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Imaginary Numbers Thought


Lightmeow
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Hello, I have a question. So we all know that, in the real number system, you can only have positive and negative numbers, right?

Because

 

The numbers cycle twice because there it is one dimension. And you can only have right or left.

1*-1=-1,

-1*-1=1.

 

Then you have imaginary numbers, right.

i*i=-1.

-1*i= -i.

-i*i = 1

1*i=i

 

They cycle because on the complex plane, there is four quadrants, thus can cycle four times.

 

So, now, lets go to three dimensions shall we. Is there a number that cycles 8 times?

 

Please answer... And if there isn't one, lets try making one up, shall we? :P

 

Oh, and when we make it up, how would it be used?

 

Josh

Edited by Lightmeow
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Check out the quaternions, the octonions, and the sedenions.

 

* The quaternions have four generators 1, i, j, and k. How they relate to one another was discovered in a flash of insight by William Rowan Hamilton in 1843, as he was walking across Brougham bridge in Ireland. Hamilton was so struck by his discovery that he carved the relations among i, j, and k on the bridge; and a plaque commemorates the event till this day.

 

http://en.wikipedia.org/wiki/Broom_Bridge

 

Quaternion multiplication is not commutative. That is, it is not the case that xy = yz for quaternions x and y. Just as when we go from the reals to the complex numbers we lose the ability to say when one complex number is "less than" another; when we go to quaternions we lose commutativity. As you go up you always lose something.

 

http://en.wikipedia.org/wiki/Quaternion

 

Quaternions are actually used a lot in game programming, since they're a gadget for expressing rotations of three-space. Here's an interesting-looking article I found when I Googled, "quaternions and game programming."

 

http://3dgep.com/?p=1815

 

* Next up are the octonions, with eight units. Not only is multiplication not commutative, it's not even associative. So (xy)z need not equal x(yz).

 

http://en.wikipedia.org/wiki/Octonion

 

Here is John Baez's famous article about the octonions. This is the abstract of the article.

 

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

* Then there are the sedenions, with 16 generators or units.

 

http://en.wikipedia.org/wiki/Sedenion

 

* It turns out that there is a general construction, and you can keep going up forever by powers of 2 to get various types of numbers.

 

http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction

Edited by Someguy1
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