What are some applications of the concept of complex infinity?

[shamrock5585]

hey man i made a very basic observation that i thought all math nerds knew and you guys picked it apart and told me i was an idiot... just wanted to put a small note that in real numbers you divide by zero you get undefined not infinity and here we are debating it still...

 

[matt grime]

Since the reals do not contain infinity, you cannot divide by zero in them. The purpose of this forum is to enlighten, surely, not to make incorrect and unqualified statements as if they were fact. Mathematicians are pedants, aren't they?

 

[LukeD]

hey man i made a very basic observation that i thought all math nerds knew and you guys picked it apart and told me i was an idiot... just wanted to put a small note that in real numbers you divide by zero you get undefined not infinity and here we are debating it still...

Because you've been arguing about it even though people have told you repeatedly that the topic at hand has nothing at all to do with that discussion.

By the way, when you divide by zero in the real numbers, you don't get "undefined". When you divide by zero, you've written nothing but nonsense; you've written something that's not necessarily even a number. If you're manipulating an equation, you've written something that does not follow from the axioms of the real numbers and therefore, something that you do not know to be true.

You also said that complex numbers aren't "real" because nothing "measurable" can be assigned a complex number. Well that's true, nothing that can be said to be "large" or "small" could possibly be assigned a complex number because complex numbers cannot be ordered (i.e., there is no consistent way to say that for any complex x and y that you can have x < y, x = y, or x > y. This is very easy to prove.), so you should not expect to be able to do such a thing. The complex numbers instead have other uses.

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.

 

[CRGreathouse]

I imagine these same posters wonder about your claim that real numbers exist: you haven't given any evidence yourself. Integers existing is one thing, but real numbers? An uncountable infinity of them? Almost all of which can never be constructed?

maybe you can refer to the theorem i posted above and then retract your post

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".

 

[shamrock5585]

By the way, when you divide by zero in the real numbers, you don't get "undefined". When you divide by zero, you've written nothing but nonsense; you've written something that's not necessarily even a number.

hence the reason we call it undefined haha

ok I am going to put a link here and then ill quote it...

http://en.wikipedia.org/wiki/Division_by_zero

"It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, division by zero is UNDEFINED. Division by zero must be left UNDEFINED in any mathematical system that obeys the axioms of a field. The reason is that division is defined to be the INVERSE OPERATION OF MULTIPLICATION. This means that the value of is the solution x of the equation bx = a whenever such a value exists and is unique. Otherwise the value is left UNDEFINED."

 

[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.