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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 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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