Wikipedia:Reference desk/Archives/Mathematics/2008 June 6

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June 6

Quotient of an algebraic expression?

I have no idea how to do this one, I just can't figure it out. Please help! I think it's because the coefficient in the first part can't be factored out and I'm not sure how to solve it.

${\frac {4x^{2}-8x+10}{2x+1}}$

Thanks a bunch! --71.98.9.18 (talk) 02:23, 6 June 2008 (UTC)

Are you wanting to factor the numerator? What are you wanting to solve for? There is no equation here, just an expression. Ζρς ι'β' ^¡hábleme! 02:47, 6 June 2008 (UTC)

Never mind, I think I see what you seek. Cancel the part of the top from the bottom. That should leave you with a simple polynomial to factor. Ζρς ι'β' ^¡hábleme! 02:51, 6 June 2008 (UTC)

Yes I want to simplify it...and I know I must simplfy the top and bottom, then cross out the same terms, but I don't understand how to do it because of the 4 as the coefficient in front of the x². --71.98.9.18 (talk) 03:01, 6 June 2008 (UTC)

If you're hoping this simplifies into a simple expression (like a monic linear polynomial or something), I think you'll be disappointed. At least as written, the function has a singularity at $x=-.5$. 98.221.167.113 (talk) 03:17, 6 June 2008 (UTC)I swear I forget to log in every time! Someletters<Talk> 03:18, 6 June 2008 (UTC)

Hmm, I thought so. I graphed it. It is a rational function. Ζρς ι'β' ^¡hábleme! 03:22, 6 June 2008 (UTC)

What "simplified" means may depend on your teacher's goals, but one approach would be to rewrite it as:

$a*x+b+{c \over 2x+1}$

where a, b, c are constant you find. If that is what you want to do, then a generalizable approach is to realize that the ratio:

$4x^{2}-8x+10 \over 2x+1$

can also be written as:

${A*(2x+1)^{2}+B*(2x+1)+C \over 2x+1}=A*(2x+1)+B+{C \over 2x+1}$

Where A, B, and C are things you find by expanding the terms and comparing coefficients. Dragons flight (talk) 03:31, 6 June 2008 (UTC)

Okay...ha sorry I'm just having a difficult time understanding this. So my answer sheet I have says the answer is ${\frac {2x-5+15}{2x+1}}$ but I'm not sure how to get that. Would this be correct according to the stuff you listed dragon? --71.98.9.18 (talk) 03:36, 6 June 2008 (UTC)

Btw- I really appreciate the help and responses! --71.98.9.18 (talk) 03:37, 6 June 2008 (UTC)

I think you must mean $2x-5+{\frac {15}{2x+1}}$, yes? That's what Dragon's flight's first method was doing - take a look at polynomial division to see what's happening. Confusing Manifestation(Say hi!) 03:48, 6 June 2008 (UTC)

Take a ratio of the two highest-degree terms: ${\frac {4x^{2}}{2x}}={\color {Red}2x}.$ Now multiple the denominator by itt: ${\color {Red}2x}\cdot (2x+1)={\color {OliveGreen}4x^{2}+2x}$ and exclude the result from the numerator: $(4x^{2}-8x+10)-({\color {OliveGreen}4x^{2}+2x})={\color {Blue}-10x+10}.$ That makes a simplified form of your expression:
${\tfrac {4x^{2}-8x+10}{2x+1}}={\tfrac {({\color {OliveGreen}4x^{2}+2x})+({\color {Blue}-10x+10})}{2x+1}}={\tfrac {{\color {Red}2x}\cdot (2x+1)+(-10x+10)}{2x+1}}=2x-{\tfrac {10x-10}{2x+1}}.$
Iterate the method to further reduce the numerator degree, until it gets lower than the denominator degree. --CiaPan (talk) 06:35, 6 June 2008 (UTC)

See also Polynomial long division.  --Lambiam 05:17, 7 June 2008 (UTC)

Ahh, okay, thanks guys! I think I'm getting it! :D --71.117.39.109 (talk) 17:40, 7 June 2008 (UTC)

Evolution

Why have we evolved an ability to visualize objects in three dimensions, but no more than that. If that was the case, then that means 3 dimensions has more importance than 2 dimensions as well as more importance than 4 dimensions or more.68.148.164.166 (talk) 04:26, 6 June 2008 (UTC)

Likely because being able to understand and deal with a 3D world was essential to survival. A key factor in evolution. -- Tcncv (talk) 04:35, 6 June 2008

(UTC)

One of the answer is that human can only survive in a 3D world,seeAnthropic principle--Lowerlowerhk (talk) 08:06, 6 June 2008 (UTC)

I don't think the Anthropic principle quite applies. The Anthropic Principle says that only universes which are capable of supporting intelligent life can contain intelligent observers within it, but that doesn't preclude the possibility that a higher-dimensional universe could support higher-dimensional intelligent creatures. A universe with five dimensions of space-time could theoretically have five dimensional intelligent creatures within it. We just don't happen to be in such a universe.

Au contraire! This paper from the peer-reviewed journal Classical and Quantum Gravity [1] shows that 3+1 dimensional spacetime is the only one that allows for the existence of intelligent observers who can make reasonable predictions about the future. —Keenan Pepper 18:09, 6 June 2008 (UTC)

Thanks for the interesting sounding link, Keenan, although unfortunately it appears that I can't read the paper without coughing up money. :( From the abstract it sounds like they are ruling out multiple temporal dimensions, so higher dimensional beings would have to occupy a universe with one temporal dimension and 3+ spatial dimensions. However it doesn't sound like they necessarily ruled out observers in a universe with 4+ spatial dimensions and 1 temporal dimension saying only that "in a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures." Notice the word "perhaps", implying that they left the door open on the question of having stable "atomic" structures. 63.111.163.13 (talk) 19:25, 6 June 2008 (UTC)

If you want to read it, you can give me an email address and I'll send you a PDF. —Keenan Pepper 18:09, 7 June 2008 (UTC)

Getting back to the original question, though, we can only internally visualize the universe based on the reflection of observable energy like light and sound and through movement of matter and energy in space-time. Therefore we can only internally visualize objects in terms of three spatial dimensions and one temporal dimension. The only way we would have been able to evolve to conceptualize more than three spatial dimensions would be if there was some observable form of energy transmission that travelled through a fourth spatial dimension. 63.111.163.13 (talk) 14:56, 6 June 2008 (UTC)

It's not completely the case that we can't visualize things in four spatial dimensions, and it's not completely the case that we can in three. The most natural things for us to visualize are two-dimensional, since that's the shape of our retina and our internal viewing screen. That is, however visual information is processed, it comes to us as a flat image with a bit of extra info scattered around to imply depth, and especially to imply overlap. The transformation from three dimensions to two is roughly projective, and there's a lot that gets distorted in the process. The only way to actually directly visualize three dimensions would be to have something like a three-dimensional array of memory spaces plus something to keep track of their geometric relationships. That can certainly be finangled within a single person's imagination, and can be generalized to a few more dimensions with effort. Past that, it's got to be hard to hold 20 dimensions of any kind of information in your head at once, but that doesn't mean it's impossible. And yes, it appears that most of that would be unhelpful in ordinary life, which is probably why we don't do it. Two dimensions and a bit is more than enough to get by. I have a problem with the previous post, incidentally. I don't think it's the case that we can only visualize things we've sensed, or could sense. I get more the impression that we have some capacities, and some needs, and try to fit them together as efficiently as is practical. For instance, I've heard (and could probably track down) that people blind from birth still use the same parts of the cortex to keep track of spatial information, suggesting that they experience the world the same way we do (other than the not-colored part). What happens in our heads has to be to some extent independent of what happens in the world around us. For a more mathematically-inclined example, how about curved space? To many people for a very long time, it was flat-out inconceivable. Not just that it couldn't happen in the real world, but that it couldn't even be imagined. Now we know that it is real, and can be imagined, but so can flat space, or space curved differently. Likewise continuous versus discrete. We can imagine all kinds of things that can't exist at the same time because they're mutually exclusive. We're much more flexible than the real world seems to be. Black Carrot (talk) 18:06, 6 June 2008 (UTC)

Your point that humans have an easier time envisioning two dimensional projections is valid, and we have more difficulty envisioning 3D spaces. However the original question was why humans evolved to visualize objects in 3 dimensions and not more, and the answer is that we have no 4D spatial objects and energy does not travel in 4 spatial dimension. Everything we encounter and all the lines of transmission are 3D.

As an analogy, consider that we could hypothetically program a computer to work with 4D objects, and such a program could in essence internally view such objects completely accurately, given proper sensory input for all the object's 4D spatial coordinates. But that would be an "intelligent design", you might say, of an AI capable of envisioning things in higher spatial dimensions. By contrast there would be no need to be able to envision things in four dimensions in nature because we never encounter such objects. So mutations which might assist a human to envision things four dimensionally offer no obvious benefit to the organism and therefore are probably not likely to be successful evolutionary off-shoots. 63.95.36.13 (talk) 19:43, 6 June 2008 (UTC)

That's fair, but my point is also that part of our difficulty visualizing things in four dimensions comes from not doing it very often, rather than a natural deficit. So, evolution has not completely weeded out an unnecessary capability. It's still there, ready to be used by anyone who cares enough. Black Carrot (talk) 23:34, 10 June 2008 (UTC)

I need to use this equality but I can't without knowing it's true.

This page says that $\scriptstyle \sin ^{4}\theta ={\frac {3-4\cos 2\theta +\cos 4\theta }{8}}$, but there is no proof, even here, where a proof would logically be. I'm unsure that it is in fact the case. Please supply a proof, so I can be sure it's true. Thanks in advance, 71.220.219.115 (talk) 19:00, 6 June 2008 (UTC)

Alas, we do not provide proofs for every single degree of n in $\sin ^{n}\theta$. Nevertheless, I suggest you start on the right-hand-side and use $\cos 2\theta =1-2\sin ^{2}\theta$ and $\sin 2\theta =2\sin \theta \cos \theta$ several times. x42bn6 Talk Mess 19:09, 6 June 2008 (UTC)

(ec) Have you tried breaking it down in terms of sin(θ)’s and cos(θ)’s? GromXXVII (talk) 19:11, 6 June 2008 (UTC)

It's definitely true, but try and prove it as Grom suggests. You should be able to do it in 5 lines. -mattbuck (Talk) 20:28, 6 June 2008 (UTC)

Just toss in some numbers and check if it works.--Fangz (talk) 20:52, 6 June 2008 (UTC)

Yeah, "proof by example" isn't particularly rigorous... --Tango (talk) 22:18, 6 June 2008 (UTC)

Well, sure. But for something like trig functions, it's vanishingly unlikely that it will work for 3 or 4 integers by pure chance. If you just want to confirm something that probably can be proved with more effort, five minutes on a calculator can save you lots of work at something that is pretty unimportant for your problem. _{But then again I work in statistics.} --Fangz (talk) 22:27, 6 June 2008 (UTC)

Because precision is fun: trigonometric functions are analytic, and thus if two trigonometric expressions are different, then they are equal at at most countably many points. Thus for any sensible random choice of argument, the values of the two expression will almost surely be different. Less precisely: if you plug in a random value into two trig expressions and get the same value, this is very very good evidence that the expressions are equal. Algebraist 22:35, 6 June 2008 (UTC)

Thinking, in fact, there is a reasoning here - cos and sin are analytic functions. Hence, this means (roughly) that either it is constantly zero, or the set over which it is zero has zero measure, making it hard to hit it by chance - provided you are selecting numbers actually randomly. Sadly, I doubt your teacher will be pleased with this approach.Dammit, I was going to say the same thing--Fangz (talk) 22:42, 6 June 2008 (UTC)

More to the point, the zeros of an analytic function do not have limit points. This is important since it implies that, for example, the probability of an analytic function being zero at n / 10^p, for a given integer p and a uniformly chosen integer 0 ≤ n ≤ 10^p (which is what you get in practice if you ask someone to pick a random number between zero and one), tends to zero as p tends to infinity. Mere countability of the set of zeros is not enough to show that, since e.g. the set of all rational numbers is countable. —Ilmari Karonen (talk) 23:35, 6 June 2008 (UTC)

"Alas, we do not provide proofs for every single degree of n in $\sin ^{n}\theta$. " Well we should! As it is a general formula with a general answer. $\sin ^{n}\theta ={\frac {1}{(2i)^{n}}}(e^{i\theta }-e^{-i\theta })^{n}$. so then its just a matter of binomial expansion in this case $\sin ^{4}\theta ={\frac {1}{(2i)^{4}}}(e^{i\theta }-e^{-i\theta })^{4}={\frac {1}{16}}(e^{i4\theta }-4e^{i2\theta }+6-4e^{i2\theta }+e^{-i4\theta })={\frac {1}{16}}(2\cos 4\theta -8\cos 2\theta +6)$ Hope this answers your question. Sorry its taken several days. Philc 0780 23:49, 6 June 2008 (UTC)

I've added the general formula here. Please check to see if I've done the right thing. Thanks, --hydnjo talk 12:23, 7 June 2008 (UTC) Guess not! --hydnjo talk 19:01, 7 June 2008 (UTC)

I'll get it in terms of only trig functions and add it there for you. Philc 0780 18:47, 9 June 2008 (UTC)

Thanks Philc 0780 - another excellent example of article improvement through the RD. --hydnjo talk 21:17, 10 June 2008 (UTC)

.999...=1

Alright, my dad and I have differing opinions on this. He refuses this fact and I have tried to convince him of it by presenting the algebraic and fraction proofs. This question is about the proof by a fraction.

${\frac {1}{3}}=.{\overset {-}{3}}$

${\frac {2}{3}}=.{\overset {-}{6}}$

$.{\overset {-}{3}}+.{\overset {-}{6}}=.{\overset {-}{9}}$

${\frac {1}{3}}+{\frac {2}{3}}=1$

Therefore:

$.{\overset {-}{9}}=.{\overset {-}{3}}+.{\overset {-}{6}}={\frac {1}{3}}+{\frac {2}{3}}=1$

My dad refuses this proof because he says the repeating decimal never actually exactly equals the fraction; therefore, .999... never quite equals 1. I tell him that "after and infinite number of decimal places" (for lack of a better term) the decimal is exactly equal to fraction. He refuses this by claiming I am treating an infinite number of places as if it were finite. In this situation, which case is correct, and what is the way to disprove the contrary? Thank you, Ζρς ι'β' ^¡hábleme! 23:05, 6 June 2008 (UTC)Oops. I changed "refute" to "refuse" now.

See 0.999.... --Prestidigitator (talk) 23:20, 6 June 2008 (UTC)

Just by the way, "refute" means "prove false". If you disagree with him, then it follows that you don't think he's refuted the arguments. --Trovatore (talk) 23:56, 6 June 2008 (UTC)

Your dad believes that 0.9999 etc. does not equal 1. Here's the standard proof:

$x=0.{\overset {-}{9}}$

$10x=9.{\overset {-}{9}}$

$10x-x=9x=9.{\overset {-}{9}}$ - $0.{\overset {-}{9}}$

$9x=9$

$x=1$

Wikiant (talk) 00:26, 7 June 2008 (UTC)

Yes, I have read the article and shown him that proof too; however, I would prefer something regarding the proof by fractions. Thanks, Ζρς ι'β' ^¡hábleme! 01:50, 7 June 2008 (UTC)

11 X 1/11=.09090909.... X 11

11/11=.9999...

same with 1/3 and .333...--Xtothe3rd (talk) 02:10, 7 June 2008 (UTC)

Sometimes it helps if you ask the sufferer of this delusion what they think the result is of subtracting 0.999... from 1.  --Lambiam 05:05, 7 June 2008 (UTC)

He says $1-.{\overset {-}{9}}=.{\overset {-}{0}}1$ Ζρς ι'β' ^¡hábleme! 07:14, 7 June 2008 (UTC)

Then what is the result of multiplying that number by 10?  --Lambiam 09:24, 7 June 2008 (UTC)

No, no, no, wrong question! "That number" is, in the "sufferer"'s mind, the smallest number there is (greater than 0, of course). So there's no problem multiplying it by ten. If you really want to explode their mind, ask them to divide that number by ten! —Steve Summit (talk) 03:18, 10 June 2008 (UTC)

Unfortunately, that combination of symbols has no meaning. Take a look at decimal expansion. When you see something like $x=0.d_{1}d_{2}d_{3}\ldots d_{n}\ldots$, you have to understand that there is a precise definition lurking underneath. In this case, the decimal expansion article tells you that, by definition,

$0.{\overline {9}}=\lim _{N\to \infty }\sum _{j=1}^{N}{\frac {9}{10^{j}}}$.

A limit is not "moving" or approaching anything. It has a formal definition, and when the limit exists it specifies a single number (or element of the topological space, or whatever). Applying basic limit rules,

$\lim _{N\to \infty }\sum _{j=1}^{N}{\frac {9}{10^{j}}}=9*\lim _{N\to \infty }\sum _{j=1}^{N}{\frac {1}{10^{j}}}=9*{\frac {1/10}{1-1/10}}=1.$

The derivation of the second to last equality is in geometric series, and it only relies on some basic algebra and properties of the limit. Any other explanation is simply an attempt at persuasion. If someone is uninterested in being persuaded (that is, uninterested in accepting a plausibility argument), then they are obligated to acquire a sufficient understanding of the formal definitions. 24.8.49.212 (talk) 07:52, 7 June 2008 (UTC)

Yeah, I'm also personally uncomfortable with the original fractions proof, (and to an extent, the algebraic proof), since both of them rely on theorems like the fact that you can add, deduct and multiply convergent sequences. It's far better to return to how 0.999... is defined, and say it itself isn't 'a number' (because numbers can be denoted using any number system), but rather it is a statement referring to a number which has 0.9999... as its decimal approximations. Perhaps something that works would be to show him how some 'nice numbers' have an infinite expansion in e.g. base 2.--Fangz (talk) 08:04, 7 June 2008 (UTC)

24..212 is right, the notation $.{\overset {-}{0}}1$ has no formal meaning. But let's just sorta ignore that for a moment, and try to figure out what it might plausibly mean. Evidently it's a decimal point, followed by an infinite number of 0's, followed by a 1. So it's equal to $1\times 10^{-\infty }$, or ${\frac {1}{10^{\infty }}}$.

Now, I believe (and mathematicians will agree) that ${\frac {1}{\infty }}=0$. The OP's dad might not agree, but at any rate, we're not talking about ${\frac {1}{\infty }}$, we're actually talking about ${\frac {1}{10^{\infty }}}$, which is clearly vastly smaller, so it's just got to be 0!

(Hmm. Come to think of it, though, this isn't going to convince the OP's dad at all. The OP's dad is gonna think that ${\frac {1}{\infty }}=.{\overset {-}{0}}1$, where $.{\overset {-}{0}}1>0$, and we're right back where we started.) —Steve Summit (talk) 04:28, 10 June 2008 (UTC)

Maybe you should try thinking of it this way. Your dad says that 0.333... never exactly equals 1/3. But just assume it is another symbol for 1/3, one that is very unintuitive to humans, but does ultimately represent 1/3 because we say so. Then once he accept that 0.333...=1/3 he might accept the proof —Preceding unsigned comment added by RMFan1 (talk • contribs) 17:41, 7 June 2008 (UTC)

--

Here's the most intuitive way I know of to bridge the gap, if he's not willing to learn very much.

1. Terminating (finite-length) decimals are fractions, by definition. They're a somewhat roundabout way of expressing some fractions. Nonterminating (infinite-length) decimals are completely different. They're shorthand for a sequence of fractions, the terminating decimals produced by slicing off the nonterminating decimal farther and farther along. So, the only way you can deal with the thing is by extension from its finite-length counterparts.
2. A nonterminating decimal is a list of directions, not the number itself. (Though, like anything in math, you can fuse a representation with the thing it's representing when it's convenient.) You read the directions as a sort of warmer-colder thing. The number represented is "the least number greater than everything in the list." This is meant to be a giant flashing sign pointing at the number in question. I emphasize than nothing in the list is supposed to actually equal that number.
3. Some numbers, inconveniently, have more than one representation in this system, just as all fractions have more than one representation as a quotient of integers. For instance, the terminating decimal 1 equals the fraction 1. On the other hand, the smallest number greater than .9, .99, .999, .9999, and .99999 (etc) is also 1. 5/2 is too big, pi is too big, 2/3 is too small. Anything greater than 1 is too big, since it's also greater than anything less than 1, and anything less than 1 is too small, since it's smaller than at least one number in the list.

One other thing. Anything less than all positive fractions, but greater than zero, is called infinitecimal. So 0.00...01, if it existed, would be infinitecimal. It may help to look at any of the wide variety of numerical, algebraic, and geometric systems that incorporate infinitecimals, and try to see what makes them so completely different from the real numbers. The phrase "nonarchimedean geometry" may send you in the right direction. Black Carrot (talk) 23:59, 10 June 2008 (UTC)

I don't think that's a good way to think about it. A lot of the problems people have with 0.999... come from thinking of it as a sequence (they say things like "0.999... gets closer and closer to 1 but never actually gets there"), when it isn't, it's a representation of a single number, which happens to be "one". I think learning about non-archimedean number systems is an unnecessary confusion, just stick to the real numbers and accept that infinitesimals simply don't exist (see archimedean property for a proof). --Tango (talk) 20:43, 11 June 2008 (UTC)

But that misunderstanding is so close to the right answer that it's hardly worth correcting. A nonterminating decimal exists already in its entirety, without any infinite-time process to produce it. However, in order to extract a value for this decimal and compare it to others, you have to unzip it anyway and deal with finite substrings. It seems like there's a wasted step there, if you aren't planning to go any farther into the theory. The decimal represents, by definition, the supremum of its associated set of terminating decimals. Right? And what people see when they look at a nonterminating decimal, is a family of terminating decimals of unbounded length. Right? So what's the problem? The definition of the number is "the number you'll never quite reach by extending this out a long way." There's no need to take a detour through things that actually have infinite length. Black Carrot (talk) 18:25, 12 June 2008 (UTC)