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The exponential function can be defined by the series expansionwhere it is important to note that for a definite magnitude |x| we have convergence. Therefore, we can define the exponential function of the elements of a geometric algebra. In Cl3, we have four important types of elements (numbers): nilpotents (dual numbers, square to zero), imaginary (square to −1), hyperimaginary (square to 1), and idempotents (square to itself). Nilpotents are the easiest to deal with: the expansion (∗) gives zeros, except for the first two terms, and consequently, we have
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