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Square Roots Of Complex Numbers

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Date: 02/22/99 at 00:47:05
From: Max Newman
Subject: Square Roots Of Complex Numbers

Find the square roots of 5-12i.
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Date: 02/22/99 at 05:02:43
From: Doctor Mitteldorf
Subject: Re: Square Roots Of Complex Numbers

tedious way is this: suppose some number (a+bi) satisfies the condition
that multiplied by itself it gives 5-12i. Well, you can actually
multiply out a+bi times itself, and get a^2-b^2 for the real part and
2ab for the imaginary part. Therefore, a^2-b^2 = 5 and 2ab = -12. If you
solve two equations with two unknowns, you get answers for a and b.
Note that you are working with a quadratic equation, so you will have
two solutions. There is always a positive and a negative square root of
every number, even a complex one.

Here is a clever way to think about the question that may be a bit over
your head for now. A complex number x+iy can be represented as a point
in the x-y plane. However, it is often useful to represent that same
point with polar coordinates r and t. "r" is the distance from the
origin (sqrt(x^2+y^2)), which is the modulus of the complex number.
"t" is the angle from the x-axis, satisfying tan(t)=y/x. When you take
the square root of the number, you are taking the square root of the
modulus, so r -> sqrt(r). Interestingly (I will not prove this today)
the angle t is cut in half: t -> t/2.

Another way to take the square root of a complex number is to use the
tangent half-angle formula, or actually find the angle, halve it, and
take the tangent. Of course, you will probably want to express the
answer as real + imaginary parts when you are done, so you will reverse
the process. In your example, it works like this:

r = sqrt(5^2+(-12)^2) = 13

so the modulus of the square root is sqrt(13). tan(t) = -12/5, so
t = -1.176 radians. The t for the square root is -0.588 radians. This

sqrt(13)*(cos(-.588) + i*sin(-.588))
= 3 - 2i

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Exponents
High School Imaginary/Complex Numbers

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