November 2009 - mathfun@mailman.xmission.com

Re: [math-fun] Cube root of a complex number
by Dan Asimov 12 Nov '09

12 Nov '09

No, that's perfect, since the equations for x and y can be solved by the solution to the general cubic . . . and will *hopefully* give real answers -- leading to expressions for the real and imaginary parts of (a + bi)^(1/3) in terms of radicals, which is what I was looking for. --Dan James wrote: << Oops, I seem to have x and y reversed below. >> On Wed, Nov 11, 2009 at 9:01 PM, James Buddenhagen <jbuddenh(a)gmail.com> wrote: Not as pretty, and not really a formula, but sort of interesting: Let the cube root of a+bi be x^(1/3) + y^(1/3) i. Then x is a root of 64x^3 - 48bx^2 - (27a^2+15b^2)x - b^3 = 0, and y is a root of 64y^3 + 48ay^2 - (27b^2+15a^2)y + a^3 = 0. On Wed, Nov 11, 2009 at 7:13 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: << It's a pleasant exercise to calculate the square root of the complex number a + bi directly (without making use of polar form or trig functions). (Be sure to consider all cases.) But is there a similar formula for the cube root of a + bi ??? >> >> >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] The Goat hall problem
by Landon Curt Noll 12 Nov '09

12 Nov '09

> Date: Sat, 31 Oct 2009 19:42:27 -0000 > From: "Guy Haworth" <g.haworth(a)reading.ac.uk> > To: <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Monty Hall ... > > I am much more interested in the Dual of the Monty Hall Problem: > > Behind three doors there are two cardboard cutouts of Monty Hall and a > car. > > A goat is hosting the show ... and a contestant walks in front of Camera > 1 .... > > Discuss ... The contestant finds themselves in a Goat Hall game containing: host: a goat 3 doors hiding: 2 cardboard cutouts of Monty Hall and a car The contestant selects a door at random, the goat bleats once and another door showing a cardboard cutout of Monty Hall is revealed. The contestant either keeps their original door or switches to the other unopened door. When their final choice door is opened: * If the door reveals the car: the contestant wins the car and the game stops * If the door reveals a cardboard cutout of Monty Hall: then one of 3 things happen with equal probability: 1) The contestant wins the zonk prize, a cardboard cutout of Monty Hall, and the game stops 2) The contestant begins playing a classic Monty Hall game containing: host: Monty Hall 3 doors hiding: 2 goats and a car prize NOTE: This game works in the standard way: The contestant picks a door and Monty opens a different door showing a goat. They may keep their original door or switch to the other unopened door. If their final choice door reveals the car, then they win the car and the game stops. If their final choice reveals a goat, then the goat bleats once, they win the goat and the game stops. 3) The contestant is teleported to another Goat Hall game containing: host: a goat 3 doors hiding: 2 cardboard cutouts of Monty Hall and a car NOTE: This game runs under the same Goat Hall rules as previously indicated. Questions: * What is the average number of times the goat bleats for a contestant? * If the contestant always switches, what is the probability of them eventually winning the car? * If the contestant always keeps their original door, what is the probability of them eventually winning the car? chongo (Landon Curt Noll) /\oo/\

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Re: [math-fun] Cube root of a complex number
by Dan Asimov 12 Nov '09

12 Nov '09

Yes, I included your earlier comment to that effect. Apologies if my interjection "No" was confusing; it was a *positive* "No". --Dan << << Yes but my first line should read: Let the cube root of a+bi be y^(1/3) + x^(1/3) i. (or else swap x,y in the two cubics). On Wed, Nov 11, 2009 at 9:38 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > No, that's perfect, since the equations for x and y > can be solved by the solution to the general cubic . . . > and will *hopefully* give real answers -- > leading to expressions for the real and imaginary > parts of (a + bi)^(1/3) in terms of radicals, > which is what I was looking for. > > --Dan > > > James wrote: > > << > Oops, I seem to have x and y reversed below. . . . >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Cube root of a complex number
by Dan Asimov 12 Nov '09

12 Nov '09

It's a pleasant exercise to calculate the square root of the complex number a + bi directly (without making use of polar form or trig functions). (Be sure to consider all cases.) But is there a similar formula for the cube root of a + bi ??? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Carmichael orrery?
by Jason 11 Nov '09

11 Nov '09

It occurs to me that it ought to be possible to demonstrate pseudoprimality with a purely mechanical device. For instance, we could demonstrate 2^560%561 with 560 2:1 (or equivalently, 35 65536:1) gear amplifications followed by a 1:561 reduction, and rotate the input shaft once to see the output gear rest at 1. Of course, 2^560 ~ 3.8E168, so you'd have to turn the crank fairly hard. You could also do a single 65536:1:561 train cranked n times for individual 65536n (mod 561) reductions, and repeat 35 iterations for 65536^35%561. That would provide a (false) Fermat witness of 2. Simile for the other integers < 561. Not very interesting so far, but perhaps there's something more elegant. CARMICHAEL ORRERY: MERRY ARCHAIC ROLE. CARMICHAEL ORRERY? CARRY A RELIC, HOMER.

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[math-fun] New geodesic dome
by Henry Baker 09 Nov '09

09 Nov '09

More modern version of Fuller's geodesic dome: http://capitalcityweekly.com/stories/070109/new_457424071.shtml I haven't been able to find details on this dome, but it appears that it is a "rounded" version of the original. I assume that many of the pieces are identical, but haven't been able to confirm this.

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[math-fun] My pumpkin is smarter than yours
by Hilarie Orman 02 Nov '09

02 Nov '09

It's a dessert, it's a summation: http://hilarie.orman.name:8013/photos/hw2009/pkn2.jpg http://hilarie.orman.name:8013/photos/hw2009/pkn1.jpg Hilarie

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[math-fun] best published explanation of the Monty Hall Paradox?
by James Propp 01 Nov '09

01 Nov '09

What book or magazine article or website has the best (in your opinion) written explanation of the Monty Hall Paradox for the lay reader? (Obviously the best way to explain something mathematical to anyone is through a dialogue in which the teacher gets information from the student about the student's state of understanding and changes tactics accordingly; but I'm looking for the best canned explanation that you've seen.) Did Martin Gardner ever try his hand at explaining the paradox? Jim

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[math-fun] FlashForward (mathematician's edition)
by James Propp 01 Nov '09

01 Nov '09

On November 11, 2009, hundreds of millions of people all over the world will engage in a particular activity and suddenly experience premonitions of engaging in the same activity exactly two years in the future. Jim Propp

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[math-fun] computers and mathematics, binary system
by Simon Plouffe 01 Nov '09

01 Nov '09

Dear all, here is an interesting article : http://www.techradar.com/news/computing/why-computers-suck-at-maths-644771 what makes a problem is the conversion of decimal numbers into binary numbers. have a nice 0.99999... day. Simon Plouffe

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