Fantastic Quaternions - Numberphile
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Date: 2022-04-08
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Comments and reviews: 9
Sudhakaran
Let power(base, exponent)
Then no. of dimension of numbers required to describe 'n' physical dimension is power(2, n-1.
If n = 1, that is 1D physical line. The. No. Of dimension of numbers is 2-(1-1) = 2-0 = 1. Real numbers
For n=2, 2D physical plane. Dimension of numbers is 2-(2-1) = 2-1 = 2. Complex numbers
For n=3, 3D physical space. Dimension of numbers is 2-(3-1) = 2-2 = 4. Quaternions
For n=4, 4D physical space. Dimension of numbers is 2-3 = 8.
5D space requires 16D numbers
6D space requires 32D numbers. Goes on.
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Let power(base, exponent)
Then no. of dimension of numbers required to describe 'n' physical dimension is power(2, n-1.
If n = 1, that is 1D physical line. The. No. Of dimension of numbers is 2-(1-1) = 2-0 = 1. Real numbers
For n=2, 2D physical plane. Dimension of numbers is 2-(2-1) = 2-1 = 2. Complex numbers
For n=3, 3D physical space. Dimension of numbers is 2-(3-1) = 2-2 = 4. Quaternions
For n=4, 4D physical space. Dimension of numbers is 2-3 = 8.
5D space requires 16D numbers
6D space requires 32D numbers. Goes on.
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Jake
It seems to me that 2d rotation also requires 3 dimensions but that the 3rd dimension is implicit - i. e. perpendicular to the paper.
EDIT: Having watched another video it looks like 2d rotation is actually a + bi + cj where 'a' is the 3rd dimensions and i and j are the rotation. But beacuse 'a' is perpendicular to the paper and also constant the 'j' cancels out.
TBH I don't know enough to double check as I'm just looking at the patterns but I suspect i-2 = j-2 = ij = -1
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It seems to me that 2d rotation also requires 3 dimensions but that the 3rd dimension is implicit - i. e. perpendicular to the paper.
EDIT: Having watched another video it looks like 2d rotation is actually a + bi + cj where 'a' is the 3rd dimensions and i and j are the rotation. But beacuse 'a' is perpendicular to the paper and also constant the 'j' cancels out.
TBH I don't know enough to double check as I'm just looking at the patterns but I suspect i-2 = j-2 = ij = -1
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Tyson
TL DR: For the quaternions A + B + C still = A + (B + C, however for quaternions A + B +C does not equal B + A +C. and for the octonions even less is true. Not only does A + B + C not equal B + A + C but A + B + C does not equal A + (B + C. This is why as far as we understand, we can't go any further to a higher level of numbers. Too much is no longer capable once we reach octonions. yet we do get to utilize the octonions as functional existing set of numbers.
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TL DR: For the quaternions A + B + C still = A + (B + C, however for quaternions A + B +C does not equal B + A +C. and for the octonions even less is true. Not only does A + B + C not equal B + A + C but A + B + C does not equal A + (B + C. This is why as far as we understand, we can't go any further to a higher level of numbers. Too much is no longer capable once we reach octonions. yet we do get to utilize the octonions as functional existing set of numbers.
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Jean
I recently watched a conversation between Eric Weinstein and Sir Roger Penrose. IIRC Eric said there are no other numbers after the octonions because we have no more properties to lose, but now I learn there also exist sedenions! Is he wrong then? You also say that octonions are used for 8dimensions but shouldn't it be 4? I really enjoy all these free math lessons but with these contradictions spotted, who am I to believe is an authority on this subject?
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I recently watched a conversation between Eric Weinstein and Sir Roger Penrose. IIRC Eric said there are no other numbers after the octonions because we have no more properties to lose, but now I learn there also exist sedenions! Is he wrong then? You also say that octonions are used for 8dimensions but shouldn't it be 4? I really enjoy all these free math lessons but with these contradictions spotted, who am I to believe is an authority on this subject?
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Willie
Quaternions is why Wiles is wrong and Fermat's conjecture is negated.
Quaternions follow the rules of natural logs.
One rule of quaternions is q-n=nq
Fermat's conjecture states n cannot be greater than 2 in a-n+b-n=c-n.
We allow unit quaternions a, b, c to equal integers 2, 3, 5 respectively and n=3.
Therefore for unit quaternions, (2-3)+(3-3)=5-3.
Therefore (3x2)+(3x3)=3x5.
Therefore Fermat's conjecture is negated.
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Quaternions is why Wiles is wrong and Fermat's conjecture is negated.
Quaternions follow the rules of natural logs.
One rule of quaternions is q-n=nq
Fermat's conjecture states n cannot be greater than 2 in a-n+b-n=c-n.
We allow unit quaternions a, b, c to equal integers 2, 3, 5 respectively and n=3.
Therefore for unit quaternions, (2-3)+(3-3)=5-3.
Therefore (3x2)+(3x3)=3x5.
Therefore Fermat's conjecture is negated.
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education
I redescovered Quanternions by just asking below question:
-If base of natural number can be 1. n, but what if the base of a number was a comple number or i? -
Later Wikipedia showed me about Quaternions. I believe human technology has not understood how and where to apply quaternions like we did in electrical field of science. One day i believe quaternion will solve mysteries of world.
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I redescovered Quanternions by just asking below question:
-If base of natural number can be 1. n, but what if the base of a number was a comple number or i? -
Later Wikipedia showed me about Quaternions. I believe human technology has not understood how and where to apply quaternions like we did in electrical field of science. One day i believe quaternion will solve mysteries of world.
reply
Anmol
Watching the original video, it is counter intuitive to think that it loses some property, since we are adding dimensions, hence there is some more infomation, not less. But, after seeing this makes sense that they are decresing in specificity and increasing in generality of as we are going to higher over number system, and thus the immediate use of the numbers are not apparent in real life.
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Watching the original video, it is counter intuitive to think that it loses some property, since we are adding dimensions, hence there is some more infomation, not less. But, after seeing this makes sense that they are decresing in specificity and increasing in generality of as we are going to higher over number system, and thus the immediate use of the numbers are not apparent in real life.
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no
why in the equation he does from 3: 24 to 3: 44 does it appear he ends up with a point that should be graphed on the right and lower side of the euclidean plane because it appears to be a positive 90 degree rotation but he plots it backwards representing a negative -90 degree rotation. anyone else see this? i am bad at math so i am trying to learn this but it appears he messed up
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why in the equation he does from 3: 24 to 3: 44 does it appear he ends up with a point that should be graphed on the right and lower side of the euclidean plane because it appears to be a positive 90 degree rotation but he plots it backwards representing a negative -90 degree rotation. anyone else see this? i am bad at math so i am trying to learn this but it appears he messed up
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Jake
Why does 4 + 6i equal the same height as -6 + 4i? Because if i is the height on the graph, then 6i is higher than 4i, but on the graph they are on the same height because -6 + 4i is just a translation of 4 + 6i but on the negative x axis. So how are they the same height if the i value is different? Because 4i does not equal 6i. Does any of this make sense? Someone please help?
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Why does 4 + 6i equal the same height as -6 + 4i? Because if i is the height on the graph, then 6i is higher than 4i, but on the graph they are on the same height because -6 + 4i is just a translation of 4 + 6i but on the negative x axis. So how are they the same height if the i value is different? Because 4i does not equal 6i. Does any of this make sense? Someone please help?
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