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November 10th, 2017, 07:27 AM  #1 
Senior Member Joined: Nov 2011 Posts: 110 Thanks: 1  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, 08:25 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,145 Thanks: 1418 
It's a convention that 4$^{1/2}$ = √4 = 2. The two square roots of 4 are ±√4 = ±2.

November 10th, 2017, 12:41 PM  #3 
Senior Member Joined: Nov 2011 Posts: 110 Thanks: 1 
Is it convention by historic reasons, Or rational reasons?

November 10th, 2017, 01:03 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,378 Thanks: 542 
It depends on context. For example, the quadratic solution uses both and is displayed explicitly.

November 10th, 2017, 03:06 PM  #5 
Senior Member Joined: May 2016 From: USA Posts: 823 Thanks: 335  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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