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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.

Complex Numbers: Level 3 Challenges

         

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 \(x^2+x\)?

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Basil Kuriakose by Basil Kuriakose

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 \(i=\sqrt{-1}\).

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Sandeep Bhardwaj by Sandeep Bhardwaj

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

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Utkarsh Bansal by Utkarsh Bansal

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 \(p+q\).

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Zhou ZeHao by Zhou ZeHao

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

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

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Saurav Pal by Saurav Pal
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