What are some applications of the concept of complex infinity?

[shamrock5585]

You mean when you said "F = ma"? So would

$$C^{mn}_{pq}=g^{ma}\epsilon_{abpq}\sqrt{-g}$$

(related to Hodge duality of Maxwell's equations, apparently) convince you that complex numbers exist? If not, why? This is no less physical than "F = ma".

yes you can define an imaginary number for practical purposes so that when you are finished deriving you end up with real numbers... if you multiply i * i you get -1.

 

[CRGreathouse]

Even in modeling real life things. In fact, even the extended complex plane (that's right, where 1/0 = infinity) is used to model things in physics. The catch is that is not used to "measure" anything; it is instead used to model certain properties of a system.

Slight aside: that's the one-point compactification of $\mathbb{C}.$ The real numbers admit (at least) a one-point (projective) and a two-point (extended); complex numbers also admit one with infinite points at infinity (S-C). Is there a good classification of the (useful? interesting?) compactifications (up to isomorphism)? Or is that just silly to ask because there are so many?

 

[CRGreathouse]

yes you can define an imaginary number for practical purposes so that when you are finished deriving you end up with real numbers... if you multiply i * i you get -1.

A person could say the same about real numbers: "You can calculate in the reals, but in the end you'll end up with a rational number. Real numbers are just intermediate steps."

Kronecker seemed to think that only integers were real, and thus even rational numbers would be (for him) only intermediate calculation steps. There's surely sympathy for that viewpoint today in discrete mathematics. "All is number", said the Pythagoreans; "all is bit" say computer scientists and reductionist quantum physicists (e.g.).

 

[matt grime]

Shamrock, you're posting more and more things that aren't saying anything useful. Please, we all know the definition of a field, we all know everything in that wiki article. Nothing there will contradict anything we've said. No one is attempting to define 'division by zero as the inverse of multiplication by zero', no one is claiming that the Riemann sphere is field. What on Earth is your point?

Your phrase 'when you divide by zero you get undefined' is semantically incorrect. Division by zero in the real numbers is undefined. Saying that 'you get undefined' implies that division by zero is defined in the real numbers, and that the answer is the symbol 'undefined', which is a very strange real number indeed.

Can we move on from this high school level discussion? What next infinity plus one is bigger than infinity?

 

[CRGreathouse]

Your phrase 'when you divide by zero you get undefined' is semantically incorrect. Division by zero in the real numbers is undefined. Saying that 'you get undefined' implies that division by zero is defined in the real numbers, and that the answer is the symbol 'undefined', which is a very strange real number indeed.

It sounds like shamrock is thinking of a system like IEEE math, where NaN is an element.

 

[matt grime]

ok well i just wanted to say that 1/0 is not infinity... it is undefined... as you pointed out that infinity times 0 does not give you back 1... it technically would give you zero because if you multiply anything by zero it gives you zero... so that would make the equation wrong in the first place so in a math sense 1/0 has no value it is undefined.

Here's your original post, with emphasis added. Where do you mention the real numbers? In a thread about the extended complex plane, what did you expect us to think you mean?

"In a maths sense" 1/0 has a very god definition, in the extended complex plane, which is, sorry, was, the entire point of the thread.

 

[matt grime]

ok dude what you just said had nothing to do with what i was talking about... you don't need to explain to me how to use infinity... i understand its application... when applied to 1/0 it seems that this would equal infinity but you would be incorrect in saying this because 1/0 can never be equal to anything when analyzed algebraically... it is undefined

Here's your second (I think) post, which again contains the fallacious assertion that 1/0 has no meaning ever, when "analysed algebraically". That is to my mind false. The group of Mobius transformations would appear to be a very useful and well defined algebraic gadget, and that requires 'division by 0'.

 

[matt grime]

The OP asks about 1/0, and your second post asserts that

1/0 can never be equal to anything when analyzed algebraically

which is incorrect; no one has said that the symbol 1/0 has any meaning in the study of R. The 'wording' is always important in mathematics, otherwise you risk making false statements, as you did in at least your first two replies.

You have repeatedly written off topic and at odds with the central point of this thread, and been insultingly abusive, and now homophobic as well.

 

[Hurkyl]

I'm closing this thread, since it has devolved into a demonstration of shamrock5585's stubbornness.


Beer w/Straw: I'm assuming that your question has been answered to your satisfaction. If that is not the case then feel free to tell me via private message, and I will transfer the tangential discussion with shamrock5585 into a separate thread, and I will reopen your thread.