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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

1. Consider 0 as the complex number 0+0i. Then 0 is **both real and imaginary**!

2. Let *X* be a topological space. Then the empty set and *X* are sets that are **both open and closed**!

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**Ricky****Moderator**- Registered: 2005-12-04
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I don't quite see how 0 is imaginary.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

It is the only member of the intersection of the set of all purely real complex numbers and the set of all purely imaginary complex numbers.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Zero again ... I swear it is more trouble than infinity!

But yes, is the (0,0) point on the complex plane real, imaginary, both or neither? ... for that matter, on the simple number line is zero positive or negative, or both or neither (I go for neither)?

It may be better to define zero as NOT having a certain property, just so we can define sets of numbers more easily, or else the set of real and imaginary numbers intersect.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**JaneFairfax****Member**- Registered: 2007-02-23
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To me, I take the definition of a pure imaginary complex number as any complex number *z* for which Re(*z*) = 0. In other woeds, its the set

Clearly 0 belongs to this set and its the only purely real number that belongs to this set. This would have to be the case unless you have another defintion of a purely imaginary number but I dont see anything wrong with this definition I am using.

*Last edited by JaneFairfax (2007-03-30 10:23:17)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

The second part is more interesting. A set that is closed is the complement of an open set. A set that is both open and closed may be described as clopen.

Given a topological space *X*, the empty set and *X* itself are always clopen sets. If *X* has no other clopen sets, it is said to be connected.

On the other hand, a topological space is said to be disconnected iff it is the disjoint union of two nonempty open sets.

If a topological space *Y* is disconnected, then *Y* is the disjoint union of nonempty open sets *S* and *T*. Then *S* and *T* are closed (since they are each others complement), and they are also proper subsets of *Y* (since neither of them is empty). Hence *Y*, having clopen subsets other than the empty set and itself, is not connected.

Conversely, if *Y* is not connected, then there exists a clopen set *S* which is neither the empty set nor *Y*. Since *S* is closed, its complement in *Y* is open, and since *S* is not the whole of *Y*, its complement in *Y* is not empty. Hence *Y*, being the disjoint union of *S* and its complement in *Y*, both of which are open and non-empty, is disconnected.

And, whadayah know? This **proves** that a topological space is disconnected if and only if it is not connected.

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

What is your point, Jane? Where are the terminological contradictions? I see none here.

Take a look at http://www.mathsisfun.com/forum/viewtopic.php?id=4134

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**Ricky****Moderator**- Registered: 2005-12-04
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I don't believe that Jane is talking about true contradictions. Rather metaphorical ones. We name mathematical things with metaphors. For example, we say something commutes because we can move it around in an equation.

And we use the same thing for open and closed sets, and real and imaginary numbers. But there is a place where the metaphor breaks down. In real life, something can't be both open an closed. Something can't be real and imaginary.

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**George,Y****Member**- Registered: 2006-03-12
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0 means vanity, nothing, according to the Ancient Indians.

**X'(y-Xβ)=0**

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**George,Y****Member**- Registered: 2006-03-12
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So 0 means super vanity, regardless of real or complex, one dimentional or some dimentional.

Always nothing, though different meaning in different contexts.

**X'(y-Xβ)=0**

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

Ricky wrote:

I don't believe that Jane is talking about true contradictions. Rather metaphorical ones. We name mathematical things with metaphors.

As we do with any language, English, German or mathematics. We can get philosophical if you want - in fact there is a whole area of mathematics called category theory which straddles math and philosophy.

Let me just say this, though. It is my belief that mathematics defines terms with rather more precision that everyday language does: what is the connection between an open door, an open mind, an open judicial case etc.? It's hard to pin down without using a further metaphor. But in topology, the terms open and closed are very precisely defined, and they don't really conflict with ordinary usage.

I note that Jane uses the horrid word "clopen" to mean both open and closed. This has as its only virtue the fact that it doesn't collide with the common usage of open and closed.

But regarding zero, there is a problem. First there seems to be a confusion here between imaginary and complex numbers. An imaginary number is of the form bi, where i is defined as i² = -1. A complex number is defined as a + bi, for all real a and b. A real number is defined to be a subset of the complex numbers when b = 0, i.e. a = a + 0i.

Now it is in the nature of any set with identity (monoid, group, ring,.....) that it must share the identity with its subset. So that fact that R and C share the additive identity, 0, comes as no surprise, and is certainly not a contradiction metaphorical or otherwise. This would be like thinking of 3 as both real and prime as a contradiction!

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**JaneFairfax****Member**- Registered: 2007-02-23
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I should point out at this stage that whole point of this thread is to take a light-hearted look at the way mathematical language is used in particular, at how certain mathematical terms differ in meaning from the way those terms are used in ordinary language. This thread is **not** meant to be a serious critique of mathetical language itself it is only a light-hearted comparison of mathematical and non-mathematical language.

Ben, what you say is entirely correct. I agree with you. What I mean by contradictory is contradictory (or apparently so) only in the everyday-language sense, not in the mathematical sense. My point is that mathematics sometimes defines different concepts using terms which may seem contradictory in terms of everyday language. Thus, closed and open sets are defined in mathematics in not the same way as how one would ordinarily think of, say, a box which is why a set can be both open and closed in not the same way as a box can be. There is nothing wrong with the mathematical definitions themselves, and I am not saying that there is.

Lets take a hypothetical example. Suppose I am developing a brand-new branch of mathematics to study some newly discovered phenomena in the universe. I find that some phenemena satisfy some set of conditions, say *A*, and some phenomena satisfy some other set of conditions, say *B*. Now I wish to make some definitions. If a phenomenon satisfies conditions *A*, I define it as alive. If it satisfies conditions *B*, I call it dead. Now what if a phenomenon satisfies both sets of conditions *A* and *B*? Then I am perfectly entitled to call it both alive and dead!

Then I could add to the list in my first post:

3. Let *x* be a phenomenon (e.g. Schrödingers cat ) satisfying both conditions *A* and conditions *B*. Then *x* is **both alive and dead**!

This is just a hypothetical example. But as long as the terms alive/dead are interpreted in the hypothetical technical context, not as in their ordinary-language sense, there is no problem with the underlying semantics at all.

Contradictory is perhaps not a good word to use in the title of this thread. Oxymoronic might be better, I believe. It is not contradiction that I wish to explore, but rather oxymoron: http://z8.invisionfree.com/DYK/index.php?showtopic=69

*Last edited by JaneFairfax (2007-04-02 04:45:29)*

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

JaneFairfax wrote:

I should point out at this stage that whole point of this thread is to take a light-hearted look at the way mathematical language is used in particular, at how certain mathematical terms differ in meaning from the way those terms are used in ordinary language. This thread is

notmeant to be a serious critique of mathetical language itself it is only a light-hearted comparison of mathematical and non-mathematical language.

In that case I apologize (I'm always the last to get a joke in a crowd).

You are right. In fact, when I first started in science, it really pissed me off at what I thought was the hijacking or "ordinary" words for technical purposes. I now see it as inevitable - as I said, ordinary usage is so imprecise as to be virtually useless.

Good, we agree.

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**JaneFairfax****Member**- Registered: 2007-02-23
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It is great that we both agree.

And there is nothing to apologize. I did not make it clear in the first place that it was not meant to be taken too seriously, so you were entitled to have your own views about it.

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**Ricky****Moderator**- Registered: 2005-12-04
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So we all agree then. English is horrible, math rocks.

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

JaneFairfax wrote:

Then I could add to the list in my first post:

3. Letxbe a phenomenon (e.g. Schrödingers cat ) satisfying both conditionsAand conditionsB. Thenxisboth alive and dead!

OK Jane, nice post. Just for fun, note this. I am not a physicist, but was recently told that Schroedinger devised the "cat-in-a-box" thought experiment to illustrate what he regarded as the illogicality of quantum mechanics (or do I mean the Copenhagen interpretation? Not sure)

And now the cat in its box is in all the texts as a *perfect* illustration of how QM works. Poor old Schroedinger!

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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I found his (translated?) quote "One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, ..."

Poor guy indeed, now Wikipedia uses it as the theme picture, see http://en.wikipedia.org/wiki/Quantum_mechanics

I think the point is that the "cat in a box" is NOT reasonable, because something the scale of a cat is not probabilistic.

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**Ricky****Moderator**- Registered: 2005-12-04
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I think the point is that the "cat in a box" is NOT reasonable, because something the scale of a cat is not probabilistic.

The cat isn't what the probabilistic parts of quantum mechanics act on. Rather, we use a probabilistic thing such as a wave function to set off a trigger. When this trigger goes off, it does something (releases gas I believe is the most common) to kill the cat.

The trigger is small enough. However, it directly effects the macro world.

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

Good point Ricky, I hadn't thought of that (but why would I?)

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

Ricky wrote:

Rather, we use a probabilistic thing such as a wave function to set off a trigger.

Well actually, it's the absolute square of the wave function psi, which is itself an eigenfunction. For, wave functions have, by definition, codomain [-1,1], which makes no sense probabilistically.

Ha! I am tempted to start a thread on linear vector spaces, where all this might be explained. But, judging from the responses to my previous "lectures" I am not confident it would be well received.

Anybody have any thoughts (it is a cool subject, and only involves simple arithmetic)?

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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We could craft a main website page on it.

Ricky has a few pages on this site (example: Introduction to Sets) and Ganesh has made a few pages, too.

It does take a bit of effort, though, with drafts, redrafts, graphics, etc. But worthwhile if it opens up the subject to people.

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

I think the biggest problem with getting good responses to your "lectures" is simply the audience here. There aren't so many people at the undergraduate level or above here, or they aren't interested in mathematics at that point. I for one enjoy your threads, even if I don't comment on them. It was sad to see your thread on Lie Groups end before you had even given a definition, as they are truly an interesting topic. Just know that even if I don't say anything I'm watching, I read everything posted in this section.

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**ben****Member**- Registered: 2006-07-12
- Posts: 106

First, Zhylliolom: yes maybe I was being a bit of a prima donna, but it's so hard to know without some come-back. Maybe one of these days I'll resurrect the Lie Group thread. We'll see, but I take your point about your members here; there is also the worry that, if people log onto a site that declares math to be fun, and are faced with a barrage of mystifying stuff, they might conclude that math is no fun at all!

MathIsFun: Gimme a day or two to get a Vector Space thread together. I'm not sure I'm the right person to make a website page on it, but if I do start a thread you will all be free to make use of it for that purpose.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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OK. look forward to it.

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JaneFairfax wrote:

Contradictory is perhaps not a good word to use in the title of this thread. Oxymoronic might be better, I believe. It is not contradiction that I wish to explore, but rather oxymoron: http://z8.invisionfree.com/DYK/index.php?showtopic=69

If I said that someone was an **unpleasant kind** of person, I would be uttering an oxymoron.

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