So the operator has both another implied operator and an implied operand?

And what are the operands of the multiplication operator? An operator and a constant?

Sure seems a LOT cleaner to classify it as a constant and be done with it. Is there anything that is not perfectly well defined with that classification?

Yes.

Yes.

As you wish.

 

It's not a matter of wishing. It's an honest question: Is there anything that is not perfectly well defined by just using 'j' as a literal constant?

I have no problem with someone saying, "I like to think of it as an operator because it let's me visualize things in this useful way." But it's worth exploring the fine implications of doing so and if you need to extend the definitions of all kinds of other things in a kludgy way, such as redefining multiplication so that one or both operands can be an operator (but not any operator, just this one specific operator) to make it all hang together, then that's fine and that's your choice -- it's a choice I don't want to make and that's my choice. There's no absolute right or wrong approach.

But that doesn't mean that I'm not genuinely interested in knowing if there is some hole in MY preferred approach, which is to define it simply as a literal constant. Is there some unwitting kludginess that I'm implicitly defining by doing so?

 

True it IS a constant, but interestingly enough this constant does not exist and hence is called imaginary. I find calling it an operator useful when graphing terms in Cartesian coordinates with the meaning of the necessary 90 degree rotation of the "j" terms.

It may not be correct in a formal sense but it is useful to me.

 

So the operator has both another implied operator and an implied operand?

And what are the operands of the multiplication operator? An operator and a constant?

Sure seems a LOT cleaner to classify it as a constant and be done with it. Is there anything that is not perfectly well defined with that classification?

Hello there,

Usually if things are called two different names or have two different definitions there is a reason for that. We have words in language that have multiple meanings that sometimes have nothing to do with each other.

In this case we are thinking about uses for the 'j' symbol, and so what if it has two uses rather than just one.

For example, in a LITERAL expression we might have something like this:
y=(1+j)*(1-j)

and this resolves down to:
y=1*1+1*-j+j*1+j*-j=1-j+j-j^2

which resolves down to:
y=1-j^2=1-(-1)=1+1=2

and there we pretty much considered 'j' a constant.

But what about this:
y=a+b*j

There's no further resolution for that, and 'j' looks like it has an operand too which here would be 'b'. We could have written this as j(b):
y=a+j(b)

but since j(b) resolves down to j*b we can write that instead and keep it a little simpler.

If we change j to a constant we get:
y=a+b*sqrt(-1)

but that doesnt seem to help. So changing it into a constant did not seem to help in this case.

Even if we just have:
y=2+j

the implied operand is '1' and we cant really reduce that any further except again as:
y=2+sqrt(-1)

but again sqrt(-1) can not be reduced (yet) so it doesnt do us much good and we still have the implied operand of '1'. I dont think we can say that:
y=2+sqrt(-1)

is any cleaner than:
y=2+j

So it seems to have the function of an operator until such time as the entire expression can be resolved into a simple expression and then it may take on the role of a constant.

Of course we also have:
y=2+K

where K is a constant, but then that is a little different too i think, and we also have:
y=2+3*K

and K is not called an operator, but then K is more general too and not as specific as 'j'.

We also have 'j' as a rotation as in 3d space:
e^(jwt)

and j or -j dictates the direction of rotation.

The following resolution:
cos(x)+j*sin(x)

does not get better with:
cos(x)+sqrt(-1)*sin(x)

and we can see plainly that now K would be different:
cos(x)+K*sin(x)

because now everything looks real rather than part real and part imaginary. And then we can also have:
cos(x)+j*K*sin(x)

and clearly j and K have different implications than:
cos(x)+N*K*sin(x).

Also, in programming with the use of 'j' i have to allocate two storage locations for 'j' not just one like with a regular constant, where j=(0,1) and a regular constant is just K=3.
The math routines have to be different too in order to multiply and divide and add and subtract, because two numbers are always used to represent one complex number.

All this tells me that adding a 'j' to our formula takes it into a different world than when we just use a regular constant like 3, 5, 6.3, etc.

Maybe calling 'j' an operator isnt right, but calling it just a constant cant be perfectly right either i dont think, so maybe that is why it is called an operator.

Also, usually when we bulk things together we loose something in the process. Sometimes we want to do that as when we lump resistors together to form one resistor, but then we loose the value of each individual resistor so we cant go back and recalculate unless we have each value. For example, i think we can call both trees and bushes 'shrubbery' but then if i tell the landscaping guy on the phone that we have 'shrubbery' he wont know if we have trees and bushes or just bushes or just trees.

 

j is a constant.

I think of it as an operator. When I multiply by this operator it takes me into a different space. Multiplying a second time brings me back.
Always.

How about "constant operator"?

 

I still think the notion of "multiplying" by an "operator" makes very little since. If it is an operator, then it operates on an operand. If it you multiply something by it, then it is an operand.

So what different space does multiplying 2∠45° by this operator j take you into that you weren't in already?

 

I still think the notion of "multiplying" by an "operator" makes very little since. If it is an operator, then it operates on an operand. If it you multiply something by it, then it is an operand.

So what different space does multiplying 2∠45° by this operator j take you into that you weren't in already?

Hi,

You may have missed post #24 just to note.

It forces us to consider an additional dimension. The 'jw' axis for example takes us into another dimension in AC analysis for example, so we end up with the complex plane.

Even if you dont agree with everything else i think you have to agree that it is not exactly like other constants. It has more specific uses for one thing.

 

The symbol j represents a number in some contexts, and an operator in others. No view can be considered preferred; it all depends on the context.

As a linear operator on vector spaces,

\(j:\text{R}^2 \to \text{R}^2\)​

can be defined as the map that takes (x, y) to (-y, x), which corresponds to a 90° rotation. So if we're talking about ℂ = ℝ×ℝ, a 2-dimensional Euclidean vector space, where only vector addition and scalar multiplication are defined, it's perfectly natural to treat j as an operator.

However, if we're talking about ℂ as a ring (or field), allowing multiplication between elements, then ℂ ≠ ℝ×ℝ and \(\small{j = \sqrt{-1}}\) is simply a number, no more privileged or otherwise different than any other in ℂ.

In electronics, we tend to use ℂ computationally, as a number field, so the \(\small{\sqrt{-1}}\) perspective may be more natural. But personally, the visual of j as a rotation operator was tremendously enlightening as I was learning this stuff.