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This question already has an answer here:

We have $(+3)^2=(-3)^2=9$. But why do we define

$$\sqrt 9=+3?$$ Why $\sqrt9=-3$ is false?

Thank you

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marked as duplicate by user147263, Yiorgos S. Smyrlis, RE60K, Najib Idrissi, Davide Giraudo Oct 1 '14 at 18:31

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We want $\sqrt{\cdot}$ to be a function on nonnegative reals. To be a function, it must have exactly one value for each input, and the most natural one to choose is the positive one.

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You are right, there are two $a's$ such that $a^2=9$. Instead of saying "Please give me the positive number $a$ such that $a^2=9$, we write $\sqrt{9}$. It's short-hand for the longer sentence.

If we more often cared about getting the negative number $a$ such that $a^2=9$, we might come up with special notation for that case.

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  • $\begingroup$ To get the negative number, you'd just write $-\sqrt\cdot$ rather than $\sqrt\cdot$. $\endgroup$ – Akiva Weinberger Oct 1 '14 at 16:22

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