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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.

# Level 2

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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