation twice, always rounding up and down respectively, and increase the precision of intermediate values if the two results do not agree to the precision you desire.

(This is how Henry Briggs computed the first base-10 logarithm tables). –Henning Makholm (talk) 03:09, 30 January 2011 (UTC)[reply]

What do you mean when you say "Express the fractional part of n in binary"? Does this mean multiply by 10 ^ 8 (or whatever my precision is) to get an integer, and then express in binary? 149.169.132.90 (talk) 04:18, 30 January 2011 (UTC)[reply]

No, I mean express it as a binary fraction. –Henning Makholm (talk) 05:01, 30 January 2011 (UTC)[reply]

You're trying to find the antilogarithm to the base 10, where n is the logarithm (to the base 10). You can convert n to a natural logarithm by multiplying by the natural log of 10. Then 10^n is exp(n ln 10) which you can compute with the Taylor series for the exponential function. 71.141.88.54 (talk) 08:31, 30 January 2011 (UTC)[reply]

Arc length proof

hwo would I prove that the length of the curve defined by c(t)=(u(t), v(t)) from t=a to t=b is $\int _{a}^{b}{\sqrt {u'(t)^{2}+v'(t)^{2}}}dt$ ? 24.92.70.160 (talk) 03:34, 30 January 2011 (UTC)[reply]

Think about Riemann sums approximating this integral, along with the Pythagorean theorem. That does it. Michael Hardy (talk) 05:05, 30 January 2011 (UTC)[reply]

Some presentations even take your integral to be the definition of "length of the curve". If you don't have that as a definition, then you most likely need to work directly with whatever definition of curve length you do have. –Henning Makholm (talk) 05:09, 30 January 2011 (UTC)[reply]

I'm not sure it needs a formal proof, it follows from the definition of integration. If you insisted on a formal proof then you'd only need to prove that integration does what we're told it does. The point c(t) gives the position of a particle at time t. Then the derivative (dc/dt)(t) gives the velocity of that particle at time t. The magnitude of the velocity is the speed. Using a dot to mean differentiation with respect to t, the magnitude of (dc/dt)(t) is

||{\dot {\mathbf {c} }}(t)||={\sqrt {{\dot {u}}(t)^{2}+{\dot {v}}(t)^{2}}}\,.

Then, by the definition of integration, the integral of the speed between t = a and t = b (where a ≤ b) tells you how far the particle has travelled in that time. Thus

{\text{arc-length}}=\int _{a}^{b}\!||{\dot {\mathbf {c} }}(t)||\,\operatorname {d} t=\int _{a}^{b}\!{\sqrt {{\dot {u}}(t)^{2}+{\dot {v}}(t)^{2}}}\,\operatorname {d} t\,.

Notice that in differential geometry the phrase Euclidean arc-length is often used to tell this arc-length apart from others. That's because the arc-length in your question is a differential invariant of Euclidean transformations. There are other notions of arc-length that are invariants of different transformation groups, see for example special affine arc-length. — Fly by Night (talk) 13:49, 30 January 2011 (UTC)[reply]

"The definition of integration" in this context is usually taken to mean limits of Riemann sums. And the idea that the magnitude of a vector is the square root of the sum of the squares of the components is the Pythagorean theorem. Michael Hardy (talk) 18:11, 30 January 2011 (UTC)[reply]

Fly-by-night, you're confused about what is a definition and what is a theorem. Michael Hardy (talk) 18:12, 30 January 2011 (UTC)[reply]

Dude, go away. — Fly by Night (talk) 19:56, 30 January 2011 (UTC)[reply]

Not gonna happen. You have a mathematician helping you here, and that's all you can say to your benefactor? Michael Hardy (talk) 23:28, 30 January 2011 (UTC)[reply]

Benefactor? Helping me? Get over yourself! You're not helping, you're just being a supercilious old fool. You're more interested in correct people and pointing out minor semantic errors than you are in helping the OP. Although I shouldn't expect anything else from you: a man that spends his time chastising editors, on their own talk pages, for using - instead of − and

sin

instead of

\sin

. My post gave the OP a physical and intuitive understanding of why arc-length is expressed the way it is. But, worried that someone else's reply might actually be more useful than your own, you decide to go on the offensive. It's not a competition. We don't get rosettes for the best answer. It's a collaboration. What with your formalism and my physically intuitive reply the OP had a perfect answer; but you can't accept that. As for you being a mathematician, well, the last time I checked my pay packet, I too was employed by a mathematical research institute. Get over yourself, get a life, and get back to editing. — Fly by Night (talk) 01:39, 31 January 2011 (UTC)[reply]

Get over yourself. Constructive criticism ≠ personal attack, so stop sending the latter in response to the former. --COVIZAPIBETEFOKY (talk) 03:48, 31 January 2011 (UTC)[reply]

Wow, pots and kettles spring to mind. — Fly by Night (talk) 13:34, 31 January 2011 (UTC)[reply]

<sarcasm>Sounds like Good Will Hunting. Except more realistic.</sarcasm> Dmcq (talk) 12:31, 31 January 2011 (UTC)[reply]

"Fly-by-night", listen: I answered a question. You responded by saying "go away". Michael Hardy (talk) 06:08, 31 January 2011 (UTC)[reply]

"Michael Hardy", you told me I was confusing definitions and theorems; that's when I told you to go away. — Fly by Night (talk) 13:35, 31 January 2011 (UTC)[reply]

Which, standing alone, is a somewhat baffling accusation, because both curve lengths as well as (Riemann) integrals have several slightly different equivalent characterizations, each of which can be taken to be a definition making the others theorems. It's not objective which is which. –Henning Makholm (talk) 13:43, 31 January 2011 (UTC)[reply]

Truth to be told, I don't think FbN's comment was very illuminating, but it certainly didn't deserve that kind of attack. In fact I wonder whether Michael Hardy actually intended to attack my comment about differing definitions, but overlooked my signature and thought FbN had written it, because we both wrote at the same indentation level. –Henning Makholm (talk) 13:46, 31 January 2011 (UTC)[reply]

There's nothing wrong with my comment saying you were confusing theorems with definitions. It doesn't make sense to define arc length in a way that only works when the curve is sufficiently differentiable, when a much simpler definition involving neither derivatives nor integrals is available, nor does it make a lot of sense to define integrals by using antiderivatives unless you want to argue in favor of a new way of developing the theory, and that's not what you were doing. Your answer was rude. Michael Hardy (talk) 17:33, 31 January 2011 (UTC)[reply]

Maybe the answer would depend on whether the OP was studying an analysis or geometry course. If its analysis then the Riemann sum would be the way to go. If the course has a more geometric emphasis then that approach is overkill.--Salix (talk): 20:01, 31 January 2011 (UTC)[reply]

Since the word "prove" was used, Riemann sums didn't seem like overkill. Michael Hardy (talk) 20:14, 31 January 2011 (UTC)[reply]

See rectifiable curve. It is basically routine to prove this formula from the definition given there (at least, say, for continuously differentiable curves). 166.137.140.202 (talk) 18:23, 4 February 2011 (UTC)[reply]

Converting from rectangular to polar form

I have three impedances that need adding together. And conversion to polar form. Could someone explain how that's done?

So, for instance, the example in my book has:

${\frac {1}{0.5k\Omega }}+{\frac {1}{j10k\Omega }}+{\frac {1}{2k\Omega }}=2.5mS\angle -2.29^{\circ }$

I don't understand how they got from one to the other. Thanks, Dismas|^(talk) 09:09, 30 January 2011 (UTC)[reply]

You need to do the addition in rectangular coordinates and then find the polar form afterwards:

{\frac {1}{0.5\,\mathrm {k} \Omega }}+{\frac {1}{10i\,\mathrm {k} \Omega }}+{\frac {1}{2\,\mathrm {k} \Omega }}=2\,\mathrm {mS} -0.1i\,\mathrm {mS} +0.5\,\mathrm {mS} =(2.5-0.1i)\,\mathrm {mS}

{}={\sqrt {2.5^{2}+0.1^{2}}}\,e^{\tan ^{-1}(2.5/-0.1)i}\,\mathrm {mS} \approx 2.5e^{-2.29^{\circ }i}\,\mathrm {mS}

–Henning Makholm (talk) 10:02, 30 January 2011 (UTC)[reply]

By the way you have added the electrical impedances in parallel and the result you have is the inverse of the resultant impedance. Dmcq (talk) 10:53, 30 January 2011 (UTC)[reply]

Thanks, Henning. I understood everything up to the

e^{\tan ^{-1}(2.5/-0.1)i}

bit. But you're just finding out the angle based on the two sides of the triangle, I think. Dismas|^(talk)

Yes. I assumed you had learned about Euler's formula

e^{i\theta }=\cos \theta +i\sin \theta

. It seems that you write it

\angle \theta

. –Henning Makholm (talk) 13:33, 30 January 2011 (UTC)[reply]

I did in my calculus class last year but, with the exception of the name, don't remember it. I'm getting another degree while working, so I only have time (and energy) for one class at a time. So when my circuits teacher cruises through things, it's a bit daunting. Everyone else is taking the math courses at the same time whereas I've had to take them in past semesters. Dismas|^(talk) 21:56, 30 January 2011 (UTC)[reply]

Well, the point of classes usually is that you learn something that you might want to remember for later because it is actually useful. (Otherwise, it's not a class but a psychology experiment). The standard mathematical way to write down a parameterization of the unit circle in C is one of those things... –Henning Makholm (talk) Wow. Thanks for the chastising for not remembering a couple things from a calculus class that I took a year ago. I'll try to remember that there are people like you who never forget anything... but I'm probably too retarded. Dismas|^(talk) 10:40, 1 February 2011 (UTC)[reply]

Explanation of a few topology concents

Hi all,

I'm taking a first course in algebraic topology and I'm trying to get my head around a few of the concepts, but as many of you who were in my position at one point probably know, there's often some very weird behavior which is rather hard to picture. I'd really appreciate any aid you can give me with the following problem.

Given continuous f: $S^{n-1}\to X$ , define $X\cup _{f}B^{n}$ for the space given by gluing an n-ball to X via the (smallest possible) equivalence relation identifying x in the sphere with f(x) in X, the quotient of the disjoint union of X and $B^{n}$ over the equivalence relation. Prove that if f, g homotopic maps $S^{n-1}\to X$ then $X\cup _{f}B^{n}$ and $X\cup _{g}B^{n}$ are homotopy equivalent. By using this, show that the dunce hat, given by the cone on $\{(x,y,0)|x^{2}+y^{2}=1\}$ in $\mathbb {R} ^{3}$ with vertex (0,0,1) and identifying points $(\cos \,2\pi t,\sin \,2\pi t,0)$ with $(1-t,\,0,\,t)$ for $t\in [0,1]$ , is contractible (hint: it may help to unfurl the cone along $\{(1-t,0,t):t\in [0,1]\}$ .)

The first part of the problem is I suppose simply an issue of algebra: my thought is that we 'glue' the balls onto the spaces as explained via the f and g equivalence relations, then we don't really know anything about the spaces except that both f and g are homotopic - can we simply apply the homotopy F from f to g to X and the sphere $S^{n-1}$ in $X\cup _{f}B^{n}$ , effectively continuously deforming the set of points identified by the equivalence relation on points f(x) until we reach the set of points identified with g(x), and then extend it to an identity map on the points of $B^{n}$ which are not in X or on the sphere? Then once we have this function which is some sort of 'F patched together with the identity', we can take the reverse of F, from g to f, and somehow also patch that together with the identity. I can vaguely picture what's going on, but I'm having trouble formalising the argument - intuitively I find it hard to picture what's going on here, even in the n=2 case it isn't necessarily obvious to me.

The second part I can see they want to take the 'hat' to be the space $X\cup _{f}B^{n}$ - one of the issues I'm having is that the earlier part of the problem identifies points between the otherwise disjoint union of the ball and the space X, whereas in this case we're talking about identifying part of the cone with another part of the cone - even though it is a 1-sphere in that sense, surely it is intrinsically part of the cone already and not 'disjoint' from it, so I don't see how the first part applies. Am I right to be concerned about this? Perhaps it isn't really an issue topologically speaking, though given how one point can often ruin everything I would be surprised - but at any rate, I think the obvious map from $S^{1}$ to the image with which we're identifying points on the cone is $f:(\cos \,2\pi t,\sin \,2\pi t,0)\to (1-t,0,t)$ ; i don't honestly see how 'unfurling the cone' in the standard way to make a triangle with sides identified in the normal way helps particularly, and again, any comments would be really honestly appreciated, just so I can figure out what's going on here - once I understand, I presume it's just a case of finding a map to which f is homotopic (the identity map perhaps?) and then showing the space which it is homotopy equivalent to is obviously contractible. Thank you very much for the help, it will genuinely make a big difference to me if I can get my head around this! Typeships17 (talk) 22:13, 30 January 2011 (UTC)[reply]

The first part is a special case of Proposition 0.18 in Hatcher's (free) online book, which you can find here. As for the second part, you can view the dunce cap as a triangle with its edges identified as here. Thus it is the space obtained by attaching B² to a circle X via a map f : S¹ → X that winds around twice in one direction and then once in the opposite direction. That map is homotopic to one that winds around just once. Thus by the first part, the dunce cap is homotopy equivalent to a disk. 82.124.101.35 (talk) 08:14, 31 January 2011 (UTC)[reply]

Typeships17, as 82.124.101.35 remarked, it's in Hatcher's book. I took a look and he states it for CW pairs. When I first read your question I took X to mean an arbitrary space. After giving it some thinking I felt the need to extend certain functions on subsets of X to the whole of X. The most general extension theorem I know of is Tietze extension theorem and that only applies to functions to the reals and from normal spaces. So do you mean by X an arbitrary space or as in Hatcher's statement? Money is tight (talk) 14:35, 31 January 2011 (UTC)[reply]

As far as I'm aware, X is simply an arbitrary space. Does that make the proof incorrect? I have the book with me, but I'm not knowledgeable enough about CW complexes to know whether or not it's still okay... Typeships17 (talk) 23:39, 31 January 2011 (UTC)[reply]

Proposition 0.18 applies as is to the case where X is an arbitrary space. The point is that (Bⁿ, S^{n - 1}) is a CW-pair. But really, you don't need to bother with the case of a general CW-pair. If you refer to the proof of Proposition 0.16 (which is used in the proof of 0.18), the special case in which the CW-pair is (Bⁿ, S^{n - 1}) is covered in the first three lines (and then used to prove the general case). 82.124.101.35 (talk) 04:08, 1 February 2011 (UTC)[reply]

Okay, I understand the proof of the first part - it has quite a nice visually appealing sketch proof now that I look at it. However, the second part isn't clear to me - how is the dunce cap a situation where we can attach the B^2? I guess we define X to be the cone itself, then that identifies 2 edges of the triangle, though how we identify the bottom edge as a 'circle' as it was on the cone is beyond me, perhaps i'm mentally unfolding the object wrongly. Then the map f as given moves up the two 'sloped' edges of the triangle as t increases from 0 to 1, or equivalently moving around the circle. How do we identify the circle/triangle appropriately to obtain the right space for the homotopy equivalence? I didn't completely follow the first response, sorry - this is all fairly new to me, i'll do my best to try and follow. Typeships17 (talk) 18:41, 1 February 2011 (UTC)[reply]

The first thing is to convince yourself that the definition you gave of a dunce hat is the same as the quotient space pictured at dunce hat (topology). The cone can be pictured as a triangle with two sides identified, as you say. It is best to do this in such a way that the two sides identified correspond to the segment (1-t,0,t). The identification of the boundary circle with the segment in your original definition can then be viewed as the identification of the remaining side of the triangle with the other two.

Taking for granted that the dunce hat is as in the figure (and forgetting the cone), take X to be a circle. The triangle is a copy of B². Map its boundary onto X in the way suggested by the identifications: starting at the bottom left vertex, go twice around X in the positive direction, then once around in the negative direction. This gives a map f from the boundary of the triangle to X, and since it is surjective, the resulting construction is exactly the dunce hat. Now f is homotopic to any map g that goes around once positively, which could be taken to be the identity. (Homotopy classes of loops without basepoint correspond to conjugacy classes in π₁(X), which is isomorphic to Z, with the isomorphism counting the number of times a loop winds around X.) The space obtained by attaching B² to X via g is just B², which is contractible. 82.124.101.35 (talk) 22:33, 1 February 2011 (UTC)[reply]

Ah, I see! I was thinking about it backwards: I was taking the circle and saying that as you go around it once, you'll move up one corresponding edge of the triangle/cone, rather than looking at what happens when you move around the triangle and noticing that since our 2 'sloped' sides are in correspondence with the base which is a circle, we get a 3-to-1 mapping around the circle since every point in the triangle boundary is identified with 2 others. My problem was trying to map in the most literal sense from a circle, onto what I assumed must be the triangle as X, not from the boundary of the triangle onto another space. I really can't thank you enough :) Typeships17 (talk) 02:10, 15 February 2011 (UTC)[reply]

You're welcome. 82.124.101.35 (talk) 03:34, 3 February 2011 (UTC)[reply]