My question is Why √[(-a).(-b)] can't be written as √(-a).√(-b)?
Like, we can separate √(-4)(9) = √-36 = 6i , √4i.√9 =6i, but why can't we separate for two negative numbers inside the root?

Who says you can't?

There are no negative complex numbers.

https://en.wikipedia.org/wiki/Ordered_field

I think its largely convention/notation based on "principle" square roots. So the same reason -6 isn't considered the square root of 36 with standard notation.

This is right. There's a whole theory of Riemann surfaces where you discover that the square root of complex numbers actually creates two different "layers" of the complex plane. And when you see that, you discover that the "rule" that \sqrt{ab} = \sqrt{a} \sqrt{b} simply doesn't hold in that type of world.

You can also see this if you use polar representations of complex numbers (if you're familiar with that).

Thanks For guidance.

Thanks Aaron W. It would be grateful, if you can give some examples or more detail information.

How would you describe your level of math background? If I said "the complex plane" would you know what I'm talking about?

-5 is a negative complex number.

When you say x^2=36 the answer is always x=+-6

What is the problem here? You can do whatever separation you want if it doesnt violate any property of multiplication etc. You will get the product of two complex numbers (imaginary here) that in this case becomes real again and positive according to the convention on roots of positive numbers. What you typed in OP is fine with me and others.

Root(9) or √9 is a convention that it is 3 and not -3 when we all know that x^2=9 doesn't discard the negative solution. The convention is necessary in order to be able to represent things without ambiguity. In order for it to be an operator that is well defined , a function ( function cannot have more than one image per argument). Otherwise expressions with roots would be not well defined (uniquely defined). It is a matter of definition eg in reals that the square root is the positive number that when squared gives you the thing inside the square root. The square root is a function really that has non-negative range by construction (by choosing that solution to the original equation that motivated the square root operation). We want it on purpose to be the positive or zero thing that when squared gives you the argument or it wouldnt be a function.


The root (x) or √x is not the square root of x, it is the positive square root of x (same for complex numbers too)

We do not say i= +-(-1)^(1/2) although (-i)^2 is also -1 and would be the same as i^2=-1 in definition making i not well defined suddenly.

In reality of course the solution to x^2=-1 is +- i.

See also here;

https://en.wikipedia.org/wiki/Principal_value

"In complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and –2; of these the positive root, 2, is considered the principal root and is denoted as √4 ."

Cool.

https://en.wikipedia.org/wiki/Abuse_...se_of_language

Well if 5+3i means that i own 5 apples and 3 oranges, and moving either down or left into the negative axes means going from "own" to "owe", then only the first quadrant is the positive/own one and everything in the last three quadrants are negative versions of this. In the lower right quadrant you own 5 apples but owe 3 oranges.

Having a number below zero on the real number line makes it a negative and thus a change in reference, but being below zero on the real line is too small of a definition of negative. It means "a change in reference", and you can do that on the complex plane.

(note negative does not mean that something is smaller than nothing (zero), thats not possible).


imo

If your assumption P in the condition "if P then Q" is false, then it really doesn't matter whether Q is true or false.

The framework you suggest is not sufficient robust for other number fields, where you get lattices where your directions are no longer orthogonal to each other.

I'm not sure what happened to OP, so I'm just going to shoot for the "middle of the road" explanation and hope it lands.

All points on the complex plane can be expressed in the form z = re^{it} = r cos t + i * r sin t. (t is usually theta.) Because of this, the same point in the complex plane has infinitely many representations, since both sin t and cos t have period 2*pi.

However, when we raise z to the 1/2 power (take the square root of z), these equivalent representations do not all map to the same place. There are two distinct places these representations can end up. For example, consider the square roots of z = 1:

1 = e^(0*i) --> (e^(0*i))^(1/2) = e^(0*i) = 1
1 = e^(2*pi*i) --> (e^(2*pi*i))^(1/2) = e^(pi* i) = -1

So what's happening here is you're getting a bifurcation (branching) of values. The same point on the complex plane is being mapped to two different places under this mapping.

In order to keep things sensible, there is a "continuity" requirement. For the function f(z) = z^(1/2), if you move z in the complex plane a little bit, the output should also move a little bit instead of jumping around.

The formula \sqrt{ab} = \sqrt{a}*\sqrt{b} would potentially violate this if your three points (ab, a, and b) have the wrong representations relative to each other. You basically get that points could potentially be jumping around all over the place and break that continuity.

Doesn't make much sense. What is result of multiplying an apple by an orange?

Okay there seems to be a number of problems with this.

When does multiplying an apple by an orange ever make sense?

It doesn't. Only fishes and loaves.

This is similar to a freshman textbook problem that is supposed to teach what is and what isn't a tensor/vector to students seeing them for the first time. Something like {N}= 2 x^ + 3 y^ where the x^ and y^ are directional unit vectors is a vector, but something like {N} = 2 a^ + 3 o^ where the a^ and o^ are apple and orange "unit vectors" isn't. Despite looking the same, the apple orange expression isn't a vector because it doesn't respond to coordinate transformations (like rotations) in the way a vector (or rank 1 tensor) should. You seem to be doing something similar.

Well first i would say that i dont view the complex numbers as the same as a formal vector, because the "i" part is going to get you in trouble in calculations where it gets squared etc.

But if you take the set of complex numbers and e.g chose not not allow for other operators than addition, and you want to use them for e.g keeping track of how much you own and owe of apples and oranges, then i dont see any problems. I can draw the complex plane and use it for my purposes with addition alone and -5-3i would be a fine way of changing the reference to the opposite of what 5+3i means. Its not the most practical way of doing it, but it would work.

On your example with the vectors i dont get the difference between the two. A vector is defined by it having direction and magnitude. If i have a coordinate system where apples is right and orange is up, and the apple unit vector points up one notch, and the orange unit vector points right one notch, and both of these vectors are 2 dimensional with apples in their first slot and oranges in the second slot, then i dont get the difference. But i think i know what a vector is so maybe you are trying to toy around with some semantics or natational thing here, maybe the way you wrote it the apple and orange vector is lacking "direction" or something. But if im missing something fundamental i should have learned at a "freshman course" then i am guilty of improper understanding.

The issue is that you form an insufficient basis for understanding the actual ideas of complex numbers. If you reduce your framework to this, it's as if you're talking about complex numbers where the imaginary part is zero.

Technically, sure. What you're doing is a proper restriction of the larger idea. But the restriction you're looking at doesn't have a single one of the features that makes complex numbers what they are.

What you've done is reduced complex numbers to the accounting of two unrelated (orthogonal) classes of objects. It doesn't contain even the barest hint as to what complex numbers actually are.

I had a feeling that it was a bit too cheap of a way out, so im gonna agree with your points there Aaron.

Sure, as an accounting system is will "work" as long you define the allowed operations in a way that won't ever give wrong answers. But you don't need complex numbers and really aren't using them in any way other than a naming convention.

Yeah, this is the point of the exercise. Naively it does look like a vector, but fails when you give a vector a more precise definition. One of the standard definition of a vector has to do with how it behaves under coordinate transformations. The fruit "vector" doesn't transform correctly under 2D rotations, for example. With vectors you simply multiply by the rotation matrix, but that gives you wrong answers with fruit as you don't see apples being converted to oranges and vice versa like you do with real vectors.

Well, it depends on what your professor chose to emphasize. If he never mentioned tensors then its understandable if you didn't know about this. I TAed a freshman class where this was covered, but everybody in the class already had 2 years of calculus based physics from high school, so it might not be covered in most freshman classes.

"Naively it does look like a vector, but fails when you give a vector a more precise definition."


So what it means to be a vector gets really "tainted" when there is different stuff in every slot of the vector, e.g only addition may make much sense. But putting in something like e.g forces would make alot of operations interresting to look at, rotations etc, so it would release the full potential of being a true vector.


Is this what you mean? Still there needs to be a name for referring to such tainted vectors, not sure if there is another name for it.

That's more or less what I mean. There isn't really a reason to give that thing a name. You can just keep track of apples and oranges separately, vectors actually have a name because there are alot of operations you can do with them that you really can't do with a list of different fruits.