Algebra

Complex Numbers

Complex Numbers: Level 2 Challenges

         

What is

i2+i6+i4?\large i^2 +i^6 +i^4?

Note: ii is the imaginary number 1\sqrt{-1}.

n=01000in=?\displaystyle\sum_{n=0}^{1000} i^n=? where i=1i=\sqrt{-1}.

iz3+z2z+i=0\large i{ z }^{ 3 }+z^{ 2 }-z+i=0

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

Evaluate

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

Hint: There's an extremely fast way to solve this if you know why
1+i=2eiπ/4\large 1+i = \sqrt{2}e^{i{\pi}/{4}}

Can you evaluate this giant power of ii?

i9999\huge i^{-9999}

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