Suffice it to say that vertical lines actually sometimes are referred to as having slope of unsinged infinty. As someone once told me, wouldn’t it be useful if all lines had slope? Definitions are often extended, when the needs arise.
All one really needs to know, in order to prevent some sort of contradiction with the way, say a 9th grader is taught about slope, is that the slope of a vertical line in the plane cannot correspond to any real number. All numbers are not real, ya know :). My point is simply to acknowledge that a concept such as “infinite slope” can be quite appropriate, in some contexts. Heck, I’ve even seen it used in contexts of 9th grade math, as an informal explanation. That doesn’t mean, though, it can’t be formalized.
BTW, have you found the fallacy in your string of equations yet? OK I won’t hold you in suspense. Do you know that a/b=c implies and is implied by, b*c=a, when b=0 and c=infinity? No… That’s an implication concerning REAL numbers a,b,c with b nonzero. Arithmetic gets a little strange when infinity is involved.
Amen. The various threads engendered by Cecil’s column amply demonstrate that.
One question. Does “your string of equations” (last paragraph) refer to my earlier exercise or to ginkging’s rebuttal? If to me, no complaints. If to ginkging, I don’t quite follow you; and since he seems to have a good objection, I’d like to understand your point. Looks like you’re saying that one just can’t perform commutation on an equation of this sort. (I’ve attempted a couple of alternative examples, but find I have to assume what I’m trying to prove in order to make any headway.)
I thought it quite clear, but apparently it wasn’t clear enough, that I was addressing the remarks that I had quoted of ginkging. I also thought I made it quite clear, but apparently not clear enough, that (at least) one flaw exists in that line of reasoning. In real numbers, we have:
a/b = c ==> bc=a, b<>0
This means we can say things like 12/4=3 therefore 4*3=12. Unless I am terribly mistaken (and I really don’t think I am) what we CANNOT say, even in contexts such as extended reals where division by “infty” is defined to be 0, is:
1/infty = 0 ==> infty*0 = 1
Or, that:
12/0 = infty ===> 0*infty = 12
These simply don’t hold the way they do in the real number system (with nonzero denominators) and you should already have a good feel for why. Look back at the reference to Apostle where division by +infty and -infty are defined in the extended real number system. The numerator can be any real.
Infty0 is an indeterminate form. Not only does 1/infty=0, but so does 2/infty=0, 3/infty=0, etc, and 12/infty = 0 also. However, this does not mean that infty0 is 12.
Now on to 12/0 = infty ==> 0infty=12 as was claimed. Now, to muddy the waters a little: There are contexts such that 12/0 is unsigned infty, eg in such a context as is being debated here, by saying the slope of a vertical line is unsigned infinity. However, even in this context, unless I am terribly mistaken I really don’t believe we can say 0infty=12 ust because we may say 12/0=infty.
Actually, until ginkgeng brought it up, I don’t believe that I or anyone else ever spoke of any division by 0. Just because we may speak of infinite slope of vertical lines, does not necessarily mean we are speaking of division by 0. Remember, the usual slope definition (as with any standard division) prohibits division by 0, clearly. All that means, is the standard slope definition is simply not the definition in use (at least it’s not the ENTIRE definition in use) when speaking of infinite slope as I am.
It’s nothing to lose any sleep over, really. If it bothers one to speak of a vertical line having slope of unsigned infinity, all you have to do to put this in terms that may be more pallateable is to realize that I am speaking of a vertical line, ie no defined slope according to standard slope definition.
Ginkging tries (in good faith, I’m sure) to present an argument that 0<>12 therefore I shouldn’t speak of infinite slope. That argument is doomed since it is assumes properties that simply do not hold. Not to mention the assumption that 0infty = 0000, etc. etc. etc. off to infty, which allegedly is 0, therefore 12<>0 therefore I shouldn’t speak of infinite slope. 0*infty is indeterminate, actually, not 0.
Sorry, I forgot to mention that this particular reference to Apostle was made in the other thread 1 = .999… by Jabba which I’ll repeat for your convenience. In extended reals we have, by definition
It occurred to me this morning that Ginkging had simply botched the operation.
12/0=inf.
multiply both sides by zero and we have
0(12/0)=0(inf.)
since multiplication by zero always produces a product of zero, this quickly reduces to
0=0.
Apparently Ginkging added the intermediate steps
(0/1)(12/0)=0(inf.)
(0/0)(12/1)=0(inf.)
and assumed that the expression 0/0 equals one.
Now, that can be argued. Any number divided by itself yields one. But then, any fraction with a numerator of zero equals zero. But then, if “Bullwinkle’s Axiom” (x/o=inf.) is true, then 0/0=inf. (if not, then division by zero is simply impossible, especially after the Patriot Act).
If you think I’m going to get into a discussion of the true meaning of 0/0, you’re nuts.
Stop. First of all, when I said there are contexts where “12/0” is unsigned infinity, I did not necessarily mean so in the arithmetic sense that 12 divided by 0 is infinity. Actually, IIRC, contexts do exist where this (or something close to this) can be said, but I certainly don’t want to get into that.
Secondly, even if we were to define the arithmetic expression 12/0, we still have the multiplcative property of equality, which says that if both sides of an equation are ultiplied by some NONZERO number, the resulting equationwill be equivelent.
You really don’t have to go any further. You can stop right here, but I’ll go ahead and address a couple of your other concerns.
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0(12/0)=0(inf.)
since multiplication by zero always produces a product of zero, this quickly reduces to
0=0.
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We apparently know nothing about what “intermediate” steps he may have had in mind. Either way you look at it, it’s incorrect.
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No. Any NONZERO number divided by itself is 1. Big difference.
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No. Any fraction with a numerator of 0 and denominator NONZERO equals zero. 0/0 is another one of those pesky indeterminate forms.
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0/0 is an indeterminate limiting form. I know of no context where the expression 0/0 is well defined. You brought it up, not me. EVEN IN CONTEXTS where something like 12/0 may be defined (and I’m not really sure such contexts exist) but if they do exist (I have overheard others saying, yeah, such contexts exist) then what they are talking about by the expression 12/0 is not the same arithmetic operations you would normaly infer from standard arithmetic. Even if it WAS, this does not mean you can multiply each side of an equation by 0 and necessarily expect any sense from the result (much less any ogical implications from the result.) Just because (0)(x)=(0)(y) ==> 0=0, does not mean 0=0 ==> x=y. In real numbers, multiplying both sides of any valid equation by 0 gives the meaningless equation 0=0 (meaningless in the sense that it says nothing about the solution to the original equation.) This is the very reason why you don’t ultiply both sides of an equation by zero. Doing so results in an equation true for all x (no x’s in 0=0, so it’s true regardless of x) while the original equation may very well be true for only certain values of x.
Stop. First of all, when I said there are contexts where “12/0” is unsigned infinity, I did not necessarily mean so in the standard arithmetic sense that 12 divided by 0 is infinity. Division by 0 is not well defined.
Secondly, even if we were to define the arithmetic expression 12/0, we still have the multiplicative property of equality, which says that if both sides of an equation are multiplied by some NONZERO number, the resulting equation will be equivelent.
You really don’t have to go any further. You can stop right here, but I’ll go ahead and address a couple of your other concerns.
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We apparently know nothing about what specific “intermediate” steps, if any, he may have had in mind but either way you look at it, it’s incorrect.
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No. Any NONZERO number divided by itself is 1. Big difference.
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No. Any fraction with a numerator of 0 and denominator NONZERO equals zero. 0/0 is another one of those pesky indeterminate forms (wrt a limit). The standalone expession “0/0” AFAIK has no well defined meaning in any mathematical context I am aware of.
I’m puzzled here. If this guy is so smart, why is he spending so much effort solving puzzles that were already solved much more cleanly and easily by Newton? Next week, why doesn’t he figure out a solution to the puzzle of retrgrade motion of the planets, based on a complicated set of crystalline spheres?
I have a mathematical mystery relevant to this discussion. How come the number of posts are not increasing? I thought a graph of the number of posts had to be a positive definite number.
Darrell, you and Chronos are going to butt heads for an indefinite length of time. Dividing by zero is a hazard of physics. Take orbital mechanics. The eccentricity (e) of the orbit is a measure of the shape of the orbit. Circular orbits have an e = 0, ellipses have 0< e < 1. A particular orbital parameter related to the inverse of the eccentricity of an elliptical orbit might be undefined for a circlular orbit (the direction of the semi-major axis), or defined (position along the orbit as measured from the semi-major axis becomes position along the orbit from an arbitrary point that was the direction of the semi-major axis for a family of ellipses). Physicists don’t really care because the physics of the situation determines the interpretation of the mathematics. The orbiting body after all, knows where to go.
I am unware of any “head butting” between Chronos and I. On the contrary, our positions seem to support one another. Or at least, we don’t contradict one another.
Although I have extended the context to that of one beyond just real numbers (actually it was BJMoose who did that, I just provided a considerable amount of followup), I see nothing from Chronos that contradicts me, nor anything from me that contradicts Chronos. In real numbers, neither x/0, nor x/infity exist, and I believe I made that clear myself event. I also vaguely remember some discussion of :
I would further state we have never, nor ever will, butt heads over the statement that there is no real a,b,c with b=0 such that that a/b=c.
That’s just simply not necessarily the case (x/0 being undefined when x is real) for certain other number systems, though. Or that x/infty=0 can’t happen (since that’s also true in certain number systems.) Chronos has not presented anything contradicting this, AFAIK. All I recall him saying is something to the effect of 1/x does not exist when x=0, and there is certainly no argument from me, because I know he was talking about REAL numbers only, IOW I understood the context in which he made that statement.
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Why do you think I’m talking about physics? I understand that the subject line (Zeno) refers to a physical situation, but the discussion has been far removed from that in many cases.
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If you insist, but this is completely irrelevent to anything I was discussing before.
So? By the same token, mathematicians don’t necessarily care about physics. (BTW, eccentricity is well defined even for a purely mathematical object like an ellipse, and a circle is often considered a special case of an ellipse, the limiting case as e–>0.)
We’re talking apples and oranges here. You’re talking physics (planets, their orbits, and what not.) I’m talking (was talking) about a mathematical concept of “slope” pertaining to a mathematical object called a “line” and in certain circles, no pun intended, it is quite appropriate to refer to a vertical line has having infinite slope. It’s really that simple. How orbits of planets and what not came into the discussion, I dunno…
It is quite appropriate, in certain contexts, to say things like x/infty = 0 and yes, EVEN x/0=infty, in mathematics. Take a look at affinely extended real numbers and projectively extended real numbers, which are each summarized at http://wolfram.mathworld.com. The complete URL’s were so long that this message board mucked them up, so just look up “affinely extended real numbers” and “projectively extended real numbers” once you get there.
Don’t freak out when you see see division by zero defined, or division by infinity defined. We didn’t freak out when we learned that x^2+1=0 has solutions, so we need not freak out when we define division by 0 (or infinity) either, no more than we freaked out when we realized that no real thing resembling a triangle in shape, can have a “negative” side length.
These are perfectly valid mathematical systems. What any of this has to do (or doesn’t have to do) with any particular physical application, is not my problem but that of the physicist. no physics, whatsoever, are implied from a purely mathematical statement like x/0=infty. You said it yourself: “Physicists don’t really care because the physics of the situation determines the interpretation of the mathematics.” so in short, interpret x/0 in way you like, that is useful to a physicist for example. Or if that can’t be done, then realize that has nothing to do with the mathematical use of such a statement.