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If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?

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The answers of using de Moivre's formula are correct but it may also be instructive to try and find the square roots directly using $(a+bi)^2 = 9 + 4i$ (say) and solve for $a$ and $b$ or even just use the quadratic formula directly, which will give you an appreciation for why we use de Moivre's formula. –  Jason Polak Jun 9 '11 at 19:38
    
@automorphism: we use De Moivre's formula... Do we? For what purposes? (These are real questions.) –  Did Jun 10 '11 at 6:53
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6 Answers 6

up vote 19 down vote accepted

The square root is not a well defined function on complex numbers. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is probably to go with De Moivre's formula, that is, converting your number into the form $r(\cos(\theta) + i \sin(\theta))$, and then you will get $(r(\cos(\theta)+ i \sin(\theta)))^{1/2} = ±\sqrt{r}(\cos(\theta/2) + i \sin(\theta/2))$.

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The "two different square roots" issue applies to real numbers as well; it's just that in that case there's a consensus on which one is the "principal" square root. –  Dan Jun 9 '11 at 23:49
    
@user1374 There is also a consensus about which square root of a complex number is the principal square root--at least for almost every complex number... See my answer. –  Did Jun 10 '11 at 5:55
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Oh, and I am afraid I must object to the assertion that going with De Moivre's formula would be the easiest way. –  Did Jun 10 '11 at 6:35
    
@Did What's the rational basis for having positive square roots as the principal square roots instead of negative square roots? –  Doug Spoonwood May 22 '13 at 13:47
    
@DougSpoonwood You might want to explain what you have in mind when you call some complex numbers positive and some others negative. –  Did May 22 '13 at 15:13
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This is a three parts post. The first part was written by user Did, it provides a formula and some brief comments on it. The second part was written by user Hans Lundmark, it provides a geometric way to understand the formula. The third part was written by user t.b., it provides some explanatory pictures and some brief comments on them.


(Did) If one is able to compute the square root of every positive real number and the modulus of every complex number, a nice formula for the principal square root $\sqrt{z}$ of $z$ is $$ \sqrt{z}=\sqrt{r}\frac{z+r}{|z+r|},\quad r=|z|. $$ Try to prove it and you will see, it works...

The principal square root is the one with a positive real part. The only case when the formula fails is when there is no principal square root, that is, when $z$ is a negative real number.

No sine or cosine is involved, one does not even need to solve second degree polynomials, one just uses squares and square roots. For example, for $z=9+4\mathrm{i}$, $$ \sqrt{z}=\frac{9+\sqrt{97}+4\mathrm{i}}{\sqrt{2(9+\sqrt{97})}}. $$


(HL) There's a geometric way of understanding the formula in Did's answer. To find a square root of a given complex number $z$, you first want to find a complex number $w$ which has half the argument of $z$ (since squaring doubles the argument). Compute $r=|z|$ and let $w = z+r$; thus $w$ lies $r$ steps to the right of $z$ in the complex plane. Draw a picture of this, and it should be clear that the points $0$, $z$ and $w$ form an isosceles triangle, from which one sees that the line from $0$ to $w$ bisects the angle between the real axis and the line from $0$ to $z$. In other words, $w$ has half the argument of $z$, as desired. Now it only remains to multiply $w$ by some suitable real constant $c$ so that $|cw|^2 = |z|$; then we will have $(cw)^2=z$ and hence $cw$ is a square root of $z$. Obviously, $c=\pm\sqrt{|z|}/|w|$ will do the trick, so this method only fails when $w$ happens to be zero, i.e., if $z$ is a negative real number.


(t.b.) Following a suggestion of Did, I take the liberty of adding two pictures I originally posted as a separate answer, but it seemed better to have them here:

nicer configuration

Here's the picture for $z = 9 + 4i$:

configuration z = 9 + 4i

Remark: The construction of the square roots is geometrically exact. That is to say, they were constructed using straightedge and compass only. I decided to hide the construction, as it seems rather obfuscating the intended illustration than adding to it. Nevertheless, I suggest taking a moment and thinking about how you would achieve the geometric construction.


Added (t.b.)

Here's the construction I used: Intersect the circle around $z/2$ through $z$ with the tangent to the unit circle orthogonal to $z$. Then $h^2 = (|z|-1)\cdot 1$ and thus the red circle has radius $\sqrt{|z|}$. It remains to intersect the red circle with the angular bisector of the $x$-axis and $z$ which I constructed using the process Hans described in his part of the post.

square-root construction

The pictures were created using GeoGebra.

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@Hans: Yes. This is a question where a picture would be useful (if only I knew how to draw pictures on MSE (and if I had more time on my hands)). Thanks. –  Did Jun 10 '11 at 6:14
    
Hans: Following a suggestion of @Didier I added two pictures I originally posted as a separate answer. It seemed more reasonable to have them in one answer. I hope you don't mind. –  t.b. Jun 10 '11 at 11:44
    
Since you can't see it on my deleted post, here's @Didier's original comment: "Theo and @Hans: What do you think about the idea of making one single post from our three complementary answers? (Re reputation, if that matters, either one of you may sign the resulting post, I really do not care.)" I refrain from adding Didier's answer in since it seems a bit too aggressive an edit to me, but I'd be strongly in favor of doing it. –  t.b. Jun 10 '11 at 11:53
    
Hans, @Theo: Done (at the risk of being seen as aggressive...). As soon as Hans says he is OK with the expanded version of his post, I will delete mine. –  Did Jun 10 '11 at 12:17
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@legends2k Right, the formula gets even simpler... Thanks for noticing this. Since you checked the formula, I cancelled the mention "unless I am mistaken". –  Did Jan 6 at 14:53
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Here is a direct algebraic answer:

Suppose that $z=c+di$, and we want to find $\sqrt{z}=a+bi$ lying in the first two quadrants. So what are $a$ and $b$?

Precisely we have: $$a=\sqrt{\frac{c+\sqrt{c^{2}+d^{2}}}{2}}$$ and $$b=\frac{d}{|d|}\sqrt{\frac{-c+\sqrt{c^{2}+d^{2}}}{2}}.$$ (The factor of $\frac{d}{|d|}$ is used so that $b$ has the same sign as $d$) To find this, we can use brute force and the quadratic formula. Squaring, we would need to solve $$a^2-b^2 +2abi=c+di.$$ This gives two equations and two unknowns (seperate into real and imaginary parts) which can then be solved by substitutions and the quadratic formula.

Hope that helps,

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One way is to convert the complex number into polar form. For $z = re^{i\theta}$, $z^2 = r^2 e^{i(2\theta)}$. So to take the square root, you'll find $z^{1/2} = \pm \sqrt{r} e^{i\theta/2}$.

Added: Just as with the nonnegative real numbers, there are two complex numbers whose square will be $z$. So there are two square roots (except when $z = 0$).

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You can also do following (technique often advised at school) :

  • Let's write $z² = 9 + 4i$ with $z = a + bi$. The goal is to find $z$

    Thus we have $(a + bi)² = 9 + 4i$ and if you expend we get $a²+ 2abi - b² = 9 + 4i$

    If you identify the real and imaginary parts, you obtain :

    $a²-b² = 9$ (1)

    and

    $2ab= 4$ (2)

  • Now, as $z² = 9 + 4i$, the modulus of $z²$ and $9 + 4i$ are equal so we can write :

    $a²+b² = \sqrt{9²+4²}$

    $a²+b² = \sqrt{97}$ (3)

  • Now find $a$ and $b$ with the the equations (1) , (2) and (3) :

    (1) + (3) $\Leftrightarrow 2a² = 9+\sqrt{97} $

    so $a = \sqrt{\frac{1}{2}(9+\sqrt{97})} $ or $a = - \sqrt{\frac{1}{2}(9+\sqrt{97})} $

    With equation (2) and the previous result you can now find $b$ :

    $2ab= 4$

    $b= 2/a$

    so $b = 2\sqrt{\frac{2}{9+\sqrt{97}}} $ or $b = - 2\sqrt{\frac{2}{9+\sqrt{97}}} $

    The answer is : $z = \sqrt{\frac{1}{2}(9+\sqrt{97})} + 2i\sqrt{\frac{2}{9+\sqrt{97}}} $ or $z = - \sqrt{\frac{1}{2}(9+\sqrt{97})} - 2i\sqrt{\frac{2}{9+\sqrt{97}}} $

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x^2 - y^2 +2ixy = write the complex no.

x^2 - y^2 = write the real value---------(1)

2xy = write the imaginary value------(2)

from equation (1) and (2)

x^2 + y^2 = root over{(x^2 - y^2)^2 + 4x^2y^2}

you will get x^2 + y^2 = ?-------(3)

from equation (1) and (3)

you will get the value of x and y

Note : ^ signifies square of that no. e.g x^2 = square of x

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