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My teacher said that for $\sqrt{x}$ X must belong to integer whereas in $x^\frac{1}{2}$ X belong to entire complex plane. Is there any source for that? How are $\sqrt{x}$ and $x^{\frac{1}{2}}$ actually defined?

Aditya Pratap Singh
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user541396
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  • That is ... wrong. $\sqrt x \equiv x^{\frac 12}$ and both belong to the entire complex plane. Are you sure you haven't misunderstood your teacher? – For the love of maths Mar 27 '19 at 15:05
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    Possible duplicate of Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? – Tips Mar 27 '19 at 15:09
  • Now when I look back what he might mean is that $\frac{a}{b} \sqrt X $ has no meaning if a/b don't simplify to natural number. Which again opens a new question. Sigh. – user541396 Mar 27 '19 at 16:06
  • @saketkumar no they are completely different questions. – user541396 Mar 27 '19 at 16:06
  • Are you sure that you don’t mean real number instead of “integer” or “natural number?” The value of $\sqrt2$ is neither, but it is a real number. – amd Mar 27 '19 at 18:50
  • As an aside, whenever I see “My teacher said...” questions here, I always end up wondering why people don’t ask the teacher to clarify the statement. – amd Mar 27 '19 at 18:51
  • There is a reason for that. I don't want the teacher to make fun of me in front of whole class. You are very lucky that you can ask your teacher a stupid question without hesitation. Not everyone is as lucky as you.( I mean no disrespect) – user541396 Mar 28 '19 at 00:53

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There is no difference between $\sqrt x$ and $x^{1/2}$.

However, for the second part of your question, sometimes $f=\sqrt{\cdot}$ is understood as a function, e.g. $f:\mathbb R_0^+\to\mathbb R_0^+$ or $f:\mathbb N\to\mathbb R$. Sometimes (and more infrequently), however, it is understood as a relation. The latter is often needed in the complex plane, where there are multiple branches of the square root.

log_math
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I would say that without further specification, they both require that $x$ is a non-negative two number to be well-defined.

Ultimately, they are interchangeable. The difference is mainly about what you want to convey. For $\sqrt x$, the $\sqrt{\phantom x}$ part is usually quite fixed, while for $x^{1/2}$, the $^{1/2}$ part is very much something that can partake in any arithmetic that might happen. At least that's how I feel about them.

Arthur
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  • How is $\sqrt •$ fixed ? Can't we write $ \frac{a}{b} \sqrt x $. – user541396 Mar 27 '19 at 16:08
  • @user654700 What do you mean by $\frac ab\sqrt x$? Do you mean $\sqrt[a/b]x$, the $\frac ab$th root of $x$? Sure, we can write that. That's not what I mean. What I mean is that when I write $\sqrt x$, I usually intend for it to stay as $\sqrt{\phantom x}$. I may change what's outside the root and what's inside the root, or the root might disappear, but I usually wouldn't change the root itself. If I intend for it to change, then I would use fractional exponents instead (for instance, I wouldn't write $\sqrt x\sqrt[3]x=\sqrt[6/5]x$, but instead go for $x^{1/2}x^{1/3}=x^{5/6}$). – Arthur Mar 27 '19 at 16:30