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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.

Arithmetic

         

What is the absolute value of the complex number \[12+5i?\]

Details and assumptions

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

Is it true that \(\text{Re}(\text{Im}(z)) = \text{Im}(\text{Re}(z))\) for all complex numbers \(z\)?

What is the value of \[ \left(9 + \sqrt{-81}\right)\left(20 - \sqrt{-400}\right)? \]

Simplify \(\frac{4+2i}{2-i}\).

Details and assumptions
\(i\) is the imaginary number that satisfies \(i^2 = -1\).

Which set of complex numbers satisfy the equation \(iz + \overline{z} = 0\)?

Hint: Start by letting \(z = x + iy\).

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