Thermionic Conductivity Advantages ??? Little Help, Please ...

[Ratchit]

Cowboybob,

""i", or actually "j", as it's used in electrical engineering, is most assuredly considered a imaginary number. Note:

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

And, as a second source, Diefenderfer's "Principles of Eledtronic Instrumentation" (1979), in the appendix, page 438.

Both emphasize that j is, indeed, imaginary in that no such number exists."

I would fault the Wiki article because not once does it mention that i or j is a mathematical operator. I cannot evaluate the book because I don't possess it.

This topic has had the hell beat out of it at the link below. I invite you to read it and get back to me with good reasons why the sqrt(-1) is not a valid number. Please don't just parrot what others who should know better say about this. I am looking for good reasons why "i" does not exist, or why I am wrong in averring that it does.

Ratch

https://www.electro-tech-online.com/threads/why-dont-imaginary-numbers-make-so-much-sense.121331/

 

[cowboybob]

From the Wikipedia site: "The term "".

For instance, where you to be tasked with solving for the real number X in the formula: X = , you could not, other than stating that = , which would be, because of the demands of the question, incorrect.

Or, put another way, there is no real number that is equal to . Thus it is an imaginary number by definition.

Now, admittedly j,used in the context for which is was created and further manipulated (multiplied, divided, raised, etc..), is a "real" number, in that its use leads to real solutions.

But, in and of itself, strictly speaking and due to the fact that it has no real solution, it is imaginary.

Sematics, Ratch. Always a slippery slope...

We are, of course, majorly off-topic now. So I'll leave this as is.

 

[JimB]

We are, of course, majorly off-topic now. So I'll leave this as is.

Possibly a wise decision.

This thread started with a post which did not really make any sense.
Drifted through the minutiae of electrons, currents and charge transfer.
And is now debating the meaning of i or j.

If the OP would like to comment in which direction he would like the thread to proceed, that would be useful.

JimB

 

[Ratchit]

Cowboybob,

It is specious to say that numbers which cannot be represented in one-dimensional space are "imaginary". Especially when they have the same relation to physical phenomena as one-dimensional numbers do. Calling a multidimensional number "imaginary" is not semantics, it is a misnomer.

Ratch

 

[steveB]

The combination of imaginary and real numbers provides a 2 dimensional representation called complex numbers that provide better unity within our mathematical formalisms. When you accept complex numbers as valid numbers, mathematics gets much easier.

In the end, all numbers are abstractions. They don't exist in reality, but only in our minds. As mental concepts they exist as ideas. In that sense the number i (or j) is a number with existence as an idea and is just as valid as 2, -1, pi, 1/2 or zero.

Positive numbers are tied to a one to one mapping of number of objects to a symbol. That is an abstraction that is so simple that it makes the numbers seem more "real" than other numbers. Positive integers are abstractions only, but we make them so useful and we get so used to them that effectively they are real. A positive integer is a real concept with clear meaning once we define what it means.

The number zero was suspect at one point. Our ancestors debated the reality of this number. It is a number that represents nothing, and how can "nothing" be "real", or how can it "exist". But, the concept of nothing is real in our minds. When you lack food, you know it and having zero amount of food is something real to a human or even an animal.

Then the negative numbers were viewed as somehow "not real" or as having no existence, but they represent something real such as a lack of a quantity, or something owed. If you can conceive of a number of things, then the thought of those things not existing has meaning too.

Rational numbers were created as a ratio of these integers and allowed the concept of partial amounts of something to be represented. Half and apple is just as real as 1 whole apple, so the number has meaning and a reality as an abstract concept. But really, when you have half an apple, you have 1 "half apple" not 1/2 apples. But, since you can't really know that it is exactly one half, you really have 1 partial-apple.

The irrational numbers filled the gaps we found with rational numbers, and allowed the continuum we call Real numbers (note the capital R now). The term "Real number" is a misnomer and creates confusion. The term "Imaginary number" is also a misnomer and many people have noted that it is a poor name because it misleads people. Words can mislead people. The term "Irrational number" could lead people to think the numbers are somehow not rational meaning they are not based on reason or logic. But, the word simply means that the are not ratios and can not be represented as a ratio of two integers. Still people accept them as having "reality" or as "existing" even though our ancestors debated about the reality, existence and meaning of irrational numbers.

Now, Imaginary numbers somehow seem to be numbers without existence to some. All of these numbers are abstractions. They are as substantial or insubstantial as we conceive them to be and exist as ideas if we define existence to include ideas. To me, they have meaning in my mind making them concepts or ideas, and hence are all valid numbers with substance, existence and reality. It has been said, "I think, therefore I am". So, let's say, "I think about a number, therefore that number exists."

The operator concept is a nice idea to make the number easier to accept as something "not imaginary". However, to me it is unnecessary to even invoke the operator concept to argue the reality or existence of imaginary numbers. The ability to say "you have 2 apples" is not what make the 2 a number something with a real existence because if you believe that then the numbers 0, -1, pi, 1/2 and sqrt(-1) do not exist either. Why? Because ...

1. You can't have 0 apples (How can you "have" 0 apples when you don't have anything? If you have nothing, then you have 0 apples, 0 oranges, 0 pears and 0 of everything else in the universe. What are you really trying to say?)

2. You can't have -1 apples (How can you "have" a lack of something? If you have no apples then saying you have -1 apples is no different than saying you have -2 apples, or -3 apples ... and so on. ")

3. You can't have pi apples (but you can have apple pie).

4. You can't have 1/2 apples (but you can have 1 partial-apple)

5. You can't have sqrt(-1) apples (what can you even say about this one? Note that if you rotate the apple by 90 degrees, it still makes no sense. )

However, do I understand why people say i (or j) is not a number that exists? Of course. It is a mind bending concept and I don't feel the I have the right to say they are wrong if they feel that way. I only know that after almost 40 years of using imaginary and complex numbers, they are just as valid to me and exist in my mind as concepts along with 2, 0, -1, 1/2 and pi.

As a child my body went spinning around a merry-go-round in a playground, but as an adult my mind has been spinning around the unit circle in the complex plane, at least 10,000 times more.