I recently herd that saying $i= \sqrt{-1}$ is wrong (did not understated very well).
So my problem is that:
The simplification of $\sqrt{-4}=\sqrt{4}\times\sqrt{-1}=2i$, is this correct?
Or $\sqrt{-4}=\pm2i$ is this correct?
Or both wrong?
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2Maybe this question helps you to get some insight. – mrtaurho Sep 01 '19 at 19:24
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3Every non-zero complex number has two square roots. It is unwise to write $\sqrt z$ unless you have decided a definite convention on which one to take. – Angina Seng Sep 01 '19 at 19:25
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@LordSharktheUnknown so $\sqrt{-4}=\pm2i$? – Bad English Sep 01 '19 at 19:28
2 Answers
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$x^2=4$ is an equation with two solutions, $2$ and $-2$. We can write $\sqrt{4}=2$ because mathematicians have decided (as a convention) that the square root symbol will always mean the positive solution.
$x^2=-4$ is also an equation with two solutions, in this case, $2i$ and $-2i$. However, for complex numbers no convention similar to the one above is in place and so you should not use the square root symbol.
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You may factor the given equation in order to obtain \begin{align*} x = \sqrt{-4} \Longleftrightarrow x^{2} = -4 \Longleftrightarrow x^{2} + 4 = 0 \Longleftrightarrow (x+2i)(x-2i) = 0 \Longleftrightarrow x = \pm 2i \end{align*}
Here, it is explicitly assumed we are dealing with complex numbers.
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