I am confused with what Hamilton's equation means. I understand that the coordinates span 4 coordinates but I don't see how the quaternion product formula works. For example, if we break apart the quaternion product formula into simply i^2 = ijk, this intuitively does not make sense. Is there a way to visualize this?

nihar

I found this video
https://www.youtube.com/watch?v=d4EgbgTm0Bg&t=730s

nihar

Also what does the real part of the quaternion represent? Is it always zero?

adilets

An interesting topic I found was about how increasing the dimensions of these complex number systems results in losing some algebraic property.

Real -> Complex : Loss of ordering

Complex -> Quaternions : Loss of commutativity

Quaternions -> Octonions : Loss of associativity

More in depth explanation can be found on this Wikipedia page.

I am confused with what Hamilton's equation means. I understand that the coordinates span 4 coordinates but I don't see how the quaternion product formula works. For example, if we break apart the quaternion product formula into simply i^2 = ijk, this intuitively does not make sense. Is there a way to visualize this?

I found this video https://www.youtube.com/watch?v=d4EgbgTm0Bg&t=730s

Also what does the real part of the quaternion represent? Is it always zero?

An interesting topic I found was about how increasing the dimensions of these complex number systems results in losing some algebraic property.

Real -> Complex : Loss of ordering

Complex -> Quaternions : Loss of commutativity

Quaternions -> Octonions : Loss of associativity

More in depth explanation can be found on this Wikipedia page.