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## Main Question or Discussion Point

Actually, can you take a root of

__any__negative number?- Thread starter Holocene
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Actually, can you take a root of __any__ negative number?

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symbolipoint

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Yes. You simply must permit the use of complex numbers as solutions.Actually, can you take a root ofanynegative number?

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Ben Niehoff

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

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

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

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

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

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DaveC426913

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CRGreathouse

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How do you know you exist? Why do you think real numbers exist?how do you know that these complex number exists :P?

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All complex numbers can be written on the form [tex]a + ib[/tex] where [tex]a[/tex] and [tex]b[/tex] are real numbers, and [tex]i[/tex] is the imaginary unit, DEFINED as [tex]i=\sqrt{-1}[/tex]. 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 [tex] \sqrt[n]{-x} [/tex] within the complex numbers, and also that there is no solution within the real numbers.

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HallsofIvy

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There are technical difficulties with that. Since complex numbers, just like real numbers, haveAll complex numbers can be written on the form [tex]a + ib[/tex] where [tex]a[/tex] and [tex]b[/tex] are real numbers, and [tex]i[/tex] is the imaginary unit, DEFINED as [tex]i=\sqrt{-1}[/tex]. This is just definitions.

What we can do is define the complex numbers as

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 i

Are you asserting that real number "exist in a physical sense"? Can you draw a line of length [itex]\pi[/itex]? For that matter, can you draw a line of length 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.

I'm not complaining (too much). I agree with most of what you say.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 [tex] \sqrt[n]{-x} [/tex] within the complex numbers, and also that there is no solution within the real numbers.

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Actually, there is a completely unambiguous way of defining them as such:There are technical difficulties with that. Since complex numbers, just like real numbers, havetwosquare roots, defining i as [itex]\sqrt{-1}[/itex], or as "the number such that i^{2}= -1", is ambiguous.

What we can do is define the complex numbers aspairsof 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).

You can define [tex]\mathbb{C}[/tex] as being the elements of [tex]\mathbb{R}

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

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No, I'm not.Are you asserting that real number "exist in a physical sense"? Can you draw a line of length [itex]\pi[/itex]? For that matter, can you draw a line of length 1?

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 [tex]\sqrt{-5}[/tex]cm.

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HallsofIvy

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

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Ben Niehoff

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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 [itex]e^{i\theta}[/itex]), 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 [itex]i[/itex]. To draw a line of length [itex]3+4i[/itex] to the north, we multiply:

[tex]i \cdot (3 + 4i) = -4 + 3i[/tex]

and draw a line to this point. As you can see, a "line of length [itex]3+4i[/itex] 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

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