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How do I get the square root of a complex number?

I'm currently playing with complex numbers and I realized that I don't understand how to compute $\sqrt{i + 1}$. My math program, sage, returns

$$1.09868411346781 + 0.455089860562227*i$$

How does sage compute that number? I don't even see how one could rewrite $\sqrt{i + 1}$ in a number of the form $a+bi$.

Community
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Konstantin Weitz
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5 Answers5

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For finding $\sqrt{x+yi}$ put $$\sqrt{x+yi}=a+bi$$ so $$(\sqrt{x+yi})^2=(a+bi)^2$$ If you do the latter identity, you will find $$(1): a^2-b^2=x;(2):2ab=y$$ and $$(3):(a^2+b^2)^2=x^2+y^2$$ Adding (1) and (3) gives you $$2a^2=\sqrt{x^2+y^2}+x$$ and subtracting (1) of (3) gives you $$2b^2=\sqrt{x^2+y^2}-x$$. Now, I think you can do your question by yourself. :)

Mikasa
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  • Remark that this is a special case of the Simple Denesting Rule linked in my answer. – Bill Dubuque Aug 28 '12 at 14:26
  • @BillDubuque: Thanks for noting me that. I was not aware of your clever answer here. Mine looks so elementary. :) – Mikasa Aug 28 '12 at 14:32
  • The Simple Denesting Rule may be viewed as a convenient way to remember the solution of the above system of equations. – Bill Dubuque Aug 28 '12 at 15:05
  • Sir,i see that $(a^2+b^2)^2=(a^2-b^2)^2+(2ab)^2=x^2+y^2$.So $(3)$ equ. should be edited. –  Feb 04 '13 at 17:14
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$i + 1 = \sqrt2 \left ( {1 \over \sqrt 2} + {1 \over \sqrt 2} i\right ) \\ = \sqrt 2 \left( \cos \left( \pi \over 4 \right) + i \sin \left( \pi \over 4 \right) \right ) \\ = \sqrt 2 e^{i \pi \over 4}$

$\sqrt{i +1} =\left( 2\right)^{1 \over 4}e^{i \pi \over 8} = \left( 2\right)^{1 \over 4}\left( \cos \left( \pi \over 8 \right) + i \sin \left( \pi \over 8 \right)\right)$

Well, this is how Wolframalpha calculates. The other root would be $\left( 2\right)^{1 \over 4}\left( \cos \left( 9\pi \over 8 \right) + i \sin \left(9 \pi \over 8 \right)\right)$

S L
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  • But you might add that one could also write it as $\sqrt2e^{i(\pi/4+2\pi)}$, which yields the other square root, ${(2)}^{1/4}e^{9i\pi/8}$. – MJD Aug 28 '12 at 09:45
  • Yep , thanks for that!! – S L Aug 28 '12 at 10:18
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Hint $\ $ Applying my Simple Denesting Rule and rationalizing denominators yields

$$\rm \sqrt{1 + {\it i}\,}\, =\, \sqrt{\alpha}+ {\it i}\,\sqrt{-\alpha'}\quad for\quad \alpha,\alpha' \,=\, \frac{1\pm\sqrt{2}}2$$

Indeed, squaring it yields $\rm\,\ \alpha+\alpha'\! + 2 \sqrt{-\alpha\alpha'}\:{\it i}\: =\, 1 + {\it i}$

Community
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Bill Dubuque
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First, the notation $\sqrt{1+i}$ is misleading, because $i+1$ has two squared roots and there is no canonical way to distinguish one from the other. Indeed, $\displaystyle i+1= \sqrt{2} \left( \frac{ \sqrt{2}}{2}+ i\frac{\sqrt{2}}{2} \right)= \sqrt{2} e^{i \pi /4}$. So $\pm \ 2^{1/4} e^{i \pi /8}$ are the squared roots of $1+i$.

Seirios
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  • I think the canonical choice is the one with positive real part, unless the real part is $0$, then it's the one with positive imaginary part. Or more generally, if you write the number as $r\mathrm e^{\mathrm i\phi}$ with $-\pi < \phi \le \pi$, then the canonical choice is $\sqrt{r}\mathrm e^{\mathrm i\phi/2}$ (and $r^{1/n}\mathrm e^{\mathrm i\phi/n}$ for the $n$th root). Note that without a canonical choice, $\sqrt{2}$ would be just as ambiguous. – celtschk Aug 28 '12 at 12:40
  • It corresponds to the lexicographic order, but since $\mathbb{C}$ is not an ordered field, I am not sure that it is really canonical. – Seirios Aug 28 '12 at 13:48
  • That it corresponds to the lexicographic order is coincidential. What is not coincidental is that for positive reals it corresponds with the choice made in real numbers. Also a wanted property is that it is continuous except for the non-positive reals. And I guess it's also a wanted property that for all numbers except of the negative reals, the square root of the conjugate is the conjugate of the square root (for the negative reals, it's not possible to achieve that). – celtschk Aug 28 '12 at 14:51
  • Also note that the square root (as I wrote in the first comment) is just the special case of the more general $n$th root, where there's the special property that all $n$th roots of $1$ can be obtained as powers of the canonical one (which is not the case for all possible choices if $n$ is compound). – celtschk Aug 28 '12 at 14:56
  • @celtschk What is not canonical is to decide that one wants a continuus square root on the complex plane except the negative real axis. One could also consider the complex plane except any ray issued from zero, or any other line issued from zero. – Did Mar 13 '13 at 23:14
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Using the fact that $ {\rm e}^{i2k\pi} = 1 $ for any integer $k$, we have

$$z^2 = {(1+i)} = \sqrt{2}\, { \rm e}^{i \frac{\pi}{4}} = \sqrt{2}\,{ \rm e}^{i \frac{\pi}{4}} { \rm e}^{2 k\pi i} = \sqrt{2}\,{ \rm e}^{i \left(\frac{\pi}{4} + 2k\pi\right)} $$

$$ \Rightarrow z = 2^{\frac{1}{4}} {\rm e}^{ \left(\frac{i\pi}{8} + k \pi i\right)} \,.$$

Now, you can see that, if you take $k=0$, you will get the first root, and when you take $k=1$, you will have the second root. If you take $k \geq 2$, you will have the same first two roots. Then the only choices for $k$ is $0,1$.

Mhenni Benghorbal
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