0.99999~ = 1 ?

Quite. Why do the most idiotic of discussions always seem to dominate otherwise sensible threads???

nwr writes:

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In mathematics, "equal" means "identical".

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Incorrect. Two things being identical is a stronger relationship than equality.Things that are identical are the same no matter what. Things that are equal are only so in certain circumstances. That's why the identity operators are called such: When they operate, they return the exact same object you started off with no matter what that object is. With an additive identity of 0, then x + 0 = x no matter what x is. With a multiplicative identity of 1, then x * 1 = x no matter what x is.^*a * x/x is only equivalent to a when when x <> 0. Therefore, a * x/x is not identical to a. It at best is equal to a.0.999... is not merely equal to 1, it is identical.^*As an example of why we must not confuse the symbol for what it refers to: The additive identity 0 differs depending upon the object it works on. For a number, 0 = 0. For a matrix, it is a matrix filled with 0s. But which matrix 0 is depends upon the original matrix. If you have an m x n matrix, then 0 is also m x n. But if you have an n x m matrix, then 0 is n x m. Similarly, 1 for a number is 1 but for a matrix, it is a square matrix that is as wide as the original matrix. This is if you have an m x n matrix, then 1 is an n x n matrix with 1s down the diagonal and 0s everywhere else. But if your original matrix is n x m, then 1 is an m x m matrix with 1's down the diagonal and 0s everywhere else.And yet, we use the symbols 0 and 1 to refer to all of these different things. The have equivalent concepts, but are not equivalent themselves.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon writes:

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Identical meaning does not make the figures 0.9999| and 1 identical; it does, however, make them equal.

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Incorrect. The two have the same meaning makes them not merely equal but identical. That's the point behind identity: They are the same object.Do not confuse notation with the object.

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So you are incapable of telling me whether two things are identical without knowing their symbolic meaning?

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Precisely. Notation is an artificial construct we place upon meaning. It is not the object itself. The reference to an object does not change the object in any way. Two objects are identical not because they are symbolized the same way but rather because they function the same way.That's why the sky doesn't change color just because one person calls it "blue" while another calls it "azul."It's why calculus use's Newton's methods but not his notation. The calculus did not change simply because Liebniz had a better way of writing it down.It's why "one" doesn't change just because sometimes we write it as "1" and other times we write it as "0.999...."Do not confuse the notation for the object it represents.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

The Wikipedia entry on mathematical equality agrees with me and disagrees with you.Incidentally, I do have a Ph.D. in mathematics.

Jon responds to me:

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And it is because we have the different conventions that we end up with non-identical modes of representation.

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Non sequitur. Please rephrase.Representations are, by definition, conventional. But representations of an object are not the object itself. What makes things identical are that they have the same meaning. Notation is irrelevant.It's why the sky doesn't change color just because one person calls it "blue" while another person calls it "azul."

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I have a feeling your answer is artificial, which means you are lying.

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(*chuckle*)You mean it hasn't occurred to you that there is another possibility?Hint: Consider the possibility that I am not lying.

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Or, your answer is honest, in which case you cannot definitively tell me that a big green blocky blob is not identical to a smaller more fluid white blob.

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Without knowing what they mean, no. By your logic, 1 is not identical to 1 simply because the font for the first one isn't the same as the font for the second one.You're trying to play a game of gotcha and I'm not honoring your demand. Notation is irrelevant when it comes to meaning."A nose by any other name would still smell."

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but I will finish with saying the following, a final attempt to show your understanding of identicalness to be flawed

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Oh, this should be good. Yes, please do give your lecture to the mathematician about his own subject.

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Only in mathematics can you refer to two things that are equal as being identical

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Are you implying that we aren't discussing mathematics? Silly me thought that "one" was a mathematical object. Are you implying it is something else?

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In other words, Math does not set 0.9999| and 1 as identical, but sets their meanings as such

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Incorrect. Math proves that 1 and 0.999... are identical. They are not so by fiat. We start with the property of "identity" and then see if objects meet the definition of the property. If they do, then they are identical. Not because we want them to be but because they are so all on their own.

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When referencing things outside of mathematics, your definition is non-applicable

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When did we stop talking about math?

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in which case you should have concluded the two blobs to be non-identical.

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Except I don't know what you mean by those things. Just as calling the sky "blue" is identical to calling it "azul," you might mean the exact same thing by those two so I don't know if they're identical or not.Until I know the meaning, I cannot say.

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Your application of specialized definitions to generalized reasoning is creating a miscommunication problem.

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Strange, I would say that it is your insistence on playing games of gotcha rather than paying attention to the subject at hand.Can you say, "1 is identical to 0.999..." with a straight face?

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Unless you aren't just using the term in a specialized way but actually do believe identicalness to be dependent on meaning

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Seeing as how I have repeatedly said as such, one wonders why you think there is any question.

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Are we to assume that creatures without the mental capacity for symbolic representation cannot distinguish things?

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What makes you think there is a distinction here? If you can distinguish things, then you can make symbolic representations.

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Does a cat not know a mouse from a dog?

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What makes you think a cat can't make symbolic representations? Certainly it isn't a sophisticated as the ones we make, but sophisticated symbols aren't "better" than unsophisticated ones.

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The problem with this understanding is that it rests identicalness on the existence of Symbologies.

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Incorrect. Instead, it divorces the two: Identicalness is independent of symbology. That's why two different symbols can be identical: What makes them identical is not the ability to physically overlay one symbolic notation on the other and find complete overlap.It's that they mean the same thing. The notation is irrelevant.It's why the sky doesn't change color just because one person calls it "blue" while another calls it "azul."

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Identicalness is a property dependent on Meaning.

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True.

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Meaning is a property dependent on Symbologies.

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False.The exact opposite is true. Meaning is a property completely independent of symbology. That's why the sky doesn't change color just because one person calls it "blue" and another person calls it "azul."

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Symbologies are dependent on humans.

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True.

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To maintain your logic as true, making H false (getting rid of humans) would have to render, ultimately, identicalness a non-property.

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Incorrect. To maintain my logic as true, I simply have to point out that your train is not valid. And I have.Meaning is not dependent on symbols. Meaning is completely independent of the notation.

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But this is not what we see in the Real world.

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Precisely. Since your logic has led you to a contradiction, this means that something in your premise was false.Check step two where you insist that meaning is dependent upon the notation. Consider the possibility that the two are independent. That absolutely different symbols can be identical. That the sky doesn't change color because one person calls it "blue" while another person calls it "azul."

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When two identical things are in identical environments, identical things do happen to them.

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Indeed, but that has nothing to do with the notation we use to describe that action. We didn't always use "+" to refer to addition or "=" to refer to equality. The use of "+" to mean addition came from Orseme in the 1300s. We used to write the Latin word "et" (meaning "and"). Egyptians would use a symbol of a pair of legs walking. If the legs walked in one direction, it was addition. If it was the other way, it was subtraction. Arithmetic did not change just because the symbology did.

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The Natural World (Reality) does recognize and function subject to identicalness, and would continue to do so without the grace of our bipedal selves.

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Exactly. So what does that mean for your argument? Is meaning, such as "identity," dependent upon the symbols or is it independent of the notation?

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Being identical is a property of the Real world, and is not dependent on whether or not we agree to interpret something as identical or not.

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Precisely. That's my argument. The symbols that we use to represent objects have nothing to do with the objects themselves.Hint: You assumed that in a discussion of symbols, you could introduce some new symbols and that everyone would understand that you were introducing objects. This is why I don't know if what you presented was an example of identity or not. You cannot determine if the objects referred to by various symbols are identical until you find out what they mean.

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The laws of Nature will not change if you call the blobs identicalthey will still be treated as different.

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You have that backwards. The laws of nature will not change if you call the blobs different. That's why the sky doesn't change color just because one person calls it "blue" while another person calls it "azul." The name we give to the color of the sky has no effect upon the color."A nose by any other name would still smell."

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This definition of identical as being ultimately dependent on Symbologies is just far far removed from what would be given to the word by any average speaker of the English language.

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Which is why your entire train of thought makes no sense.Meaning is independent of symbols. The sooner you realize this, the more sense you will make.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Which is why you should stop resting your entire argument on the assumption that this is the case.Look, consider the following two individuals:(a) George W. Bush(b) The 43rd President of the United StatesAre they identical? Yes or no?Now, the sequence of symbols "George W. Bush" and "The 43rd President of the United States" are not identical. But George W. Bush and the 43rd President of the United States are identical, because identity is not a matter of the symbols we use.And exactly the same is true of 1 and 0.999...The symbols are different. But this has nothing to do with the question of whether the two things are identical.

nwr responds to me:

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The Wikipedia entry on mathematical equality agrees with me and disagrees with you.

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Oy. Wikipedia?OK...did you bother to look up "identity"? Hmmm...what was it I was saying? Oh, that's right: Since you seem to put so much faith in Wikipedia, why is it my description of mathematical identity matches?Ooh! They even give examples! Hmmm...and what was the example I gave? Wow. It's like they're reading my mind. Even my examples match up.

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Incidentally, I do have a Ph.D. in mathematics.

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Which only goes to show that even doctorates make mistakes.I've got my degree in math, too. Shall we wave our sticks of chalk at each other?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

For those confused by Rrhain's diversion, here is the first line of the cited Wikipedia entry.

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Equality, or more formally the identity relation, is the binary relation on a set X defined by .

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For those who understand the notation, it is quite explicit. Roughly, it says that "x is equal to x, no matter what you take x to be in the domain of interest (numbers, for example). Moreover, that gives all possible cases of equality in the domain of interest. And there, Rrhain begins a big diversion.Sure, the word "identity" has many meanings. In this case, context requires that it be the identity relation (because it is defining the equals relation). And the identity relation is just that things are identical to themselves and to nothing else.If you don't like Wikipedia, then try a google search for equal in mathematics. Or ask your local research mathematician. And expect there to be some disagreement - you never get 100% agreement on anything.

nwr responds to me:

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For those confused by Rrhain's diversion

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(*chuckle*)In a discussion about "identity," you go off to find the definition of "equality" and you claim that I'm the one engaged in a diversion?Nice try.Of course, the definition you provide immediately jumps to identity: And indeed, things that are identical are also equal. However, things that are equal may not necessarily be identical. The arithmetic of infinity shows this easily. Have you already forgotten the example I gave how infinity - infinity is undefined? Even though the sizes of the infinities are equal, they may not be identical and thus we cannot say what the result of the equation is until we do some more investigation:Let a = x and b = x.
Solve a - b as x -> infinity.
As x -> infinity, we have infinity - infinity, but this equals 0 because a and b are identical.Let a = x and b = x².
Solve a - b as x -> infinity.
Again, as x -> infinity, we have infinity - infinity, but this does not equal 0 because a and b, though both equal to infinity, are not identical.

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OK...did you bother to look up "identity"?

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And there, Rrhain begins a big diversion.

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(*chuckle*)And here we have it again. In a discussion of "identity," you seem to find discussing the definition of "identity" to be a diversion and instead insist upon focusing on "equality."And you say I'm the one off on a distraction.Nice try.

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And the identity relation is just that things are identical to themselves and to nothing else.

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Incorrect. It was one of the first problems we had in Linear Algebra: Prove that A = A. For that, you have to use the identity matrix and use its properties to show that each element of the resulting matrix is the same as the original matrix. That is, A x I = B and then you show that B = A which allows you to say that A = A.

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If you don't like Wikipedia, then try a google search for equal in mathematics.

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Or I can just go through my old textbooks. But since you seem to be so keen, let's go through various definitions from the web: Hmm...just like I said. Alas, they don't define equality. Also like I said. 1 and 0.999... may appear to be different, but they are actually identical. Their definition of equality: Not very helpful. But they go on in the definition of "equal" to discuss identity: Which hearkens back to my point: Identity is a stronger relationship than just equality. When things are identical, you can substitute them without any change in result. I certainly hope you are so far removed that you have forgotten all the trig identities you had to memorize. Or perhaps you recall the assignments where you had to convert one Laplace transform into another. It's because the various things were identical that you could sub one for another and thus change the equation into a more calculatable form.This is precisely why the standard foundation of derivative calculus works: You're working with identical objects. To determine the derivative of a function f(x), you calculate:[f(x) - f(x+h)]/h as h -> 0Well, as h -> 0, you end up with a division by 0 which is undefined and thus, you have to figure out how to remove the h from the equation. For example, let f(x) = x².f(x) = x²
f(x+h) = (x+h)² = x² + 2xh + h²
f(x) - f(x+h) = x² - (x² + 2xh + h²) = 2xh + h² = h(2x + h)
[f(x) - f(x+h)]/h = [h(2x + h)]/hNow, here's the crucial step: We can cancel the h's. Why? Because they're identical. Thus:[f(x) - f(x+h)]/h = [h(2x + h)]/h = 2x + hAnd as h -> 0, 2x + h -> 2x.Therefore, the derivative of x² = 2x (though, to be complete, you have to do it from the other side (x-h) and get the same answer.)Again, back to the example shown above:a = x
b = xWhen x = infinity, what is a - b? Well, a = infinity and b = infinity and thus are equal to each other. But because a and b are not merely equal but identical, we can show that a - b = 0.a = x
b = x²When x = infinity, what is a - b? Well, a = infinity and b = infinity and thus are equal to each other. But because a and b are only equal to each other but not identical, we cannot show that a - b = 0.That's why we say that infinity - infinity is not defined. Things that are identical are necessarily equal. The reverse is not necessarily true.The size of the integers is equal to the size of the rationals, yes? Thus, we can put the two into a one-to-one correspondance with each other, yes?Now, suppose we have the set of rationals (Q) and we remove from it all x where x Z (the integers). What is the size of the set that remains (Q - Z)?It's infinite. We are left with the set of all rationals a/b where a <> b and without 0. And yet, Q and Z are equal in size.Indeed. But just because they're equal doesn't mean they're identical.

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Or ask your local research mathematician.

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Ahem. My research was on the logistic map.

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And expect there to be some disagreement - you never get 100% agreement on anything.

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I know. Not every mathematician is a Platonist.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Sorry, but that is gibberish. There is no number named "infinity" and there is no meaning for "infinity - infinity". I am guessing that you misremembered something, for that sure does look garbled. That's a definition of congruence. I'm not sure why you would think that says anything about "equal". It sure seems a reach.The remainder of your post does not even seem to be related to anything else in this thread. And we have drifted far from the topic of the OP. So I'll end my participation in the "equal" side issue.

nwr responds to me:

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Sorry, but that is gibberish.

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Strange, it's out of my textbook. Indeed, I have removed all of the limit bits, but that's hardly problematic.

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There is no number named "infinity" and there is no meaning for "infinity - infinity".

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Really? So why on earth did I spend all that time in Fundamental Concepts going over find the values of functions as they head off toward infinity, toward 0, and to other various numbers? Why on earth do we talk about infinite sums?1:00 always comes around.And do tell me what on earth the point of Cantor's transfinite sets are if we don't have infinity? What on earth do you think the "cardinality" of Q is?Yes, infinity is not the same type of number as, say, "5" is, but it is part and parcel of the Real number system. From Principles of Mathematical Analysis by Walter Rudin, Chapter 1, "The Real and Complex Number Systems": [continued in next post because apparently you can only have 16 items of LaTeX in a post]Edited by Rrhain, : Forgot to mention the author.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

[continued from above]Chapter 4, "Continuity": So do please tell me, why is it that my textbooks are talking about the concepts of infinity - infinity if it has "no meaning"? Please, let's not pretend that you don't understand what "not defined" means. It isn't that there is no concept as to how to interpret it. It means that there is no consistent result. It very much depends upon how you arrive at the infinities. It's why x - x = 0 as x -> infinity but x - x² = -infinity as x -> infinity, even though x = x² as x -> infinity.

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I am guessing that you misremembered something, for that sure does look garbled.

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Or, I could be looking right at my notes from the lecture. I kept everything. But then there's this from Elementary Linear Algebra by Bernard Kolman, Chapter 1, section 2, "Matrices; Matrix Operations":

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That's a definition of congruence. I'm not sure why you would think that says anything about "equal".

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Oh, I don't know...the fact that they used the word "equality" in the description: Hmm...are you about to pull a Legend and claim that because the sequence of letters is e-q-u-a-l-i-t-y rather than e-q-u-a-l, that means "equality" isn't talking about things that are "equal"?

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It sure seems a reach.

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Right. A definition that talks about "equality" is completely unrelated to a discussion about what something being "equal" to something else means. After all, it uses the word "equality," not "equal," so clearly it is of no relation at all.Oy.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.