What is product (1-i)(1-i^2)(1-i^3)...(1-i^2008)?

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You need to remember the complex number theory telling that `i=sqrt(-1), i^2=-1,i^3=-i,i^4=1` .

Notice that the product contains the factor `(1-i^4), ` hence plugging `i^4=1`  in `1 - i^4`  yields 1 - 1 = 0.

Since the factor `(1-i^4) = 0` , this factor cancels all product, hence `(1-i)(1-i^2)(1-i^3)...(1-i^2008) = 0.`

Hence, evaluating the product yields `(1-i)(1-i^2)(1-i^3)...(1-i^2008) = 0.`

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