# why square root of a positive number is positive? [duplicate]

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

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

Thank you

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

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