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1 and the imaginary unit
From the definition of the imaginary unit i follows immediately
i 4 = 1
and that is somewhat surprising.
Explanation
The complex number i has only an imaginary part, and has no real value. But you can perform calculations with it, as
i 0 = 1 , i 1 = i , i 2 = – 1 , i 3 = – i
i 4 = 1 , i 5 = i , i 6 = – 1 , i 7 = – i
and so on. With great regularity the real numbers 1 and –1 occur.