Expert: Scott A Wilson - 5/11/2009

Question
QUESTION: Can you remind how to take the root of a complex number
i will post what i know and you correct me ok?
let say
5+3i
we will say we assume that the sqrt of 5+3i is a+bi right? then we say
5+3i=(a+bi)^2
a^2+b^2=5(1)
2ab=3(2)
by squaring both equations
a^4+2a^2*b^2+b^4=25
4a^2*b^2=9
a^4-2a^2*b^2+b^4=16
(a^2-b^2)^2=16
a^2-b^2=4 (3)
(1)+(3)
2a^2=9
(can you continue from this part and fix errors?)


ANSWER: To start with, when (a + bi) is squared, you get a² + 2abi - b².
This means that 5 = a² - b² ( not plus, but minus )
and 2ab = 2(3)(5) = 30.
But this is not the proper approach to finding the squareroot.

When computing the squareroot of imaginary numbers,
it is best if they are converted from (x,y) to (r,Θ)
where r is the radius and Θ is the angle.

Once this is done, to compute the squareroot,
take the squareroot of r and half the angle Θ.

In other words, the squareroot of -81 is the same as computing the squareroot of (81 at an angle of 180°).

The squareroot of 81 is 9 and half of the angle is 90°.
In complex arithmatic, that is known as 3 straight up,
which is 9i.  9i*9i = 81i² = -81.

Convert to polar coordinates.  Here are some examples:

Now the squareroot of 9i is a different story.
It is r=9 and Θ=90°.  So the squareroot is 3, Θ=45°.

That is (3/√2, (3/√2)i).
To check it out and make sure, square it again.
(3/√2, (3/√2)i)² = 9/2 + (9/2)i + (9/2)i + (9/2)i + (9/2)i².
Since i² = -1, we have 9/2 in from minus 9/2 at the end.
They cancel.  That leaves us with (9/2 + 9/2)i,
which is 9i.  That's what we found the squareroot of.

To compute the squareroot of 5 + 3i,
we need to converting to to angular corrdinates.
r = √(5²+3²) = √25+9 = √34.
Θ = arctan(3/5) = ardctan(3/5) = 0.5404195 radians
That's the same as Θ = 30.96375653°.

To compute the squareroot, take the √√34
( that's the sqaureroot of the squareroot of 34)
as the value of r for the squareroot.
Take Θ/2 and get 15.48187827° = 0.27020975 radians.

The other way could probably be done, but it would be a whole lot more complicated.  The nice thing about polar coordinates is the angle.  If you're computing the squareroot, half the angle.  
Computing the cube root means divide the angle by 3.
Computing the 4th root means divide the angle by 4.

Now there is something else that needs to be known.

When computing the squareroot, use the number converted to polar coordinates.  Add 360°/n to the number for n-1 times.
If it's the squareroot, add 180° to the answer once.
This means that the squareroot of 36 is 6 and 6 at 180°.
Now 180° is on the negative axis, so it is also -6.
Applying this to a higher root, like the 4th root of 1,296 is 6.
In angles, though, add on 90° (4-1) times, or 3 times.
In other words, the 4th root of 1,296 is 6, 6i, -6, and -6i.
Go ahead.  Take any of those to the fourth and you get 1,296.

Now if you're looking for, say, the 5th root of 32, the real part of the number is 2 since 2*2*2*2*2 = 32.  However, there are four more numbers to consider in complex solutions.  They all have length 2, and in polar coordinates they are 2 at 0°, 72°, 144°, 216°, and 288°.
If you want to convert them to real, x = r cosΘ and y = r sinΘ.
Sometimes polar answers are the way to go.


---------- FOLLOW-UP ----------

QUESTION: -_- you keep saying hard stuff in my school when in 11th grade we were only taught to get the square root by assuming what i assumed are you sure your way is for my level?

ANSWER: Converting to polar coordinates is the easiest way to compute roots of complex numbers.  It is much easier than the method the last letter started out with.  I'm not even sure the other method would work.

Do you know cartesian coordinates (x-y)?

To convert to polar coordinates, just think of going on down the x-axis as 0°.  90° is then straight up the y axis.  180° is then off to the negatives on the x axis.  270° is then off to the negative down the y axis.  I'll sesnd you a chart to show what I mean.

Anyway, just as I have said, that's the easiest way to compute the nth root is to compute the nth root on r and divide Θ by n.

The radius r is easy to find - it is a hypoteneuse on a right triangle with base x and height y.  This means the r² = x² + y².
So r is the √(x² + y²).  The angle Θ is defined to be the angle from the x axis that this line r is at.  The first quadrant is where x and y are both positive.  Θ is between 0° and 90°.  The second quadrant is where x is negative and y is positive.  Θ is between 90° and 180°.  The third quadrant is where x is negative and y is negative.  That is where Θ is between 180° and 270°.  The fourth quadrant is x is positive and y is negative.  That is where Θ is between 270° and 360°.

This said, a 75° angle is in quadrant 1.
A 146° angle is in quadrant 2.
A 200° angle is in quadrant 3.
A 335° is in quadrant 4.

To say it a different way:
0° - 90°: quadrant 1,
90° - 180°: quadrant 2,
180° - 270°: quadrant 3, and
270° - 360°: quadrant 4.

If you would like further explanation, ask some more questions.

Do you know what the sin, cos, and tan curve
look like between 0° and 360°?

If you're expanding you're horizons by learning about how to take squareroots of complex numbers, this is where to start.
First you need to grasp what polar coordinates are.


---------- FOLLOW-UP ----------

QUESTION: in trigonometry i studied till laws of sine - cosine and the graph of function sine - cosine - tan also the sin -cosine -tan of (a+b) (a-b) (2a) (1/2a) but i am not sure about  this polar thing and i am not expanding my horizon i am studying the complex numbers but they did not teach me about this polar coordinates only the first way which i didnt completely get .

so can you tell me more about this polar coordinates thingy?

Answer
Suppose that the start is the origin.

Also suppose it's in the center of an open field.

Set the location in that field to be x = yards to the west
and y = yards to the north.

If you walked 40 yards north and 30 yards west,
what would really be said to convert it to (x, y) would be
you walked 30 yards west and 40 yards north.
This point you walked to was where x = 30 and y = 40.

Now to convert that to polar coordinates, you would have to find out how far from the starting point you were and what the angle was from straight east.

The distance is r, the hypoteneuse of a triangle.
It is known that r² = x² + y².
In this case, r² = 30² + 40² = 900 + 1600 = 2500.
Now we both know that 25 = 5² and 100 = 10², and r² = 25 * 100,
so we can say that r = 5 * 10 = 50.

The angle would be how far from west you were.
That would be Θ where tan(Θ) = 40/30 = 4/3.
That would mean Θ was 0.9273 radians,
which is around a little over 53°.

To convert from radians to degrees, not what a whole circle is.
A whole circle is 360°.  It is also 2π radians.  
That means 360° = 2π radians, or 180° = π radians.

To get r, note that r² = x² + y².
Once you have found x² + y², take the square root to find r.
To get the angle, note that tan(Θ) = y/x.  Given the tan() of the angle, though,  you need to know which side of the graph the point is on, for (-y)/(-x) = y/x.  This means that you could be 180° (or π radians) off in the other direction.