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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 3

         

If the square of \(x\) and the square root of \(x\) are equal to each other, but not equal to \(x\), then what is the value of \(x^2+x\)?

Let \(z=x+iy\) be a complex number where \(x\) and \(y\) are integers. Find the area of the rectangle whose vertices are the roots of the equation \[\large z \bar{z}^3+\bar{z}z^3=350.\]

Details: \( \bar{z}=x-iy\) and \(i=\sqrt{-1}\).

How many complex numbers are the conjugate of their own cube?

The complex numbers \(p\) and \(q\) satisfy \(p^3=5+i\sqrt{2}\) and \(q^3=5-i\sqrt{2}\).

Find the only possible integer value of \(p+q\).

\[\large e ^{ i\theta }e^{ 2i\theta} e^{ 3i\theta }...\large e^{ ni\theta } = 1\]

Find the value of \(\theta\) satisfying the above equation for \(m\in\mathbb{N}\).

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