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# Complex Numbers

"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 \(\sqrt{-1}\).

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

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

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

Evaluate

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

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

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

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

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

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