Editing Square root (section)

===Notes===

In the following, the complex {{mvar|z}} and {{mvar|w}} may be expressed as:

* <math> z = |z| e^{i \theta_z}</math>
* <math> w = |w| e^{i \theta_w}</math>

where <math>-\pi<\theta_z\le\pi</math> and <math>-\pi<\theta_w\le\pi</math>.

Because of the discontinuous nature of the square root function in the complex plane, the following laws are '''not true''' in general.

*  <math>\sqrt{zw} = \sqrt{z} \sqrt{w}</math> {{pb}} Counterexample for the principal square root: {{math|1=''z'' = −1}} and {{math|1=''w'' = −1}} {{pb}} This equality is valid only when <math>-\pi<\theta_z+\theta_w\le\pi</math>
* <math>\frac{\sqrt{w}}{\sqrt z} = \sqrt{\frac{w}{z}}</math> {{pb}} Counterexample for the principal square root: {{math|1=''w'' = 1}} and {{math|1=''z'' = −1}} {{pb}} This equality is valid only when <math>-\pi<\theta_w-\theta_z\le\pi</math>
*<math>\sqrt{z^*} = \left( \sqrt z \right)^*</math> {{pb}} Counterexample for the principal square root: {{math|1=''z'' = −1}}) {{pb}} This equality is valid only when <math>\theta_z\ne\pi</math>

A similar problem appears with other complex functions with branch cuts, e.g., the [[complex logarithm]] and the relations {{math|1=log''z'' + log''w'' = log(''zw'')}} or {{math|1=log(''z''<sup>*</sup>) = log(''z'')<sup>*</sup>}} which are not true in general.

Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that {{math|1=−1 = 1}}:
<math display="block">\begin{align}
-1 &= i \cdot i \\
&= \sqrt{-1} \cdot \sqrt{-1} \\
&= \sqrt{\left(-1\right)\cdot\left(-1\right)} \\
&= \sqrt{1} \\
&= 1.
\end{align}</math>

The third equality cannot be justified (see [[invalid proof]]).<ref>{{cite book |last1=Maxwell |first1=E. A. |url=https://archive.org/details/fallaciesinmathe0000maxw_z1t0/page/n5/mode/2up |title=Fallacies in Mathematics |publisher=Cambridge University Press |year=1959 |isbn=9780511569739}}</ref>{{rp|at=Chapter VI, Section I, Subsection 2 ''The fallacy that +1 = −1''}} It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains <math>\sqrt{1}\cdot\sqrt{-1}.</math> The left-hand side becomes either<math display="block">\sqrt{-1} \cdot \sqrt{-1}=i \cdot i=-1</math>
if the branch includes {{math|+''i''}} or<math display="block">\sqrt{-1} \cdot \sqrt{-1}=(-i) \cdot (-i)=-1</math>
if the branch includes {{math|−''i''}}, while the right-hand side becomes<math display="block">\sqrt{\left(-1\right)\cdot\left(-1\right)} = \sqrt{1} = -1,</math>
where the last equality, <math>\sqrt{1} = -1,</math> is a consequence of the choice of branch in the redefinition of {{math|√}}.