Comments by "L.W. Paradis" (@l.w.paradis2108) on "9.999... really is equal to 10" video.

  1. OMG, heard about this from a more recent video. Had a super obscure prof for differential equations, was verging on getting a C in it, and infinite series solutions saved me. I was the best in the class for those. To me, this is the easy part. Never got a limit problem wrong, ever. (But I remember the one I almost got wrong, fixed it at the last minute.) This is actually because I'm not gifted in math at all. I have super high discursive intelligence, and poor spatial intuition to the point where my brain isn't quite normal. I still can't talk about it in real life, really painful. Kept me from becoming a chemist. ;(
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  2.  @graemehook3667  You can't put in a digit "after" Infinitely many digits. That fails to specify its place in the string.
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  3.  @jeepien  Sorry! You SO beat me to it! LOL
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  4.  @moondust2365  Look at it this way, then: 1/11 = 0.090909 . . . 10/11 = 0.909090 . . . 1/11 + 10/11 = 0.9999 . . . = ((1 + 10)/11) = 11/11 = 1.
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  5.  @charlesriley2717  Zeno's paradoxes are actually highly sophisticated, and were only solved in the 20th century. The brief "resolutions," such as the observation that certain infinite series have finite sums, that you commonly see was not lost on the Greeks. The deeper solution has to do with measure theory.
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  6.  @danjbundrick  No. Do not listen to Numberphile. Really. Do not.
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  7. @Hassan Akhtar Suppose each pause is one-half the length of the previous pause?
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  8.  @GodzillaGoesGaga  Modern calculus does not use infinitesimals. Mathematicians saw they were contradictory in standard analysis.
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  9. Only on the Internet could people argue about mathematics.
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  10.  @stanbondarev9256  Numberphile! Never, never, never, never, NEVER listen to those jerks. Never. Really. Never.
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  11.  @ejrupp9555  Yeesh. Believe it or not, Aristotle got tangled up in this. The set of natural numbers, which are in one-to-one correspondence with the digits of the infinite expansion 9.999 . . . , is not being generated somewhere. It is already complete. To say it "goes on forever" is a metaphor. It isn't "going" anywhere. And it is not temporal in that sense, either, as might be thought of as implicit in the metaphor. There is no unfolding through eternity. It simply IS.
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  12.  @ejrupp9555  You are confusing notation with what the notation refers to. 9.9999 . . . is precisely equal to 10, just as 0.3333 . . . is precisely 1/3. 9.9999 . . . and 10 are equal, as are 0.3333 . . . and 1/3.
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  13.  @ejrupp9555  I already showed you how you're wrong. You're assuming 9.999 . . . "can never reach 10." It IS 10. All the digits are THERE. They are not waiting for you (or anyone) to pluck them from some netherworld and to write them down or conceive them. They are all there, just like the set of all natural numbers already IS. This is your error. I can't force you to see it.
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  14.  @ejrupp9555  Read Shadman Shahriar's explanation of why REPEATING DECIMALS are rational numbers. He explains it perfectly.
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  15.  @ejrupp9555  Infinity is neither rational nor irrational. It is not a number. It is, first of all, a property of certain sets.
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  16.  @ejrupp9555  There is no such real number. Between any two distinct real numbers there are infinitely many real numbers. Read the first chapter of any decent topology book, or the first few chapters of a real analysis book. You don't need abstract algebra or calculus to understand the initial description of the real number line. Your desire to speculate about it is enough. IOW, you are wrong in interesting ways.
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  17.  @ejrupp9555  What's the smallest positive real number? (Trick question)
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  18. I guess you'd have to subtract "the smallest positive real number" from 10 to get . . . Oh wait.
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  19.  @Chris-5318  This is why most teachers quit within five years.
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  20.  @ejrupp9555  There isn't a largest real number between any two real numbers. Nor a smallest. If a, b are any two real numbers, a < b, there are infinitely many real numbers r such that a < r < b. It's kind of like a one-dimensional fractal, the very same structure repeats on every interval.
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  21.  @ejrupp9555  I think you need to read some math books. The way you formulate these questions is naive. I don't know what you mean by making up a number or getting to a number. Do you think the natural numbers stop somewhere? If it would take too long to write a number (the earth would end long before anyone or any group could write or recite all the zeros in a googleplex), does that number somehow not exist the way 28 exists? If I can write it 10^(10^100) have I just "gotten to it?"
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  22.  @ejrupp9555  I just did, by indicating the sum of an infinite series. Sorry if you don't like them. I'll never understand. No different from using N as a symbol for the natural numbers. That is infinite in precisely the SAME way, perfect one-to-one correspondence. In fact, the elements of N are the exponents of the 1/10. I think you're just goofing around. Divide successive integers by 9 and see whacha get.
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  23.  @ejrupp9555  You know, you're not benign. I'm perfectly serious when I say this is why teachers quit. Anyone who can do this math isn't going to be teaching your kids. OTOH, teaching is a respected profession in many, many places. Basically all, except, well . . . N. B. The dots indicate ellipsis, not an infinite sequence. I know you think this is funny, but I assure you it's not. Not anymore.
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  24.  @ejrupp9555  Say. I have an idea. Why don't you produce an/the "infinitesimal" between 9.99999 . . . and 10.
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  25.  @ejrupp9555  Sorry, this is not coherent. But it does point up the problem with the whole notion of "infinitesimals" -- namely, they blink in and out of existence. Your description of an asymptote is like the difference between open and closed intervals on the real number line. Both [a, b] and [a, b) have the same supremum. The first interval contains its supremum, the second does not. The existence of the second interval does not have any bearing on the possibility of the first. TAKE A TOPOLOGY CLASS.
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  26.  @ejrupp9555  Read a topology book. Infinitesimals do not play any role in standard analysis, none. Absolutely none. Not even a wee bit. And the analogy between asymptotes and open intervals on the real number line was lost on you. The sum of the infinite series 9/10 + 9/100 + ... + 9/10^n + ... really does exist and really is 1. You are thinking of the sequence of partial sums, every one of which is less than one (but not "infinitesimally" less; on the contrary, less to a precise number we can calculate), and you are pretending that that is all that can exist. I'm sorry, but that is wrong. It's like saying open intervals "prove" closed intervals cannot exist. But suit yourself. No one is paying me to post. I'm done now.
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  27.  @ejrupp9555  In fact, I'm giving you too much credit. Most of what you said is not just wrong but muddled. For example, your first statement after the question mark is not coherent.
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  28.  @ejrupp9555  "If 10 were an asymptote" -- except that it isn't. Like, whatever. There are uncountably infinitely many real numbers between 10 and (10 minus epsilon), for any epsilon.
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  29.  @ejrupp9555  Actually, I did a thesis on this for a master's in Ancient Philosophy of Mathematics. In Paris.
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  30.  @ejrupp9555  Don't believe me, LOL!
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  31.  @ejrupp9555  The least upper bound of y is 10. No number less than 10 is in the set of upper bounds of y.
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  32.  @ejrupp9555  Why don't you actually want to know things? The real math is more fun than your ersatz version.
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  33.  @ejrupp9555  I gave you that answer a long time ago. There is no such number. Just like there is no set of all sets. Of course it is simple English. If it were not well defined, we could not say, as we can, definitively, that it does not exist. Look, I understand. YOU can't get to these numbers. I can. I thought mathematics consisted of universal truths. Turns out it's just another part of lived experience. Hope you'll have more luck in the future.
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  34.  @ejrupp9555  Oh, BTW. Blocked and muted.
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  35.  @ejrupp9555  Sure I can. 1/11 = 0.090909 . . . 10/11 = 0.909090 . . . Rational numbers are closed with respect to addition. Hence 9.99999 . . . = (9 + 1/11 + 10/11) = 10.
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  36.  @stanbondarev9256  Then you don't know what transcendental numbers are. Pick up Ivan Niven's book, Numbers: Rational and Irrational (Random House, 1961). Not exactly new stuff.
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  37. @Vorraboms They keep confusing the series with the sequence of its partial sums. The partial sums tend toward the limit. The entire sequence IS equal to its limit. The entire series does not have to be "generated;" it is already there. If it were not, we could not produce its partial sums, of which there are as many as there are positive integers. The 9s in 0.9999 . . . are all already there.
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