Comments for The Math Less Traveled http://www.mathlesstraveled.com Explorations in mathematical beauty Wed, 08 Sep 2010 14:28:08 -0400 http://wordpress.org/?v=2.9.1 hourly 1 Comment on Minus times minus is plus by Errol Roberts http://www.mathlesstraveled.com/?p=439&cpage=1#comment-62652 Errol Roberts Wed, 08 Sep 2010 14:28:08 +0000 http://www.mathlesstraveled.com/?p=439#comment-62652 Sorry! [Row1(1,0), Row2(0,-1)]^1/2=i Doesn't it? Sorry!
[Row1(1,0), Row2(0,-1)]^1/2=i
Doesn’t it?

]]> Comment on Minus times minus is plus by Errol Roberts http://www.mathlesstraveled.com/?p=439&cpage=1#comment-62651 Errol Roberts Wed, 08 Sep 2010 14:26:09 +0000 http://www.mathlesstraveled.com/?p=439#comment-62651 Overnight I had some further thoughts on my own question, maybe somebody can suggest if this has any value? The reflection matrix Ref(A) for a reflection in an axis at angle A to a reference axis is given by [Row1(cos2A,sin2A), Row2(sin2A,-cos2A)]. For the y-axis angle A =PI/2 The x-axis is the line of real numbers and so the reflection of +ive numbers through the origin is trivial and will form the -ive branch of real numbers. Thus Ref(PI/2)=[Row1(cosPI, sinPI), Row2(sinPI,-cosPI)] =[Row1(1,0), Row2(0,-1)] = -1 Thus (-1)*(-1)=[Row1(1,0), Row2(0,-1)]*[Row1(1,0), Row2(0,-1)] =+1 I suppose that this also means Overnight I had some further thoughts on my own question, maybe somebody can suggest if this has any value?
The reflection matrix Ref(A) for a reflection in an axis at angle A to a reference axis is given by [Row1(cos2A,sin2A), Row2(sin2A,-cos2A)].
For the y-axis angle A =PI/2
The x-axis is the line of real numbers and so the reflection of +ive numbers through the origin is trivial and will form the -ive branch of real numbers.
Thus Ref(PI/2)=[Row1(cosPI, sinPI), Row2(sinPI,-cosPI)]
=[Row1(1,0), Row2(0,-1)]
= -1
Thus (-1)*(-1)=[Row1(1,0), Row2(0,-1)]*[Row1(1,0), Row2(0,-1)]
=+1
I suppose that this also means

]]> Comment on Minus times minus is plus by Errol Roberts http://www.mathlesstraveled.com/?p=439&cpage=1#comment-62579 Errol Roberts Tue, 07 Sep 2010 09:44:00 +0000 http://www.mathlesstraveled.com/?p=439#comment-62579 Konrad Voelkel "I think the next question to your explanation would be: why is multiplication with -1 a reflection on the number line?" In several tutoring posts bloggers state that, along the number line, one can face positive (or negative) direction, count numbers, turn round and repeat a count etc. This implies that the rotation and facing a different way is actiually a reflection as Konrad implies. So, presumably -1 is +1+pi and then at -1 a repeated reflection brings the point back to the original +1. Since (-m)=(-1)*(+m) any number can be so represented. Multiplication then is the repetion of the basic relections and the positive number multiplications ie (-n)*(-m)=((-1)*(n))*((-1)*(m))= ((-1)*(-1))*((n)*(m)). Thus the axiom devolves into looking at multiplication by (-1) as a reflection. Is this right or have I got this completely wrong? Konrad Voelkel
“I think the next question to your explanation would be: why is multiplication with -1 a reflection on the number line?”
In several tutoring posts bloggers state that, along the number line, one can face positive (or negative) direction, count numbers, turn round and repeat a count etc. This implies that the rotation and facing a different way is actiually a reflection as Konrad implies. So, presumably -1 is +1+pi and then at -1 a repeated reflection brings the point back to the original +1. Since (-m)=(-1)*(+m) any number can be so represented. Multiplication then is the repetion of the basic relections and the positive number multiplications ie (-n)*(-m)=((-1)*(n))*((-1)*(m))= ((-1)*(-1))*((n)*(m)). Thus the axiom devolves into looking at multiplication by (-1) as a reflection.
Is this right or have I got this completely wrong?

]]> Comment on Book reviews: Math Jokes 4 Mathy Folks and Easy as Pi by Book reviews: Math Jokes 4 Mathy Folks and Easy as Pi « The Math … | eBook Reviews http://www.mathlesstraveled.com/?p=724&cpage=1#comment-62552 Book reviews: Math Jokes 4 Mathy Folks and Easy as Pi « The Math … | eBook Reviews Mon, 06 Sep 2010 22:07:22 +0000 http://www.mathlesstraveled.com/?p=724#comment-62552 [...] Original post by Brent [...] [...] Original post by Brent [...]

]]> Comment on Manufactoria by ignatius http://www.mathlesstraveled.com/?p=719&cpage=1#comment-62325 ignatius Fri, 03 Sep 2010 23:22:32 +0000 http://www.mathlesstraveled.com/?p=719#comment-62325 > a ‘Malevolence Engine’ which works out if the machine is flawless That would be equivalent to solving the halting problem - I guess that it uses some scripted heuristics, such as to check if loops are used, probes the complexity behaviour or simply tries strings which are too long to accommodate with a feed-forward network. > you have to play on specific sites for that though You can also download the Manufactoria.swf and play with the standalone flashplayer. > I just have a few levels left. If you've finished, you might check out one of those (playtested, but cannot check the links here): <a href="http://pleasingfungus.com?ctm=Robojournalist!;Produce_as_many_yellows_as_told_by_the_binary_input_(blue=1,_red=0);:|r:|rrr:|b:y|br:yy|rbb:yyy|brbb:yyyyyyyyyyy|brrrrr:yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy;13;3;1;" rel="nofollow">Robojournalists!</a> Our robotic investigators bring you today the news of tomorrow - and for all other days, too. <a href="http://pleasingfungus.com/?ctm=Roboeditors!;Accept_only_strings_containing_a_red_sequence_longer_than_any_blue_sequence!;:x|b:x|r:*|br:x|brrbr:*|bbrbbrrrb:*|rbbbrrrbrrr:x|bbrrrbbrrr:*;13;3;0;" rel="nofollow">Roboeditors!</a> ROX News, the Robotic Objective eXaminer. fair & balanced [tm] and 100% cognitive consonant. <a href="http://pleasingfungus.com/?ctm=Robolawyers!;With_BLUE_as_opening_and_RED_as_closing_bracket,_check_syntax_of_strings!;:*|br:*|bbrbrr:*|bbrbrrbrbbrbbbrrrr:*|rb:x|bbr:x|brrbbr:x|brbbrbbrbrrbrrr:x;13;3;0;" rel="nofollow">Robolawyers!</a> RAG, the robotic advocate corps, for child molesters, bankers and YOU. Now with 100% less ethics! > a ‘Malevolence Engine’ which works out if the machine is flawless

That would be equivalent to solving the halting problem – I guess that it uses some scripted heuristics, such as to check if loops are used, probes the complexity behaviour or simply tries strings which are too long to accommodate with a feed-forward network.

> you have to play on specific sites for that though

You can also download the Manufactoria.swf and play with the standalone flashplayer.

> I just have a few levels left.

If you’ve finished, you might check out one of those (playtested, but cannot check the links here):

Robojournalists!
Our robotic investigators bring you today the news of tomorrow – and for all other days, too.

Roboeditors!
ROX News, the Robotic Objective eXaminer. fair & balanced [tm] and 100% cognitive consonant.

Robolawyers!
RAG, the robotic advocate corps, for child molesters, bankers and YOU. Now with 100% less ethics!

]]> Comment on The Most Beautiful Equation in the World by David McElroy http://www.mathlesstraveled.com/?p=63&cpage=1#comment-62321 David McElroy Fri, 03 Sep 2010 21:51:52 +0000 http://www.mathlesstraveled.com/?p=63#comment-62321 2^63 grains i like the equation but not the headache Does it mean that there is a universal pattern to the world? Like the circle I think the universe works like a circle. When suns explode and compress under their own weight I think that they compress beyond our dimension and go into the 4th dimension of time. I think like most critical breaks there is a critical point or value at which it will break into the 4th dimension. I think such an event could produce a large event at one point in time. We know of one such event 'the big bang' Does this mean that the universes creation was from it's eventual destruction. That would be cool. I think too much. Sorry I have just never met smart people before I really don't know what to say 2^63 grains
i like the equation but not the headache
Does it mean that there is a universal pattern to the world?
Like the circle
I think the universe works like a circle.
When suns explode and compress under their own weight I think that they compress beyond our dimension and go into the 4th dimension of time. I think like most critical breaks there is a critical point or value at which it will break into the 4th dimension. I think such an event could produce a large event at one point in time. We know of one such event ‘the big bang’
Does this mean that the universes creation was from it’s eventual destruction.
That would be cool.
I think too much.
Sorry I have just never met smart people before I really don’t know what to say

]]> Comment on P vs NP: What’s the problem? by Brent http://www.mathlesstraveled.com/?p=755&cpage=1#comment-62298 Brent Fri, 03 Sep 2010 12:46:21 +0000 http://www.mathlesstraveled.com/?p=755#comment-62298 Well, solutions to polynomials are never transcendental (by definition -- transcendental numbers are exactly those numbers which are not roots of polynomials with integer coefficients). So I actually lied a bit -- there IS a nice, finite way to describe the real root of x^5 – 3x + 2. Here it is: "the real root of x^5 – 3x + 2". Now, you might think that is cheating a bit, but you have to admit it is precise, unambiguous, and finite! (It so happens that x^5 – 3x + 2 has only one real root, but for polynomials with more you could say something like "the largest real root..." or "the second largest..." or whatever.) As for other nice ways to write it down -- there may be some, but I don't know what they are. What I do know (and why I chose degree 5) is that in general the roots of polynomials of degree 5 and above cannot be written down using just arithmetic operations and roots (see http://en.wikipedia.org/wiki/Insolubility_of_the_quintic), except in special cases. Well, solutions to polynomials are never transcendental (by definition — transcendental numbers are exactly those numbers which are not roots of polynomials with integer coefficients). So I actually lied a bit — there IS a nice, finite way to describe the real root of x^5 – 3x + 2. Here it is: “the real root of x^5 – 3x + 2″. Now, you might think that is cheating a bit, but you have to admit it is precise, unambiguous, and finite! (It so happens that x^5 – 3x + 2 has only one real root, but for polynomials with more you could say something like “the largest real root…” or “the second largest…” or whatever.) As for other nice ways to write it down — there may be some, but I don’t know what they are. What I do know (and why I chose degree 5) is that in general the roots of polynomials of degree 5 and above cannot be written down using just arithmetic operations and roots (see http://en.wikipedia.org/wiki/Insolubility_of_the_quintic), except in special cases.

]]> Comment on P vs NP: What’s the problem? by Tom http://www.mathlesstraveled.com/?p=755&cpage=1#comment-62287 Tom Fri, 03 Sep 2010 06:44:29 +0000 http://www.mathlesstraveled.com/?p=755#comment-62287 Haha, I figured the definition of 'solve' isn't quite universal. The example with a 5th degree polynomial makes sense... though I still feel that finding a million decimal places isn't quite 'solving it.' (On a semi-related side note, is it even possible that there is an exact, expressible solution out there that we haven't found? Or are solutions to these polynomials transcendental?) For what it counts, as another (junior) undergrad, I agree 100% with Sam :) Most math blogs are either by Terrence Tao et. al. and completely incomprehensible, or in some way not as interesting as this blog. Cheers Haha, I figured the definition of ’solve’ isn’t quite universal. The example with a 5th degree polynomial makes sense… though I still feel that finding a million decimal places isn’t quite ’solving it.’ (On a semi-related side note, is it even possible that there is an exact, expressible solution out there that we haven’t found? Or are solutions to these polynomials transcendental?)

For what it counts, as another (junior) undergrad, I agree 100% with Sam :) Most math blogs are either by Terrence Tao et. al. and completely incomprehensible, or in some way not as interesting as this blog. Cheers

]]> Comment on P vs NP: What’s the problem? by Brent http://www.mathlesstraveled.com/?p=755&cpage=1#comment-62259 Brent Thu, 02 Sep 2010 16:27:47 +0000 http://www.mathlesstraveled.com/?p=755#comment-62259 Hi Tom, great question. There's actually nothing special about computer science here; the issue arises in a purely mathematical setting as well. The fact is that not every problem CAN be solved exactly. Something like x^2 - 2x + 1 = 0 is a very special case since it has the EXACT solution x = 1 (and computers can solve such problems exactly, just as you would). But consider an equation like x^5 - 3x + 2 = 0. We know there must be a real solution for x (the graph of any odd-degree polynomial goes off to infinity in different directions and hence must cross the x-axis at least once), and the solution is approximately 0.632835... but it turns out there is no better way to write the solution for x other than to write an approximation to as many decimal places as you want. Likewise, the shortest time from A to B might not be a "nice" number -- any solution you care to write down will necessarily be an approximation. Ultimately what it means to "solve" a problem depends on the problem and on what sorts of things you are willing to accept as solutions. Hmm... hopefully this helps, I'm just skimming the surface here. There's probably an interesting blog post or two in there, come to think of it. Hi Tom, great question. There’s actually nothing special about computer science here; the issue arises in a purely mathematical setting as well. The fact is that not every problem CAN be solved exactly. Something like x^2 – 2x + 1 = 0 is a very special case since it has the EXACT solution x = 1 (and computers can solve such problems exactly, just as you would). But consider an equation like x^5 – 3x + 2 = 0. We know there must be a real solution for x (the graph of any odd-degree polynomial goes off to infinity in different directions and hence must cross the x-axis at least once), and the solution is approximately 0.632835… but it turns out there is no better way to write the solution for x other than to write an approximation to as many decimal places as you want. Likewise, the shortest time from A to B might not be a “nice” number — any solution you care to write down will necessarily be an approximation. Ultimately what it means to “solve” a problem depends on the problem and on what sorts of things you are willing to accept as solutions.

Hmm… hopefully this helps, I’m just skimming the surface here. There’s probably an interesting blog post or two in there, come to think of it.

]]> Comment on P vs NP: What’s the problem? by Brent http://www.mathlesstraveled.com/?p=755&cpage=1#comment-62258 Brent Thu, 02 Sep 2010 16:14:10 +0000 http://www.mathlesstraveled.com/?p=755#comment-62258 Sam, thanks for the encouragement! I'm very glad you enjoy it. Sam, thanks for the encouragement! I’m very glad you enjoy it.

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