Bicomplex number

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In mathematics, a bicomplex number (from the multicomplex numbers, see e.g. G. B. Price) is a number written in the form, a + bi₁ + ci₂ + dj, where i₁, i₂ and j are imaginary units. Based on the rules for multiplying the imaginary units, then if A = a + bi₁ and B = c + di₁, then the bicomplex number may be written A + Bi₂. Thus, bicomplex numbers are similar to complex numbers, but the two parts are complex rather than real. Bicomplex numbers reduce to complex numbers when A and B are real numbers.

The set of all bicomplex numbers forms a commutative ring with identity; thus multiplication of bicomplex numbers is both commutative and associative and distributes over addition. Given this and rules for multiplying the imaginary units, any two bicomplex numbers may be multiplied. Multiplication of the imaginary units is given by:

i₁ · i₁ = −1
i₂ · i₂ = −1
j · j = 1
i₁ · i₂ = j
i₁ · j = −i₂
i₂ · j = −i₁

Division is not defined for some bicomplex numbers, as some are factors of zero, which cannot be divided by. Examples of these are 1 + j and i₁ + i₂.

[edit] References

G. Baley Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker Inc., New York, 1991
Dominic Rochon, A Bloch Constant for Hyperholomorphic Functions June, 2000
Clyde M. Davenport, Commutative Hypercomplex Mathematics, 2003

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