7 ms·

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are /nonassociative/."

throwawaymath 8y ago
For the curious, John Baez has written about octonions pretty extensively: http://math.ucr.edu/home/baez/octonions/ http://math.ucr.edu/home/baez/octonions/
- leecarraher 8y ago
  happy to see John Baez fans on here, i like his series on his favorite numbers https://www.youtube.com/watch?v=vzjbRhYjELo https://www.youtube.com/watch?v=vzjbRhYjELo
- byproxy 8y ago
  Also, Joan Baez's cousin. And her father, Albert Baez, explaining elliptictal orbits: http://www.youtube.com/watch?v=_qpKhCa64Eo http://www.youtube.com/watch?v=_qpKhCa64Eo
PurpleBoxDragon 8y ago
Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a complex number. I remember seeing some w based number system derived from the cube root of either 1 or -1 (forget which, but w and -w were the solutions that weren't 1 or -1), but it did all the same things that complex numbers do and was considered mostly uninteresting. It also seems like there is a pattern to go infinitely beyond octonions, but they all behave identical to octonions, but are the octonions even needed in the same way complex numbers are needed, or do they just make some math problems easier to work with?
- uryga 8y ago
  Quaternions, in three words: "Rotations in 3D". I like this explanation: https://probablydance.com/2017/08/05/intuitive-quaternions/ https://probablydance.com/2017/08/05/intuitive-quaternions/
  - uryga 8y ago
    One thing I'd like to add: they're not actually 4-Dimensional. (read the link for more on this)
- NotPhysicsPhd 8y ago
  Not sure about octonions. But quaternions are somewhat used in mechanics, specifically for dealing with rotations. The usage of quaternions is computationally simpler for describing arbitrary rotations in 3d dimensions.
- yaks_hairbrush 8y ago
  Historically, quaternions came about as a way to try to reason about three dimensional physics. I mean, complex numbers were obviously really nice -- two dimensional numbers you could meaningfully add, subtract, multiply and divide. But they were only 2-D and we live in a 3-D world and we want to do 3-D physics. So Hamilton was trying really hard to find a way to have 3-dimensional numbers that behaved nicely, and he couldn't do it. But he did find 4-dimensional numbers -- the quaternions. And they were really neat. You can use them quite well for classical mechanics, and electromagnetism, and even special relativity. So, why don't we? We look at Maxwell's equations of electromagnetism today, and they're really nice, single-line vector formulas. You can also write them as nice, single-line quaternion formulas. Our notion of vector didn't exist at the time the quaternions were first used, and it was a boon to have quaternion notation to simplify some of these physical laws. Vectors and quaternions competed for a bit, and vectors won since they generalize to arbitrary dimensions. Hidden inside of quaternion multiplication, you can find the three-dimensional versions of the dot product and cross product. And they do have some theoretically interesting properties for number theory and abstract algebra. In the end, however, sometimes items are discarded in favor of better items. I'd rate quaternions as one of the coolest items that ultimately wound up in the discard pile.

throwawaymath 8y ago

For the curious, John Baez has written about octonions pretty extensively: http://math.ucr.edu/home/baez/octonions/ http://math.ucr.edu/home/baez/octonions/

leecarraher 8y ago
happy to see John Baez fans on here, i like his series on his favorite numbers https://www.youtube.com/watch?v=vzjbRhYjELo https://www.youtube.com/watch?v=vzjbRhYjELo
byproxy 8y ago
Also, Joan Baez's cousin. And her father, Albert Baez, explaining elliptictal orbits: http://www.youtube.com/watch?v=_qpKhCa64Eo http://www.youtube.com/watch?v=_qpKhCa64Eo

PurpleBoxDragon 8y ago

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a complex number. I remember seeing some w based number system derived from the cube root of either 1 or -1 (forget which, but w and -w were the solutions that weren't 1 or -1), but it did all the same things that complex numbers do and was considered mostly uninteresting. It also seems like there is a pattern to go infinitely beyond octonions, but they all behave identical to octonions, but are the octonions even needed in the same way complex numbers are needed, or do they just make some math problems easier to work with?

uryga 8y ago
Quaternions, in three words: "Rotations in 3D". I like this explanation: https://probablydance.com/2017/08/05/intuitive-quaternions/ https://probablydance.com/2017/08/05/intuitive-quaternions/
- uryga 8y ago
  One thing I'd like to add: they're not actually 4-Dimensional. (read the link for more on this)
NotPhysicsPhd 8y ago
Not sure about octonions. But quaternions are somewhat used in mechanics, specifically for dealing with rotations. The usage of quaternions is computationally simpler for describing arbitrary rotations in 3d dimensions.
yaks_hairbrush 8y ago
Historically, quaternions came about as a way to try to reason about three dimensional physics. I mean, complex numbers were obviously really nice -- two dimensional numbers you could meaningfully add, subtract, multiply and divide. But they were only 2-D and we live in a 3-D world and we want to do 3-D physics. So Hamilton was trying really hard to find a way to have 3-dimensional numbers that behaved nicely, and he couldn't do it. But he did find 4-dimensional numbers -- the quaternions. And they were really neat. You can use them quite well for classical mechanics, and electromagnetism, and even special relativity. So, why don't we? We look at Maxwell's equations of electromagnetism today, and they're really nice, single-line vector formulas. You can also write them as nice, single-line quaternion formulas. Our notion of vector didn't exist at the time the quaternions were first used, and it was a boon to have quaternion notation to simplify some of these physical laws. Vectors and quaternions competed for a bit, and vectors won since they generalize to arbitrary dimensions. Hidden inside of quaternion multiplication, you can find the three-dimensional versions of the dot product and cross product. And they do have some theoretically interesting properties for number theory and abstract algebra. In the end, however, sometimes items are discarded in favor of better items. I'd rate quaternions as one of the coolest items that ultimately wound up in the discard pile.