My Math Forum root of positive number

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 November 10th, 2017, 06:27 AM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 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... Thank you for answering me. You are welcome!!!
 November 10th, 2017, 07:25 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,966 Thanks: 2216 It's a convention that 4$^{1/2}$ = √4 = 2. The two square roots of 4 are ±√4 = ±2. Thanks from greg1313
 November 10th, 2017, 11:41 AM #3 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 Is it convention by historic reasons, Or rational reasons?
 November 10th, 2017, 12:03 PM #4 Global Moderator   Joined: May 2007 Posts: 6,821 Thanks: 723 It depends on context. For example, the quadratic solution uses both and is displayed explicitly. Thanks from greg1313
November 10th, 2017, 02:06 PM   #5
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Quote:
 Originally Posted by shaharhada 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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