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Name: Tina M.
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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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