Hamilton's quaternions equation i^2=j^2=k^2=ijk=-1 is incorrect because -1 like every other real, imaginary and complex number can only have two square roots so k^2 cannot equal -1 unless it also equals either i or j. However -1 does have three cube roots which are cos60+isin60, cos180+isin180 which equals -1, and cos300+isin300.

## Roots of Quaternions

Hamilton's quaternions equation i^2=j^2=k^2=ijk=-1 is incorrect because -1 like every other real, imaginary and complex number can only have two square roots so k^2 cannot equal -1 unless it also equals either i or j. However -1 does have three cube roots which are cos60+isin60, cos180+isin180 which equals -1, and cos300+isin300.