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"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

What is

\[\large i^2 +i^6 +i^4?\]

Note: \(i\) is the imaginary number \(\sqrt{-1}\).

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\[\displaystyle\sum_{n=0}^{1000} i^n=? \] where \(i=\sqrt{-1}\).

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\[\large i{ z }^{ 3 }+z^{ 2 }-z+i=0\]

For \(i = \sqrt{-1}\), \(z\) is a complex number that satisfies the equation above. What is the value of \(\left| z \right| \)?

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Evaluate

\[\large{ \frac { { (1+i) }^{ 85 } }{ { (1-i) }^{ 83 } }}\]

**Hint**: There's an extremely fast way to solve this if you know why

\(\large 1+i = \sqrt{2}e^{i{\pi}/{4}} \)

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Can you evaluate this giant power of \(i\)?

\[\huge i^{-9999}\]

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