why square root of a positive number is positive?

Question

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

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

Thank you

score 2 · Accepted Answer · answered Oct 01 '14 at 15:56

2

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.

answered Oct 01 '14 at 15:56

vadim123

82,796

score 0 · Answer 2 · answered Oct 01 '14 at 16:19

0

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.

answered Oct 01 '14 at 16:19

MathStudent

1,352
11
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To get the negative number, you'd just write $-\sqrt\cdot$ rather than $\sqrt\cdot$. – Akiva Weinberger Oct 01 '14 at 16:22

why square root of a positive number is positive?

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