# Multicomplex number

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In mathematics, the multicomplex number systems Cn are defined inductively as follows: Let C0 be the real number system. For every n > 0 let in be a square root of −1, that is, an imaginary number. Then ${\displaystyle {\text{C}}_{n+1}=\lbrace z=x+yi_{n+1}:x,y\in {\text{C}}_{n}\rbrace }$. In the multicomplex number systems one also requires that ${\displaystyle i_{n}i_{m}=i_{m}i_{n}}$ (commutativity). Then C1 is the complex number system, C2 is the bicomplex number system, C3 is the tricomplex number system of Corrado Segre, and Cn is the multicomplex number system of order n.
The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of −1 anti-commute (${\displaystyle i_{n}i_{m}+i_{m}i_{n}=0}$ when mn for Clifford).
Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: ${\displaystyle (i_{n}-i_{m})(i_{n}+i_{m})=i_{n}^{2}-i_{m}^{2}=0}$ despite ${\displaystyle i_{n}-i_{m}\neq 0}$ and ${\displaystyle i_{n}+i_{m}\neq 0}$, and ${\displaystyle (i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0}$ despite ${\displaystyle i_{n}i_{m}\neq 1}$ and ${\displaystyle i_{n}i_{m}\neq -1}$. Any product ${\displaystyle i_{n}i_{m}}$ of two distinct multicomplex units behaves as the ${\displaystyle j}$ of the split-complex numbers, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane.