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I am writting an article and i want to fix a choice for a square root of a complex number:

The square root that i want to choose satisfies

$(\sqrt{z})^2=z$ for all $z\in\mathbb{C}$ and $\sqrt{z}=\sqrt{z}$ if $z\in\mathbb{R}+$ and $\sqrt{z}=i\sqrt{-z}$ if $z\in\mathbb{R}-$

How can i explain that in the begining of the article ? I am not very good in english and i don't now how to formulate that once a time for all the article.

Please help me. Thanks in advance.

Simply Beautiful Art
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naima
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  • Those conditions still gives you a variety of choices. For example you could still be unclear as to whether $\sqrt{-i}$ is $\frac{\sqrt{2}}2 - i \frac{\sqrt{2}}2$ or $-\frac{\sqrt{2}}2 + i \frac{\sqrt{2}}2$. One approach is to define $\sqrt{re^{i\theta}}=\sqrt{r}e^{i\theta/2}$ with $r \ge 0$ and (you choose) either $0 \le \theta \lt 2\pi$ or $-\pi \lt \theta \le \pi$ – Henry Mar 10 '18 at 14:36
  • can you please edit your comment it is not clear – naima Mar 10 '18 at 14:37
  • which one i choose to assume that $(\sqrt{-i})^2=-i$? – naima Mar 10 '18 at 14:42
  • Both will give you $(\sqrt{z})^2=z$ for all $z\in\mathbb{C}$ – Henry Mar 10 '18 at 14:46
  • How to formulate a sentence explaning that for negative real numbers we want $\sqrt{r}$ to be $i\sqrt{-r}$ for all the article? have you a an idea? – naima Mar 10 '18 at 14:52
  • this is what i wrote: The square root of a complex number $z=re^{i\phi}$ with $r\ge 0$ and $0\le \phi<2\pi$ is defined as follows: $\sqrt{z}={r}e^{i\frac{\phi}{2}}$. does that saficient to understand that i want $\sqrt{-2}=i\sqrt{2}$? – naima Mar 10 '18 at 15:31
  • You may have a typo and may have intended $\sqrt{z}=\sqrt{r}e^{i\frac{\phi}{2}}$. But that would then be sufficient. You will have $\sqrt{-2}=i\sqrt{2}$ but not $\sqrt{(-2)^2}=-2$ – Henry Mar 10 '18 at 20:05
  • that give $\sqrt{(-2)^2}=2$ and $(\sqrt{(-2)})^2=-2$ No? – naima Mar 10 '18 at 20:11

2 Answers2

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The universally accepted convention (which isn't universally known) is that for positive real numbers the square root sign means the positive root. If you must, you can tell your readers that for negative real numbers $r$ you want $\sqrt{r}$ to be $i\sqrt{-r}$. You can just say that at the start.

For complex numbers with an imaginary part it's a bad idea to use the square root sign since there is no good convention for choosing one of the square roots over the other.

Related: Order of operations with complex numbers

Ethan Bolker
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  • What do you recommend in place of the square root symbol? – Barry Cipra Mar 10 '18 at 15:16
  • @BarryCipra I don't think there's a good alternative, for good reason. You can use the square root symbol (or an alternative like the $1/2$ power) only if later on it doesn't matter which square root you take and you say so up front - perhaps because the result is squared, or input to another even function. Otherwise you really must come to grips with multivalued complex functions to some extent. – Ethan Bolker Mar 10 '18 at 15:22
  • I'm inclined to disagree. The square root symbol (and/or the "^{1/2}" notation) is quite useful as a well-defined function, as long as you are upfront and clear as to the convention being used, and then take care to adhere to the implications of the chosen convention. I did my PhD dissertation (ages ago) on modular forms of half integral weight. In his seminal 1973 paper on the subject, Goro Shimura picked a particular convention and stuck to it. His paper (and my dissertation) would have been unwieldy to write without a square root notation. – Barry Cipra Mar 10 '18 at 15:33
  • @BarryCipra OK I'm convinced, at least for professionals. But the number of questions on this site that hinge on misuse mean beginners should be cautious. – Ethan Bolker Mar 10 '18 at 15:45
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    I agree. The notation is beguiling, and even professionals sometimes succumb to its charms. – Barry Cipra Mar 10 '18 at 15:51
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The conventions $\Re(\sqrt z)\gt0$ for $z\in\mathbb{R}^+$ and $\Im(\sqrt z)\gt0$ for $z\in\mathbb{R}^-$ are almost always adopted for a square root function. After that, there are two standard conventions:

$$\Re(\sqrt z)\gt0\quad\text{for }z\not\in\mathbb{R}$$ or $$\Im(\sqrt z)\gt0\quad\text{for }z\not\in\mathbb{R}$$

The first of these introduces a discontinuity in the complex-valued function $f(z)=\sqrt z$ at every point on the negative half of the real axis; the second does the same at every point on the positive half of the real axis.

Barry Cipra
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