Algebraic Properties
Tessarines with w and z complex numbers form a commutative and associative quaternionic ring (whereas quaternions are not commutative). They allow for powers, roots, and logarithms of, which is a non-real root of 1 (see conic quaternions for examples and references). They do not form a field because the idempotents
have determinant / modulus 0 and therefore cannot be inverted multiplicatively. In addition, the arithmetic contains zero divisors
In contrast, the quaternions form a skew field without zero-divisors, and can also be represented in 2×2 matrix form.
Read more about this topic: Bicomplex Number
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