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Radical i Squared
Name: Tina M.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
Are we allowed to have i^2 under a radical?
Replies:
Yes. You are "allowed" to put any expression under a radical. Whether it
gives a meaningful result is another question. In this case, the result IS
meaningful: the square root of i^2 is i.
If not, why? If so, then would it be simply i or the absolute
value of i?
Things get a little complicated here. When you deal with imaginary and
complex numbers, the term "absolute value" needs to be made a little bit
more general than when you're just dealing with real numbers. The reasons
for this you are beginning to hit on in the questions you ask below.
The "absolute value" for any number is its distance from 0. For real
numbers, this can be conveniently expressed as the square root of its
square. For complex and imaginary numbers, the formula is a bit more
complicated, but it reduces to the same thing when you use it on "pure" real
numbers. The absolute value of a complex number a + bi, where a and b are
pure real, is sqrt{(a + bi)(a - bi)} = sqrt(a^2 - b^2 i^2) = sqrt(a^2 +
b^2). The number a - bi is known as the "complex conjugate" of a + bi.
The expression I gave gives the distance of a complex number from 0 if you
think of complex numbers lying on a "complex plane". One axis of this
complex plane is the familiar number line containing all the real numbers.
The other axis contains all the pure imaginary numbers. The reason the
formula gives the distance is the Pythagorean theorem, the standard way to
calculate distances between points in a perpendicular coordinate system.
Multiplying a complex number by i simply rotates it 90 degrees around 0, so
1i = i lies on the imaginary axis of the complex plane, a distance of 1 from
zero. The number ii = i^2, then, is the number 1 rotated 180 degrees about
zero, which puts it at -1. The number -i = -1i = i^3 is 1 rotated 270
degrees, and i^4 = 1.
You question was if sqrt(i^2) = i or -i. It turns out that the squares of
both of these numbers are the same thing, -1. By definition, sqrt(-1) = i,
just as sqrt(1) = 1 and not -1, even though (-1)^2 = 1. So how do we
determine which of two possible complex numbers is a square root? For real
numbers, we simply pick the positive one. For complex numbers, we can do
the same thing: pick the one whose real part is positive.
And if it is allowed then is sqrt(i^4) = i^2?
Well, it is true that (i^2)^2 = i^4. However, 1^2 = i^4 as well, and 1 is
positive, while i^2 = -1 is negative. So, strictly speaking, sqrt(i^4) =
-(i^2) = 1.
Because if so, this equals 1, and you have just gotten a negative
from under a square root. If not, then why?
If you define a square root as having a positive real component, everything
stays straight.
Why are you not allowed to do that? When you do power over root
that is what you get as your answer. Or do you have to put absolute
value bars on the i2 when it comes out? Or do you have to say
i4 = 1 and the sqrt(1) = 1?
It's really not such a big deal; when you solve the quadratic equation, you
need to keep track of both roots, and it doesn't matter much which one you
call the square root and which one you call its negative.
And if so, then why do you have to simplify under the
radical first when you don't with other numbers? When I sked my
teacher she said you have to simplify first here and not with other
numbers because the imaginary numbers have a different set of rules.
I don't know what you mean about simplifying first. The rules for pure real
numbers are just a special case of the general rules for complex numbers.
The rules aren't different, you just don't have to keep track of as much
when the number os pure real.
I can see that, but then how are the Real Number rules that are
allowed chosen and the ones not allowed disregarded? Did someone
prove that you must simplify under the radical first because
otherwise your answer is wrong?
I don't know how the rules governing operations with complex numbers were
developed. I guess it had something to do with making a system that is
self-consistent and still applies to the subset of the real numbers.
And when you do replace i with any other number, then both the
"power over root" and simplifying first under the radical return equal
answers. Why doesn't this work with i?
As near as I can figure, the same set of rules works for all complex
numbers, be they pure real, pure imaginary, or mixed. Could you give me
some examples of some numbers that don't follow the rules?
Richard E. Barrans Jr., Ph.D.
Assistant Director
PG Research Foundation, Darien, Illinois
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Update: June 2012
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