Some remarks on complex numbers

[ClamShell]

No "fishing" allowed

And might not the removal of factoring from the picture be
similar to the "Pythagorean Dream" of:

"NO IRRATIONAL NUMBERS"

After all, cannot every number with a finite number of digits
be represented as a ratio of integers? I'm thinking that the
removal of irrationals via the removal of factoring might
yield a peculiar integer mechanics of its own; just sayin'.

 

[Integral]

This is getting a bit tiresome. Have you learned anything in any of the posts of this thread? Are you even interested in learning? These forums are for learning, if you are not here to learn then this thread is pointless.

 

[homeomorphic]

If you don't allow factoring of some polynomials, you can't let i in because i is going to factor everything. But you have to start somewhere.

If you allow factoring of a certain polynomial, you get what's called its splitting field, which is everything you need to factor that polynomial, but no more. You have to start with something, though, so it's not just the splitting field, it's a splitting field over some base field like the real numbers. The splitting field of x^2 + 1 over the real numbers is the complex numbers. Splitting fields are nothing special, though. You can always get them by throwing in enough stuff, rather than requiring a polynomial to factor.

So, you can start with the rational numbers and throw in square roots of 2 (and all resulting multiples, etc.) or you can require that x^2-2 factors. Either way, you get the same result. So, no, there's nothing particularly special about allowing or not allowing things to factor. It's the same as throwing stuff in or kicking it out.

 

[ClamShell]

This is getting a bit tiresome. Have you learned anything in any of the posts of this thread? Are you even interested in learning? These forums are for learning, if you are not here to learn then this thread is pointless.

Yes, what I've learned so far is that substituting a two-two matrix for "I"
is a bit meaningless, so another direction is called for.

"Just because you don't know the answer, you don't have to get mad", said the
lion to the elephant. Please don't throw me into the Mediterranean, like Hippasus.
I'm not a magazine salesman, nor do I have some personal theory. I'm just trying
to figure out ways to avoid "I", like the New Scientist article wants too. I think
not factoring it out in the first place is a fertile not futile endeavor.

 

[ClamShell]

If you don't allow factoring of some polynomials, you can't let i in because i is going to factor everything. But you have to start somewhere.

If you allow factoring of a certain polynomial, you get what's called its splitting field, which is everything you need to factor that polynomial, but no more. You have to start with something, though, so it's not just the splitting field, it's a splitting field over some base field like the real numbers. The splitting field of x^2 + 1 over the real numbers is the complex numbers. Splitting fields are nothing special, though. You can always get them by throwing in enough stuff, rather than requiring a polynomial to factor.

So, you can start with the rational numbers and throw in square roots of 2 (and all resulting multiples, etc.) or you can require that x^2-2 factors. Either way, you get the same result. So, no, there's nothing particularly special about allowing or not allowing things to factor. It's the same as throwing stuff in or kicking it out.

Thank you Homeomorphic, answers are always good, even the ones with bad news.

 

[AlephZero]

I'm just trying to figure out ways to avoid "I".

Whatever floats your boat, but that seems a rather pointless endeavour. If you accept that the integers obey Peano's axioms (or start from ZFC if you prefer!), the concept of "i" exists as a consequence of that assumption (and so does the concept of an irrational number), even if you personally refuse to give it a name and/or talk about it.

 

[Integral]

Yes, what I've learned so far is that substituting a two-two matrix for "I"
is a bit meaningless, so another direction is called for.

"Just because you don't know the answer, you don't have to get mad", said the
lion to the elephant. Please don't throw me into the Mediterranean, like Hippasus.
I'm not a magazine salesman, nor do I have some personal theory. I'm just trying
to figure out ways to avoid "I", like the New Scientist article wants too. I think
not factoring it out in the first place is a fertile not futile endeavor.

You will be way better off to learn to use and appreciate complex numbers.

Thread closed.