for the clever mathematician

@DeWayfarer i isn't undefined.

And zero is not "a member of the empty set".

@SomeMichGuy i is only defined in even powers. And is always equal to -1 in even powers.

I said that.

And the very definition of zero IS the empty set!

Why division by zero IS undefined! As well as why they say in calculus "as x approaches zero".

Because zero is undefined. They usually indicate this by:
x: x>0

@DeWayfarer No. On most of your points.

1) A set with 0 in it has a member, and so cannot be the empty set.

2) 0 is not defined as the empty set {}.

3) Not having a defined sign is different from being the empty set.

4) i raised to odd powers IS defined. lol

i^1 = i
i^3 = -i
etc.

This was covered in high school when I attended.

5) Division by zero is formally undefined, but, if you consider it, it isn't the 0 which really causes the problem, it's the representation of the answer.

From calculus, lim_{x ‐> 0} 1/x = (infinity) which is "Not a Number". You get the same answer if you ask yourself what division *is*--fast subtraction.

When you ask "what is x divided by y?", you are asking "how many times can I subtract y from x?".

If y = 0, then you can subtract *nothing* all day & forever from x, because you never decrease x. So you can do this an infinite "number" of times.

@SomeMichGuy zero is an empty set! Don't argue on this...

https://www.physicsforums.com/threads/zero-is-it-the-empty-set.198940/

And your argument on i is not a real number. A real number is like the number -1. A function must resolve into REAL numbers to be solved.

@DeWayfarer Zero is an integer,it us NOT the empty set.

This is the empty set: {}

It has no members.

Perhaps you are confusing "zero members" with the number 0?

The integers are the set

{ ... -3, -2, -1, 0, 1, 2, 3, ... }

zero is a member of the set of integers,jt is an element, not a non-element.

This is really basic. If you want to believe some opinion from a forum, go ahead, but you should have studied math rather than...whatever got you to this.

---

I am quite familiar with real, imaginary, and complex numbers.

Clearly, you are not.

You are embarassing yourself with nonsense like


These are both wrong.
i^4 = +1, not -1, and 4 is an even number.

You didn't have the right course in high school or don't remember it. There are n nth roots of any real number, all of the same magnitude, but distributed equally in phase angle in the complex plane. You need i to make the operation of taking a root defined for all real numbers.

@SomeMichGuy No! You are embarrassing your own self because you have yet to give any collaborating links! By collaborating links I most definitely mean math or physics links!

I have done so!

@DeWayfarer LMAO You have an opinion link. It might even be something you wrote.

If you had taken math, you'd know these things. I don't have the book I used for that in the 1970s, but you can look at any basic text about complex numbers.

If you knew how i is defined, you'd realize that it's quite simple:

i ≡ sqrt(-1), where I'm using "sqrt()" to designate the square root function.

So
i^1= i (trivially),
i^2 = -1 by design, and
i^3 = (i^2)i
= (-1)i = -i
i^4 = (i^2)(i^2)
= (-1)(-1) = +1

Which anyone who had the appropriate algebra class in high school, or calculus class or complex functions classim college knows.

As for "0 being the empty set", that's wrong from basic set theory, but you never had it, because you don't knkw the empty set {}.

Again, ANY textbook on set theory can tell you this. You might be confusing the empty set--a set which had no elements--with the number 0, which is...a number, not a set at all.

You need some serious remedial TEXTS, not nonsense spouted by similarly-untrained people om a forum, even one with "physics" in its name.

@SomeMichGuy this is unresolvable simply because you refuse to quit!

I've said this before, yet you still refuse to quit!

@DeWayfarer

It's "corroborating", not collaborating.

A link into an opinion forum is not a link to a reference.

But here, for i:

https://en.m.wikipedia.org/wiki/Imaginary_number

This shows exactly what I wrote out and have known for years, but which has been mathematically correct for much longer.

For the empty set:

https://en.m.wikipedia.org/wiki/Empty_set

...which says what I wrote and again have known personally for years, but which has been mathematically correct for far longer.

@SomeMichGuy

@DeWayfarer It's not unresolvable.

You are demonstrably wrong on many points. It's math you should have had in high school or college. It's surprising to me that you have such holes in your background, but it's fixable.

@SomeMichGuy

@DeWayfarer You need to actually check out those pages...but let's see...

OMG *YOUR* support is laughable!

You are confusing cardinality of members with the numbers themselves!

The people in that question are mostly the blind leading the blind! lol

Go to a library, used bookstore, bookstore, etc., and check out a book about advanced algebra and one on basic set theory.

The resolution is you don't realize that googling idiots won't educate you.

@SomeMichGuy

@DeWayfarer Lol You need to evaluate your sources. That question was very poorly answered and the guy whose answer you are adopting didn't cite *his* sources, and didn'r understand them.

I could give you refs from books, but it's clear that this is lost on you.

@SomeMichGuy

@DeWayfarer lol
I'm sorry you don't understand these basics. I'm sure a high school teacher could help with textbooks/resources they use, so that you won't feel bad about being told this by someone with degrees in physics & electrical engineering.

@SomeMichGuy

@DeWayfarer So you aren't brave enough to look at a few short pages? lol

I waded through several pages of your
"source".

@SomeMichGuy

@DeWayfarer No wonder you quit programming...

@SomeMichGuy

@DeWayfarer lmao I am. I already helped steer others away from your obvious errors, so that's a definite win, so thanks for that opportunity.

@SomeMichGuy