Quaternion

Prime numbers include the imaginary space in two dimensions. William Rowan Hamilton has found the quaternions. It extends complex numbers.

H is the sum of quaternions.
R is space of numerical vectors.

H={(a,b,c,d)|a,b,c,d∈R}
H=a+bi+cj+dk

i^2=j^2=k^2=ijk=-1

-1=ijk
-i=i^2jk
-i=-jk
∴i=jk

×	1	i	j	k
1	1	i	j	k
i	i	-1	k	-j
j	j	-k	-1	i
k	k	j	-i	-1

∴
ij=k, ji=-k, jk=i, kj=-i, ki=j, ik=-j

ij≠ji

H=a+bi+cj+dk (i=jk,j=ki,k=ij)

You are dazing.