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## Main Question or Discussion Point

Only recently I discovered that there is a class of complex numbers named hypercomplex numbers.

Hamilton invented (or discovered) the quaternions in 1843.

i² = j² = k² = ijk = -1

I understand how complex numbers work, but I don't understand how hypercomplex numbers work.

Can someone explain the following:

Why?

for example, why does i² = j² = ij = -1 not work?

how did he invent i² = j² = k² = ijk = -1

ij = k

jk = i

ki = j

ji = -k

kj = -i

ik = -j

Dimsun

Hamilton invented (or discovered) the quaternions in 1843.

i² = j² = k² = ijk = -1

I understand how complex numbers work, but I don't understand how hypercomplex numbers work.

Can someone explain the following:

**First**, Hamilton tried to expand the two dimensional complex plane (Argand plane) to three dimensions. But he failed to work with tripleds.Why?

for example, why does i² = j² = ij = -1 not work?

**Second**, how did he discover that in 4 dimensions he could work with entities called quaternions?how did he invent i² = j² = k² = ijk = -1

ij = k

jk = i

ki = j

ji = -k

kj = -i

ik = -j

**Third**, can someone give me an example to work with quaternions? what can you do with them?Dimsun