Real Fuzzy Math

Assigning a polarity to zero is an accepted practice that allows otherwise insolvable problems to become tractable.

Jenn has guided the readers of this thread in that direction and she is correct in what she has said so far.  She is also correct in that this is a concept that is much easier to handle when Multi-variable Calculus is part of your routine mathematical tool box.

“BALDERDASH!!!” you say?

First we can start with how it is represented.
http://en.wikipedia.org/wiki/−0_(number)#Representations
I think it is worth noting that the concept of negative zero is useful enough that is is defined in 1+7 bit sign-and-magnitude representation, simple binary, one’s complement, and the IEEE 754 standard.   Jenn is not alone in her interest (fascination?) with signed zero.

“I don’t care that you can write negative zero, what is it’s value?”
http://en.wikipedia.org/wiki/−0_(number)#Properties_and_handling
As the link describes, there are various means of comparing (-0), (0) and (+0) and getting a value ranking and different answers that are useful.

“All very interesting but what is it good for?”
http://en.wikipedia.org/wiki/−0_(number)#Scientific_uses
This is a fairly simple case where the idea of negative zero is used to describe days where the temperature is below 0 but not -1.Â

More complex examples are available but to be bluntly truthful about it, with out that multi-variable calculus background that I mentioned and an abiding interest in the topic, your eyes will glaze over.

“But I really want to know!!!”

Good for you.

Lets put together a very simple control system.  We have a submarine with a compass and a rudder.  We tell it to steer a course using the rudder.  If the submarine points to the left of the ordered course we have the control system provide right rudder to return it toward the ordered course.  We do the opposite if the submarine points to the right of ordered course.  This gets expressed mathematically as “Rudder Angle = (Ordered course - Actual course)” where positive values for Rudder Angle correspond to right rudder.  This works great if the ordered course is 180 and the actual course is 175.  We get 5 degrees of right rudder.  As long as the control loop is faster than the response time of the vehicle this will bring the vehicle back to the ordered course.

Now lets put some stress on our control system.  Ordered course is 358.  Actual course is 002.  our rudder angle is now = +356 but this value takes us the long way around to the desired course instead of a negative (left) rudder taking us back to the ordered course the short way.  The end result is your control system occasionally has the vehicle do right hand cirlces if it gets knocked off course too far.  In addition, the controller attempted to drive the rudder to an angle that it could not reach.  This is not desirable.

In order to avoid these kinds of issues we start to do things like take derivates and integrands of actual and ordered courses.  From this we develop orders for the rudder.

http://en.wikipedia.org/wiki/Derivative
http://en.wikipedia.org/wiki/Integral

Sadly, as nice as these tools are for smoothing the wild swings of the rudder, we know have to deal with mathematical functions that will give us zeros at awkward times.  Even more confusing is we also have to take into account the performance of the vehicle at different speeds, environments and configurations.

Remember the idea that the control loop needs to be faster than the vehicle response?  There is a lot more mathematics behind that statement than I care to type.  The end result is asymptotic behaviours.  This means we need to know what the “sign” is of zero as we approach that asymptotic point in the control system.

Confused yet?

If not, consider a career in engineering.  If you are confused, don’t feel bad. I probably totally goofed on the explanation somewhere.

Jen- are you taking notes? Thats what everyone was looking for from you when they asked you to explain it. Not “you need more higher level math” or citations to some print essay that would explain nothing to the average reader anyway- just an earnest attempt to explain the concept. Even if people come away now still unsure, they know that Baghdaddy tried to explain it, and posted links to web resources for further info.

Again- no one was saying you were wrong and that they knew more then you, they were saying that your claims contradicted what knowledge of the subject they had, and wanted an explanation of why/how you were right. You don’t have to teach them entire fields of mathematics, just make some attempt to give them a basic understanding of the concept of negative and positive zero since everything they knew till that point said zero is not negative.

But Janus!!!

It is more fun to poke everyone in the eye and tell them they need to take more math classes so they too can feel superior in their intellect.

Personally, I was just tempted to post a link to something like this as proof and walk away.

http://www.ingentaconnect.com/content/els/09218890/2003/00000043/00000001/art00361

@rjclayton:

Oh and Baghdaddy - one day you are going to have to give me a lesson on how to search wikipedia.  I searched high and low for an article like that…

I cheat.  :-D

I read.

A lot.

I remember almost all that I read.

This means I remember bits and pieces of things and given a decent search engine, I can put a few words in and get the articles, web pages, or references.

I actually found that wiki article by using Google and the search term “negative zero”.  The wiki article was first on the list and after review seemed like a good starting point.

Actually, I just started the Tired of Games thread to explain why I’m not bothering posting sources.  Those in opposition will only try to say your sources are wrong, or find some obscure footnote somewhere on a completely different page in a valient and desperate attempt to discredit the source.  So there’s no point.  I list sources when they are easily grabbed from the net (as in I just have to click on the address bar, copy paste.)

As for waxing on in more detail, I already explained that I don’t have the functionality to use the proper mathematical symbols here, let alone the time to get into a thesis level discussion on the polarity of zero.

So I’ll let you keep on believing the world is flat while people like Galileo argue the opposite.  No skin off my nose.  You are not my students, you are not upper level mathematicians, and your personal opinion of me has very little, if any, bearing on my paycheck.

I just have to keep repeating in my head, 8 more years until tenure!

I’m still unconvinced.  Thanks for the details BD - you went a lot further than I did.  What it appears to me is that a -0 is necessary in a closed environment to avoid compromising the system, or to make work more efficiently.  In other applications, it may be a matter of convenience (such as stating the temperature).

Jenn claimed that zero has polarity.  I don’t think she can prove that.  To say that it may be necessary or easier to consider negative zero in some situations, I would entirely agree.  But that is not what she said.  Read the first post.

@Jermofoot:

I’m still unconvinced.  Thanks for the details BD - you went a lot further than I did.  What it appears to me is that a -0 is necessary in a closed environment to avoid compromising the system, or to make work more efficiently.  In other applications, it may be a matter of convenience (such as stating the temperature).

Jenn claimed that zero has polarity.  I don’t think she can prove that.  To say that it may be necessary or easier to consider negative zero in some situations, I would entirely agree.  But that is not what she said.  Read the first post.

“Polarity” in this sense refers to the poles in a graph that has asympotic behavior.  The example of an asymptotic equation, y = 3/(2-x) which has a division by zero at x = 2, gives us a “pole” in the graph at x = 2.  Since division by zero is what has caused that pole and the sign of zero defines which side of the pole you are on, the “polarity of zero” actually has meaning both in the esoteric world of mathematicians and in the nuts and bolts world of us poor engineers who are trying to make things work.

Polarity, when I use it, means positive or negative.  +1 has positive polarity. -1 has negative polarity.

@Baghdaddy:

“Polarity” in this sense refers to the poles in a graph that has asympotic behavior.  The example of an asymptotic equation, y = 3/(2-x) which has a division by zero at x = 2, gives us a “pole” in the graph at x = 2.  Since division by zero is what has caused that pole and the sign of zero defines which side of the pole you are on, the “polarity of zero” actually has meaning both in the esoteric world of mathematicians and in the nuts and bolts world of us poor engineers who are trying to make things work.

But what you are indicating is that this is different than the polarity associated with nonzero numbers.  Do I understand correctly?

@Jennifer:

Polarity, when I use it, means positive or negative.  +1 has positive polarity. -1 has negative polarity.

And if -0 = +0 = 0, then zero has no polarity.  That’s what I’m saying.  Considering zero as negative or positive for a particular application doesn’t make it polar.

Polarity depends on value, and zero is the absence thereof.

@Jennifer:

Actually, I just started the Tired of Games thread to explain why I’m not bothering posting sources.

And um… that did not work out quite like you intended did it Jen?
http://www.axisandallies.org/forums/index.php?topic=9379.0

Considering the results of SUD’s post (confirmed and backed up with links), I think we now know the reason why you failled to do what Baghdaddy did in posting an alternative discussion/explanaiton beyond the original essay…  because you HAD no other knowledge/understanding to convey besides the cut-and paste article that you originally posted from J J O’Connor and E F Robertson.

@Jermofoot:

Jenn claimed that zero has polarity.  I don’t think she can prove that.  To say that it may be necessary or easier to consider negative zero in some situations, I would entirely agree.  But that is not what she said.  Read the first post.

now now - be nice.
She CAN prove that - she just doesn’t want to.  For you do not have the higher learning that would put you in the same realm to be able to be worthy of the proof.

So BD - zero is assigned a “pole” for convenience then?  A group of people decided that in order to grasp an un-graspable “number” decided to grant zero polarity when it was thought useful? 
i.e.

• how much money do you have in your wallet Tim?
• zero
• well . . . that’s better than NO money . . . at least if you’re an engineer . . . :D

@ncscswitch:

@Jennifer:

Actually, I just started the Tired of Games thread to explain why I’m not bothering posting sources.

And um… that did not work out quite like you intended did it Jen?

Whatever.  I don’t need to stoop to your level and give you a free education.  I have more important things to do with my time, like earn my paychecks.

I find it humorous that the person who has now twice been nailed for copyright infringement claims they have to “stoop” to my level.

You have thrown out an argument, one apparently that at least 2 serious researchers in Mathematics feel rather strongly about.  You apparently feel strongly enough in support htat you were willing to claim their work as your own.  Based on that level of support for the theory, I would imagine you would be chomping at the bit to advocate the theory.

You threw down the gauntlet.
You B-slapped everyone on this board with your claims of superiority on this subject.
And then you got caught in an act of plagierism.
Now, you can attempt to prove that you were just being lazy by plagerizing (instead of being deceitful) by demonstrating your PERSONAL knowledge of this theory.

We are all ears to see you expound further on this facinating subject of extreme mathematically theory…

Again, I don’t need to stoop to your level, Switch.  You can think of how superior you are or not superior you are all on your own without my putting you in your place.  I just don’t have the time to deal with closed minded individuals who I am not getting paid to deal with or who are not paying me to deal with them.

@Jennifer:

Again, I don’t need to stoop to your level, Switch.  You can think of how superior you are or not superior you are all on your own without my putting you in your place.  I just don’t have the time to deal with closed minded individuals who I am not getting paid to deal with or who are not paying me to deal with them.

So you had the free time before you ripped off other people’s work, but don’t now.  OK.

Baghdaddy, if YOU have anything to add to your discussions on tihs topic, there are some interested folks here (since you seem to have at least a passing udnerstanding of hte concepts being discussed in that sut-and-pasted essay).  We are just looking for something other than some third party’s essay on it, and for someone to be able to answer questions that are asked on the subject instead of saying we are too ignorant to understand the discussion…

Go ahead with your character assassinations, Switch.  I really don’t care.  I have my opinion on you and what you post too, I’m just too mature to post it anymore.

Jennifer is wrong. 0 does not have a sign.

http://en.wikipedia.org/wiki/Sign_function
http://mathworld.wolfram.com/Sign.html

0 is neither positive nor negative.

x -> -0 is nothing more but a bad notation.

@Jennifer:

I’m not going to get into a debate about the function of zero with a man who’s highest level of math was maybe Calculus II.  Sorry.  But you just don’t have the ground work to have the debate.  Come back after you take Multilinear Equations and Logic I and II.

So how about getting into a debate with someone who has taught Multilinear Equations. May I assume you know about the following mathematical concepts:

• Some basic group theory (uniqueness of the neutral element).
• Polynomial rings and quotients over some ideal, especially R[T]/(T^2), the space of dual numbers.
• Complex analysis including the complex plane and its one point compactification, the Riemann sphere.