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November 10th, 2017, 07:27 AM   #1
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root of positive number

The root of 4 is 2 and not -2.
Why is it? Is this a convention?
Or there is another reason...
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November 10th, 2017, 08:25 AM   #2
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It's a convention that 4$^{1/2}$ = √4 = 2. The two square roots of 4 are ±√4 = ±2.
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November 10th, 2017, 12:41 PM   #3
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Is it convention by historic reasons, Or rational reasons?
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November 10th, 2017, 01:03 PM   #4
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It depends on context. For example, the quadratic solution uses both and is displayed explicitly.
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November 10th, 2017, 03:06 PM   #5
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Quote:
Originally Posted by shaharhada View Post
Is it convention by historic reasons, Or rational reasons?
It is a little bit of both.

We want $x^{1/2}$ to be a function so we want it to have a unique answer.

We want $\sqrt{x}$ to be a function so we want it to have a unique answer.

That is the purely rational part of the answer.

So we COULD define $\sqrt{x^2} = -\ |x| \le 0.$

But square roots were studied long before negative numbers were invented. And, for many practical problems, negative answers are meaningless. So the historical assumption that $\sqrt{x^2} \ge 0$ frequently makes practical sense.

So we define $\sqrt{x^2} = |x| \ge 0.$
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