Wikipedia:Reference desk/Archives/Mathematics/2008 July 20

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July 20

Vector Analysis

Hi I saw a question in a model paper which is mentioned below. I am bit confused in the way that answer got in the paper. Please get mt the answer with a explaination

If the position at time t of a particle is given by, r=(2t²,t²-4t,3t-5) Find the particle's velocity and an accelaration components to the direction i-3j+2k? —Preceding unsigned comment added by Mufleeh (talk • contribs) 13:16, 20 July 2008 (UTC)

The position is (x,y,z) so r_x=2t² , r_y=t²-4t , r_z=3t-5 (those are cartesian coordinates)

the direction i-3j+2k means the line travelling in the direction given by the line from (0,0,0) to (1,-3,2) (i,j,k are used to represent the x,y, and z axis respectively)

The velocity is given by differentiating the position with respect to time

eg v_x=d/dt(r_x)=d/dt(2t²)=4t etc

The acceleration is given by differentiating the position with respect to time twice ie the rate of change of velocity. eg a_x=d/dt(v_x)=d/dt(4t)=4 etc

To find the components of these in the direction i-3j+2k you need to find the angle between the direction of velcoity (or acceleration) and that vector.

You can use the dot product to do this.

If the angle between the two vectors is A then the component in that direction is V_total cosA along the vector 1-3k+2k. (ie consider the triangle formed by the velocity and the line i-3j+2k)

and you should already know that: V_total² = V_x²+V_y²+V_z² (pythagorus)

Is that enough to help you?87.102.86.73 (talk) 15:16, 20 July 2008 (UTC)

Random Formula

Moved to Earthan Philosopher and Earthan Philosopher talk by Philosophia X Known(Philosophia X Known) 03:06, 21 July 2008 (UTC)--Earthan Philosopher

Algebraic numbers

Is it true or false that any arithmetic expression involving roots is an algebraic number?

For instance, this expresion;

${\frac {i+{\sqrt {3}}}{{\sqrt[{7}]{2}}+{\sqrt[{5}]{7}}}}=x$

I cannot solve to an algebraic expression. Of course that doesn't prove much - most likely it is my limited mathematical abilities. I can eliminate the 7th root by making it the subject and then raising to the 7th power. The 5th root is then a problem as the equation is now quartic in the 5th root of 7. Of course, one could solve the quartic equation but I don't see how that is going to help to get an algebraic expression. SpinningSpark 18:56, 20 July 2008 (UTC)

Yes, x is a root of a polynomial, of degree dividing 140. It is in general true that adding, subtracting multiplying and dividing algebraic numbers gives algebraic numbers. The usual proofs are fairly indirect, but I'm sure you could find an explicit polynomial if you really wanted to (do you?). Algebraist 19:01, 20 July 2008 (UTC)

No, I am not trying to solve a real problem, I was just interested in knowing whether I failed to find the polynomial because it was impossible or I did not have the skill. SpinningSpark 19:07, 20 July 2008 (UTC)

If you have access to mathematica, the command RootReduce will produce a polynomial with the given root. I believe maple has similar functionality, and there are other freely available computer algebra systems which should be able to do the same thing. It possible, but slightly tricky, to code this on your own however. If you want, I have some references on how to do this. siℓℓy rabbit (talk) 19:23, 20 July 2008 (UTC)

Assuming that you know a little bit of linear algebra, then the following will make sense to you. An algebraic number x is characterized by the fact that the numbers $1,x,x^{2},\dots ,x^{n}\,$ are linearly dependent over $\mathbb {Q}$ for some n. In other words, the field $\mathbb {Q} (x)$ generated by x is finite dimensional over the rationals. If x and y are algebraic, $e_{1},\dots ,e_{k}$ is a basis for $\mathbb {Q} (x)$ over $\mathbb {Q}$ and $f_{1},\dots ,f_{m}$ is a basis for $\mathbb {Q} (y)$ over $\mathbb {Q}$, then the collection of products $e_{j}\,f_{\ell }$, where $j\in \{1,\dots ,,\}$ and $\ell \in \{1,\dots ,m\}$ contains the sum $x+y$ as well as the product $x\,y$ and is closed under addition and multiplication. From this you can deduce that x+y and $x\,y$ are algebraic. To see that also $1/x$ is algebraic you can just use the linear relation between $1,x,x^{2},\dots ,x^{n}$, multiply it by $1/x$ (a few times if necessary), and get that $1/x$ is a linear combination of $1,x,\dots ,x^{n}$. I hope this clarifies the picture. Oded (talk) 19:49, 20 July 2008 (UTC)

Whoops. I misread the above post as saying "I am not trying to solve a real problem". Indeed, it is fairly easy to see abstractly (as Oded argues) that algebraic numbers are closed under arithmetic operations. With some effort, one can write down an algorithm for determining a polynomial equation that x+y, xy, and 1/x must satisfy, given the minimal polynomials of x and y. siℓℓy rabbit (talk) 22:19, 20 July 2008 (UTC)

Who is the world's most famous mathemetician?

Past or present?........I'll give you a little longer.........See, even people into math get stumped. The souls who discover and eloquently express reality don't get the props they deserve. Will there ever be a day when mathematicians (and scientists in general) are as household nameable as Ashton Kucher, Mariah Carey, and Santa Claus? I'll bet if they ever have to calculate with a sliderule the missile trajectory to blow an Earth shattering rock into smithereens! --Hey, I'm Just Curious (talk) 19:30, 20 July 2008 (UTC)

I nominate Count von Count.87.102.86.73 (talk) 20:15, 20 July 2008 (UTC)

I don't agree with the premise that people "into math" couldn't name famous mathematicians. There are tons of possible answers, but a couple of choices are Gauss, Euler and Euclid, for historical figures, and Andrew Wiles and Terry Tao for active mathematicians. 69.106.57.217 (talk) 22:21, 20 July 2008 (UTC)

Many math historians rank Archimedes among the top mathematicians, not necessarily the most famous though. In my opinion Erdős would be among the most famous recent mathematicians – and after all, he has the most publications among them. GromXXVII (talk) 23:04, 20 July 2008 (UTC)

IMO, the most well known by the public is Pythagorus, The most famous to those who know a bit about it is probably Euclid and within the maths community, it is probably Paul Erdos -- SGBailey (talk) 22:19, 20 July 2008 (UTC)

I'd agree with Pythagoras, with a nod to Isaac Newton, although he's known more amongst the public for his work in the physical sciences than for his contributions to mathematics. Confusing Manifestation(Say hi!) 22:36, 20 July 2008 (UTC)

One of the first that sprung to my mind was Nikolai Ivanovich Lobachevsky, in part because of the Tom Lehrer song by the same name. Also because he helped discover hyperbolic geometry. Black Carrot (talk) 23:51, 20 July 2008 (UTC)

Other somewhat famous ones are Kurt Gödel, Alan Turing and John Nash. 84.239.160.166 (talk) 10:23, 21 July 2008 (UTC)

All the ones I initially thought of have already been listed, but I think Gottfried Leibniz and Blaise Pascal also deserve mention. Oliphaunt (talk) 10:52, 21 July 2008 (UTC)

Are there no patriots here - if you go to Poland it will be quite clear who the greatest mathematician is Nicolaus Copernicus.. (also the greatest scientist, philosopher, physicist, astronomer. etc)87.102.86.73 (talk) 11:15, 21 July 2008 (UTC)

He's not even the greatest Polish mathematician. That accolade probably goes to the great Stefan Banach. Algebraist 11:27, 21 July 2008 (UTC)

According to this poll, 89% of people could identify Einstein's face (although only 4% classified him as a mathematician). Although it also reported that "was pleased to see that even 2/3 of the kids under 15 could spot Einstein" so his recognizability it falling somewhat. Newton is a good pick but most wouldn't recognize him for his contributions to mathematics (sort of like Kant's contributions to gravity/science aren't widely known, but definitely not to the same degree). Descartes also is a recognizable mathematician but probably not for his mathematics.--droptone (talk) 12:21, 21 July 2008 (UTC)

Surely the combination of "89% of people can identify Einstein's face" and "2/3 of the kids under 15 could spot Einstein" only shows that the recognition increases with age, as one might expect. I think the latter figure is quite impressive, given some of the things they (and their elders) don't know about. All is not yet lost... AndrewWTaylor (talk) 15:06, 21 July 2008 (UTC)

Srinivasa Ramanujan. --LarryMac | Talk 15:24, 21 July 2008 (UTC)

(The question was "most famous", not greatest..)87.102.86.73 (talk) 21:01, 21 July 2008 (UTC)

Ramanujan was probably more famous than great. No doubt he was brilliant, but in terms of the fruitfulness of his contributions, I frankly doubt he cracks the top thousand. What people find compelling about him is more his personal story plus the mysterious nature of his thought processes. --Trovatore (talk) 05:04, 24 July 2008 (UTC)

Functions

I've just done an exam question that I can't check because I don't have the answers; if I give the question and my answer, could someone here please tell me if I'm right?

If $f(x)=3x^{2}+4x$ and $g(x)=x^{2}+p$ and it is given that there is only one value of t, a scalar parameter, for which $f(x)+tg(x)$ can be written in the form $a(x+k)^{2}$ for some constants a and k, find p.

I worked out p as $-{\tfrac {16}{9}}$. If I'm wrong, please tell me but don't give me the answer; I want to get that by myself. Thanks. 92.2.122.213 (talk) 20:51, 20 July 2008 (UTC)

Yes, that's correct. Algebraist 21:04, 20 July 2008 (UTC)

Geesh, a mere 13 seconds later - I'm still reading the question - give us a break here ;-) -hydnjo talk 02:07, 22 July 2008 (UTC)

Those are minutes, Hydnjo. Algebraist 11:34, 22 July 2008 (UTC)