Imaginary unit
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The imaginary unit, , is the fundamental component of all complex numbers. In fact, it is a complex number, itself.
Problems
One common problem involving the imaginary unit is along the lines of:
"Find the sum of ."
Let's begin by computing powers of .
We can now stop because we have come back to our original term. This means that the sequence i, -1, -i, 1 repeats. Note that this sums to 0. That means that all sequences have a sum of zero (k is a natural number). Since
, the original series sums to the first two terms of the powers of i, which equals -1+i.