Radical of a complex number?

Algebra Level 4

If the absolute value of the imaginary part of $\sqrt{1-i }$ is equal to $I$ , find $\left \lfloor 1000I \right \rfloor$ .

The answer is 455.

1 solution

Hobart Pao
May 22, 2017

You can derive the formula by yourself or look it up in Ahlfor's Complex Analysis book.

$\sqrt{a + ib } = \pm \left( \sqrt{\dfrac{a + \sqrt{a^2 + b^2}}{2}} + i \dfrac{b}{|b|} \sqrt{\dfrac{-a + \sqrt{a^2 + b^2}}{2}} \right)$

$a= 1, b = -1 \to \text{imaginary part is } -\sqrt{\dfrac{-1 + \sqrt{2}}{2}} \to \boxed{455}$ because of abs. value.

FYI I simplified it to ask for the absolute value directly, instead of making people parse out that sentence.

Calvin Lin Staff - 4 years ago

Thanks. However I thought absolute value can't be negative, but the answer is negative. How can that be right?

Hobart Pao - 4 years ago

What about people that considered 455? My answer is still wrong...

Guilherme Niedu - 4 years ago

Oh, for some reason, I didn't realize your answer was negative. I thought your note was to make the answer positive. (Similar interpretation that Brian had.)

I have updated the answer to 455.

Calvin Lin Staff - 4 years ago

Radical of a complex number?

The answer is 455.

1 solution

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