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At first the question of what $\sqrt{x^2}$ is seems silly. It looks like $x$. And for $x \in \mathbb{R}^+$ it is.

However, for $\mathbb{R}$, I'm not sure. I can think of 3 answers:

  • $abs(x)$
  • $\{x,-x\}$
  • $x$

Which one is correct and why?

bitmask
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    https://math.stackexchange.com/questions/13094/significance-of-sqrtnan – AlvinL Mar 04 '20 at 10:26
  • The symbol $\sqrt{x}$ most often means the unique positive root of $x$, which allows $\sqrt{x}$ to be a single valued function. If this is the case, your expression takes the same (positive) value for $x^2$ and $(-x)^2$ so it is option 1. Alternatively, the symbol $\sqrt{x}$ sometimes (e.g., in some British educational systems) refers to both the positive and negative roots in which case you would have option 2. But really this should be written as $\pm \sqrt{x}$. There is no case where option 3 would be chosen, since this is arbitrarily choosing positive and negative square roots. – Jam Mar 04 '20 at 11:37

2 Answers2

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$\sqrt{x^2}=x$ for $x\geq0$ and $\sqrt{x^2}=-x$ for $x<0$, which says $\sqrt{x^2}=|x|.$

Michael Rozenberg
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$\sqrt a$, for $a\in[0,\infty)$, is the non-negative real solution ofthe equation $x^2-a=0$. Therefore $\sqrt{t^2}=\lvert t\rvert$ for all $t\in\Bbb R$.