Definition
Let {1, i, j, k} be the basis for the (real) quaternions, and let u, v, w, x be complex numbers, then
- q = u 1 + v i + w j + x k
is a biquaternion. To distinguish square roots of minus one in the biquaternions, Hamilton and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h since there is an i in the quaternion group. Then
- hi = ih, hj = jh, and hk = kh since h is a scalar.
Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions, now available in the Historical Mathematical Monographs of Cornell University. The two editions of Elements of Quaternions (1866 & 1899) reduced the biquaternion coverage in favor of the real quaternions. He introduced the terms bivector, biconjugate, bitensor, and biversor.
Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor.
Read more about this topic: Biquaternion
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