Can you take an even root of a negative number?

[Holocene]

Main Question or Discussion Point

Actually, can you take a root of any negative number?

 

[symbolipoint]

Actually, can you take a root of any negative number?

Yes. You simply must permit the use of complex numbers as solutions.

 

[Ben Niehoff]

$$\sqrt[n]{-x} = \sqrt[n]{x} \cos \frac{k\pi}{n} + i \sqrt[n]{x} \sin \frac{k\pi}{n}$$

for all x > 0 and all natural numbers n, where k is any odd integer.

 

[ehj]

how do you know that these complex number exists :P?

 

[BryanP]

how do you know that these complex number exists :P?

Numbers don't "exist" physically. They were created for handling the problems we solve today. Complex numbers are really no different. There had to be a way to handle negative numbers in a way that made sense. So it's essentially no different from the "real number" system that we use.

 

[DaveC426913]

And they solve real world problems, for example in electronics and optics, so it's not like they're just mathematicians' playthings.

 

[CRGreathouse]

how do you know that these complex number exists :P?

How do you know you exist? Why do you think real numbers exist?

 

[ehj]

I just thought there was a "proof" of its existence? Like nearly everything else in mathematics, shouldn't there be an argument for it? Or maybe that's quite advanced?

 

[Tricore]

I just thought there was a "proof" of its existence? Like nearly everything else in mathematics, shouldn't there be an argument for it? Or maybe that's quite advanced?

All complex numbers can be written on the form $$a + ib$$ where $$a$$ and $$b$$ are real numbers, and $$i$$ is the imaginary unit, DEFINED as $$i=\sqrt{-1}$$. This is just definitions.

So complex numbers exist because we created them. They don't exist in a physical sense - you can't draw a line on paper with a complex/imaginary length, but perfect circles don't exist in a physical sense either.

Becuase of that, it doesn't make sense to ask for a proof that they exist. What we CAN prove however, is that there exists a solution to $$\sqrt[n]{-x}$$ within the complex numbers, and also that there is no solution within the real numbers.

 

[HallsofIvy]

All complex numbers can be written on the form $$a + ib$$ where $$a$$ and $$b$$ are real numbers, and $$i$$ is the imaginary unit, DEFINED as $$i=\sqrt{-1}$$. This is just definitions.

There are technical difficulties with that. Since complex numbers, just like real numbers, have two square roots, defining i as $\sqrt{-1}$, or as "the number such that i²= -1", is ambiguous.

What we can do is define the complex numbers as pairs of real number, (a, b), with addtion and multiplication defined by (a, b)+ (c, d)= (a+ c, b+ d), (a, b)*(c, d)= (ac- bd, ad+ bc). Of course one must then prove that this is "well defined" and forms a field (but it does NOT form an "ordered" field).

We can then write (a, b)= a(1, 0)+ b(0,1) where a and b are real numbers and identify 1 with (1, 0) and i with (0, 1). We can then show that i²= (0, 1)*(0, 1)= (0*0- 1*1, 0*1+ 1*0)= (-1, 0)= -1.

So complex numbers exist because we created them. They don't exist in a physical sense - you can't draw a line on paper with a complex/imaginary length, but perfect circles don't exist in a physical sense either.

Are you asserting that real number "exist in a physical sense"? Can you draw a line of length $\pi$? For that matter, can you draw a line of length 1?

Becuase of that, it doesn't make sense to ask for a proof that they exist. What we CAN prove however, is that there exists a solution to $$\sqrt[n]{-x}$$ within the complex numbers, and also that there is no solution within the real numbers.

I'm not complaining (too much). I agree with most of what you say.

 

[LukeD]

There are technical difficulties with that. Since complex numbers, just like real numbers, have two square roots, defining i as $\sqrt{-1}$, or as "the number such that i²= -1", is ambiguous.

What we can do is define the complex numbers as pairs of real number, (a, b), with addtion and multiplication defined by (a, b)+ (c, d)= (a+ c, b+ d), (a, b)*(c, d)= (ac- bd, ad+ bc). Of course one must then prove that this is "well defined" and forms a field (but it does NOT form an "ordered" field).

Actually, there is a completely unambiguous way of defining them as such:

You can define $$\mathbb{C}$$ as being the elements of [tex]\mathbb{R}/(i^2+1)[/tex]. It uses a little bit of more advanced machinery, but it shows that defining them by just saying $$i^2=-1$$ is completely consistent (and that there are in fact no technical difficulties)

Also, it's not ambiguous to say that $$i=\sqrt{1}$$ because there "are 2 different square roots of -1" because any models of the complex numbers are isomorphic

 

[Tricore]

Thanks for correcting me

Are you asserting that real number "exist in a physical sense"? Can you draw a line of length $\pi$? For that matter, can you draw a line of length 1?

No, I'm not.

What I was trying to say is that you can draw a line that's (almost) say 5cm, but it doesn't make sense to draw a line that's almost $$\sqrt{-5}$$cm.

 

[Vid]

It doesn't make sense to draw a line of length -5cm either, but we still use negative numbers. Different numbers are used to model different physical situations.

 

[HallsofIvy]

Thanks for correcting me



No, I'm not.

What I was trying to say is that you can draw a line that's (almost) say 5cm, but it doesn't make sense to draw a line that's almost $$\sqrt{-5}$$cm.

Can't I say that a line, of length $\sqrt{5}$, parallel to the imaginary (i.e. y-axis in an xy-coordinate system) is such a thing?

 

[Ben Niehoff]

Interestingly enough, we can actually draw a line with a length of $3+4i$ in a particular direction. Remember that the way in which to interpret different kinds of numbers is something we can define arbitrarily. For example, when counting apples, we don't necessarily have to assign the number 1 to the situation of having a single apple. We can just as easily assign the number 1³/₄, and so long as we do everything else proportionally, the mathematics will still work out under this interpretation.

When talking of drawing lines in the plane, what we normally do is consider the plane to have a 2-dimensional Cartesian coordinate system. A "direction" in the plane is a unit vector, and a "length" is a (real) scalar which is multiplied by the unit vector to give a line of a particular length.

However, due to some nice properties of complex multiplication, we can also define another convention in 2 dimensions. In this case, interpret the paper as a complex plane, with a real axis and an imaginary axis. Now, a "direction" is any complex number with magnitude 1 (of the form $e^{i\theta}$), and a "length" can be any complex number. To make a "line" in a particular direction, again you follow the same process: you multiply the "length" by the "direction", and draw a line to that point in the complex plane.

For example, the direction "north" is represented by $i$. To draw a line of length $3+4i$ to the north, we multiply:

$$i \cdot (3 + 4i) = -4 + 3i$$

and draw a line to this point. As you can see, a "line of length $3+4i$ to the north" will actually point to the northwest on our sheet of paper. (Try Googling "Argand diagram" for a nice visualization of this type of thing).

Isn't this more complicated than just representing directions and lengths the normal way? Maybe, but it depends on the application. Using complex numbers to represent points in a plane allow you to do some nifty geometrical transformations using simple arithmetic. But you wouldn't want to use this sort of representation if you were, say, giving someone directions to the store.

The point is that the meaning of numbers is separate from their pure mathematical properties. Mathematics is strictly the study of the properties of numbers; how they behave under various operations. We are free to interpret these numbers in any way we like, so long as it is consistent with those mathematical properties. One might even say that the objective of physics is to find ways to interpret mathematical systems such that the observed behavior of reality is consistent with the mathematical properties of those systems.

Sometimes multiple systems can be used in different ways to describe the same observed phenomena. For example, we can count apples one at a time, and under this system, they obey the basic laws of arithmetic. But we can also measure apples by weight; in this case, not all apples are equal, but this interpretation is still consistent with the laws of arithmetic.