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About a year ago, I started to notice that a lot of textbooks of recent date seem to favor the notation $x^\frac{1}{2}$ over $\sqrt{x}$, while I agree that notating the square root as an exponentation with one half can be beneficial in circumstances where the laws of exponentation are applied (i.e. $x^2 \cdot x^\frac{1}{2}=...$), a square root has nice visual cues that help grasping formulas quicker, especially since one can avoid using brackets for squared expressions $\sqrt{a^2+b^2} = (a^2+b^2)^\frac{1}{2}$ and within fractions, one can easily mistake the exponent 1/2 for another factor.

  • Is the perceived feeling of exponent notation becoming more popular in newer literature true? Is $\sqrt{}$ a "conservative" notation?

  • Are there typsetting suggestions regarding this question?

  • Is this a cultural thing? I have seen more power-laws in literature from an english speaking background.

wirrbel
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Generally speaking, the square-root symbol refers to the positive square root of a real number: $$\sqrt{4} = +2,$$ whereas the exponent of $\frac12$ refers to the multi-valued solution to $x^2 = \textrm{whatever}$. Note that if $x^2 = 4$, then $x = \pm 2$, but when we say $\sqrt{4}$, we always mean $2$.

Emily
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    $\sqrt{z}$ or $\sqrt{f(z)}$ are common notations in complex analysis. Of course there's always (well, almost always) the remark about the branch that is/has to be chosen, and the domain. – Daniel Fischer Feb 19 '14 at 20:39
  • @DanielFischer Really? I've never come across the notation in a modern text. Or maybe I have but I don't even notice it... – Emily Feb 19 '14 at 20:43
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    I disagree somewhat with this answer. My experience is that in most contexts the exponent 1/2 is synonymous with positive square root. For example, $e^{1/2}$ is usually interpreted as the exponential function $e^x$ evaluated at $x=1/2$, hence it means $\sqrt{e}$ (and not $-\sqrt{e}$ or the multivalued $\pm\sqrt{e}$). – Hans Lundmark Feb 19 '14 at 20:52
  • Ahlfors may not count as modern anymore, but he uses it (e.g. Chapter 2, section 3, $$z = \arccos w = - i \log (w \pm \sqrt{w^2-1})$$ and elsewhere). – Daniel Fischer Feb 19 '14 at 20:54
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I have no idea which text books you are reading. My experience is that the $\sqrt x$ notation is used extensively in algebra, where it matters whether a number has a quadratic or cubic character (for example).

On the other hand, in analysis $x^{\frac 12}$ is just one power of $x$ amongst a continuum.

Experienced mathematicians tend to use what is most convenient and natural to the context.

Mark Bennet
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