I may have just misunderstood your vid, but my first take on it is that it confuses numbers with our common representation of numbers using decimal digits. That is, "1" is not a number, it is how we represent the abstract concept of the number 1 in our written language. As your vid shows, you can (if you understand limits) also represent the number 1 as a decimal of infinitely repeating nines. (0.9999...) But that's just a quirk of the decimal system we commonly use to represent numbers. Having different representations for the same number also happens with rational numbers. For example, you can write 1 as 1 or 1/1 or 2/2 or 3/3 and so forth. Yes, we refer to all of these representations as simply "a number", but that's just a linguistic short-cut we take for convenience. If you instead always refer to it as, perhaps, "a written representation of the abstract concept of a particular number" it would get very tedious very quickly, no?
He proved that 0.999... = 1, twice, and concluded that means that 1 isn't a real number. I wonder who owned all those math text books that are behind him.
Middleschool math here. You missed an important fact in the video about the algorithm x=".99...". It is circular and designed to give the exact value it started with. Many claim this is the "proof" that ".99..." is 1. Making a minor change of adding x instead of subtracting, then dividing by 11 instead of 9. 10x="9.99..." 11x="10.99..." x=".99..." as it should. What happened.? Can infinite numbers be used in arithmetic? Adding ".11..." to ".11..." to get ".22..." seems safe. There is a one to one correspondence not affected by infinity. (Math says you can't add to infinity, °°+x=°°) Adding ".11..." to ".99..." Let's see. .9+.1=1.0, .09+.01=.10, .009=.010. So far I've got 1.1100... I give, can't get rid of the 0 at the end. (Yes, 10 divided by 9 results in "1.11..." but it is incomplete. ) 1-".99..."=".00...", where did that 1 go. Again 1.0-.9=.1, .00-.09=-.09, .000-.009=-.009, So far .1-.099=.001 1/3 step by step. 1.0/3.0= wait for it! WAIT FOR IT!! UH, No. I can't wait for it, the 3s just won't end. It will always be incomplete and ever changing, unless you say it is finite by ignoring the 3s past some mathical point. (Which is exactly what math is saying, infinity is that point, where ever that is. Where is infinity exactly? How many digits of pi are necessary for the Webb telescope to determine the width of the furthest star or piece of dust in the universe? Do we know? The continuum of Real Numbers keeps us in the game.) The valuable concept of infinity is dangerous, incomplete, inconsistent and imprecise.
bonehead: "Middleschool [sic] math here. You missed an important fact in the video about the algorithm x=".99...". It is circular ..." It is not circular. A circular argument requires using the result (or an equivalent result) to prove the result. Nowhere did the proof assume that 0.999... = 1 in order to prove that 0.999... = 1. You have been told this many times, yet you choose to ignore it because you are only here to troII. bonehead: "... and designed to give the exact value it started with." Nope. It was designed to show that 0.999... = 1 and it only used junior school arithmetic. You also have had that shown to yoou many times, but you choose to ignore it because you are only here to troII. bonehead: "Many claim this is the "proof" that ".99..." is 1." That's right. bonehead: "Making a minor change of adding x instead of subtracting, then dividing by 11 instead of 9. 10x="9.99..." 11x="10.99..." x=".99..." as it should." LOL. You keep on providing a proof that 0.999... = 1. This must be at least the tenth time you have provided this particular proof. Using your own results: 11x - 10x = 10.999... - 9.999... = 10 + 0.999... - 9 - 0.999... = 1. bonehead: "What happened.?" LOL. You happened. bonehead: "Can infinite numbers be used in arithmetic?" Infinite decimals have been used in arithmetic for hundreds of years. bonehead: "Adding ".11..." to ".11..." to get ".22..." seems safe." That's right, even though you think it can't be. bonehead: "here is a one to one correspondence not affected by infinity." What are you blathering on about? What has a one to one correspondence with what? bonehead: "(Math says you can't add to infinity, °°+x=°°)" No it doesn't. You even gave an example. However, that has nothing to do with the proof because oo was not being used as a number. bonehead: "You can "count" to ".99..." without a problem." ??? If you mean as in you can count to 1 without a problem, then I agree. You use terms and phrases so weirdly that it is had to know what you actually mean. bonehead: "Try adding ".99..." and ".11...". Is the answer 1.00...?" Off course not. 0.999... + 0.111... = 1.111... bonehead: "Let's see. .9+.1=1.0, .09+.01=.10, .009=.010. So far I've got 1.1100..." If you keep going you will get closer and closer to 1.111..., although you will never actually obtain 1.111... bonehead: "I give, can't get rid of the 0 at the end." Nobody can. So what? bonehead: "(Yes, 10 divided by 9 results in "1.11..." but it is incomplete. )" What id incomplete? 1.111... = 10/9 exactly. bonehead: "1-".99..."=".00...", where did that 1 go." What 1? bonehead: "Again 1.0-.9=.1, .00-.09=-.09, .000-.009=-.009, So far .1-.099=.001" LOL. that was weird. 0.999... - 0.9 = 0.0999..., 0.999... - 0.99 = 0.00999..., 0.999... - 0.999 = 0.000999..., ... Does that somehow mean that 0.999... != 0.999... in your muppet opinion? bonehead: "1/3 step by step. 1.0/3.0= wait for it! WAIT FOR IT!! UH, No. I can't wait for it, the 3s just won't end. It will always be incomplete and ever changing, unless you say it is finite by ignoring the 3s past some mathical point. (Which is exactly what math is saying, infinity is that point, where ever that is. Where is infinity exactly? How many digits of pi are necessary for the Webb telescope to determine the width of the furthest star or piece of dust in the universe? Do we know? The continuum of Real Numbers keeps us in the game.)" That was weird too. Nobody with a clue says that you can obtain 0.333... that way. You have been told that many times. What has physical measurements go to do with pure mathematics? Math is not constrained by such pedestrian limitations. What you'll get (using a succinct way of expressing it) is: 1/3 = 0.3 + 1/30 = 0.33 + 1/300 = 0.333 + 1/3000 = ... = .. The limit is 1/3 = 0.333... + 0 = 0.333..., but you will not get to that limit. bonehead: "The valuable concept of infinity is dangerous, incomplete, inconsistent and imprecise." You keep on repeating that mindless mantra. You never attempt to justify it. What are you trying to hide?
I'll revisit your "10x="9.99..." 11x="10.99..." x=".99..."" 10x = 9.999... = 9 + 0.999... = 9 + x => 9x = 9 => x = 1 11x = 10.999... = 10 + 0.999... = 10 + x => 10x = 10 => x = 1 Coupled with 0.999... = x = 11x - 10x = 10.999... - 9.999... = 1 Altogether that means you gave three equations that are equivalent to 0.999... = 1. Yet you somehow think that means you have proved that 0.999... != 1. What it actually proves is that you are a muppet. PS According to you, a muppet, you had already assumed that 0.999... = 1 in order to say that 10x = 9.999.... According to you that is a circular something or other (algorithm???). How did you use 0.999... = 1 in order to calculate that 10x = 9.999...? Same for 11x = 10.999.... I'll be fascinated to see how you did that. Obviously I won't hold my breath because you never answer straightforward questions because you have something to hide (not your brilliance, that's for sure).
At about 3:38 you asked for two integers whose ratio is 0.999.... The most "natural" ratio is 9/9. In fact proof wiki shows how to do a delayed long division procedure in order to derive 0.999... from 9/9. Obviously you are trying the good old appeal to personal incredulity fallacious argument there. Here's an loosely related semi general argument (that you'll hate). Let 0.(N) represent the n digit string N being repeated. Then: 10^n * 0.(N) = N.(N) = N + 0.(N) => (10^n - 1) * 0.(N) = N => 0.(N) = N / (10^n - 1) Note that 10^n - 1 as a decimal is an n digit string of 9s. Note that N and 10^n - 1 are natural numbers, so 0.(N) represents a rational number. So 0.(857142) = 857142 / 999999 = 6/7. I personally like 0.(9) = 9/9 = 0.(99) = 99/99 = 0.(999) = 999/999 = 1/ You'll hate it because you'll see it as blasphemy.
KC, note that it is disputed that your 10^n*N=N.(N). Basically it could be said that n 0s are missing. Using infinite numerals in arithmetic is mostly inaccurate. 10*".99..." does not equal "9.99...". Further the discussion is whether 1=".99..." which is related. ".99..."/1 is not rational and 9/9 is 1. ".99..."/(9/9)=".99..."*9/9=".99..."
bonehead: "KC, note that it is disputed that your 10^n*N=N.(N)." LOL. You can't even quote me accurately, yet alone reply to my comment in the same thread. Who, other than cIueIess muppets like yourself, is disputing it? I said, "Let 0.(N) represent the n digit string N being repeated. Then: 10^n * 0.(N) = N.(N) = N + 0.(N) => (10^n - 1) * 0.(N) = N => 0.(N) = N / (10^n - 1) Note that the decimal numeral for 10^n - 1 an n digit string of 9s. Also note that N and 10^n - 1 are natural numbers, so 0.(N) represents a rational number." I'll add that it immediately follows that 0.(9) = 9/9 = 1 and so it represents a rational number. bonehead: "Basically it could be said that n 0s are missing." What 0s, you muppet? bonehead: "Using infinite numerals in arithmetic is mostly inaccurate." Where's your proof of that ridiculous claim, you deIusionaI muppet? bonehead: "10*".99..." does not equal ".99...". LOL. That's right. 10 * 0.999... = 9.999... bonehead: "Further the discussion is whether 1=".99..." which is related." What discussion and what is related? bonehead: "".99..."/1 is not rational ..." Where's your proof of that false assertion? In over a year you haven't ever tried to prove any of your assertions. I have shown you that 0.999... is rational at least a hundred times. bonehead: "...and 9/9 is 1." True, so is 0.999... bonehead: "".99..."/(9/9)=".99..."*9/9=".99..."" LOL. It's also 1. ---- Yet another rambling post from you. You haven't tried to prove any of your ridiculous assertions. You are only here to troII.
You’re a very brave man, who go against most people’s believe, that 0.9 repeating is equals to 1. I think most people missed the fact that even 9 over 9 and 10 over 10 are both equal to 1 but they are not the same 1, one is base 9 and the other is base 10 system. The example you used 1 over 9 showed in base-10 system is 0.1 repeating but in base-9 system is 0.1 ( no repeating); 9 over 9 as you said should be 0.9 repeating in base-10 system but since it is base-9 system so it should be 0.9, but again, 0.9 in base-9 system is 1.
As far as IF you choose ".99..." to represent 1, then it is not a good choice. Basics in decimal based representation of real numbers, ".99..." looks like a real number. Real number integers in decimal based representation should have no non-zero digits to the right of the decimal point. Rational numbers are the ratio of 2 integers. Then ".99..."/".99..."=".99..." a rational number? What about the sum of a infinite geometric series?? The sum of any infinite series with all terms >0 is questionable and is only defined by a limit. The sum of any 2 numbers >0 is > either of the 2 numbers. A infinite series of >0 terms is strictly increasing.
bonehead: "As far as IF you choose ".99..." to represent 1, then it is not a good choice." Says you, a clueless troIIing muppet. Where's your proof? In fact it is the best choice. However, that choice is not made directly. Rather, it is a consequence of more general choices (and definitions). bonehead: "Basics in decimal based representation of real numbers, ".99..." looks like a real number." Your grammar is atrocious. I can barely decipher what you said because of it. What do real numbers look like - bear in mind that they are pure abstract concepts that only exist as ideas. bonehead: "Real number integers in decimal based representation should have no non-zero digits to the right of the decimal point." Says you, a clueless troIIing muppet. As -usual- always, you just assert, but never justify. bonehead: "Rational numbers are the ratio of 2 integers." That's nearly right. Rational numbers are numbers that can be expressed as the ratio of two integers. You have been told that many times, yet you repeat the same false "definition". That's because you are a clueless troIIing muppet. bonehead: "Then ".99..."/".99..."=".99..." a rational number?" Of course it is. bonehead: "What about the sum of a infinite geometric series??" What about it? bonehead: "The sum of any infinite series with all terms >0 is questionable ..." Yet again, as -usual- always, you haven't justified your ridiculous assertion. If the series is convergent, then it has a sum. There are plenty of ways to test for convergence. bonehead: "... and is only defined by a limit." That's right, except YOU don't know what that limit is or how it is defined, despite being told hundreds of times. If you have a point, what is it? bonehead: "The sum of any 2 numbers >0 is > either of the 2 numbers." That's right. It must be the most advanced math you know, hence your need to keep on repeating it. bonehead: "A infinite series of >0 terms is strictly increasing." Nope. It is strictly not changing. What is increasing is the value of the terms in the sequence of its partial sums as you step through it. Over a year and you still don't know what sums, limits, series or sequences are. --- That was yet another of your rambling comments where you fail to make any points, only make a series of (mainly false) assertions, and don't make the slightest attempt at justifying anything you say. That's because you are a clueless troIIing muppet.
LOL. So you proved that 0.999... =1 and then just arbitrarily claimed that 0.999... is not a real number. You didn't make any attempt to justify your claim. You just gave another proof that 0.999... = 1. You even said something like if 0.999... = 1 then why do we need 0.999... in the first place. So does that mean that 2/5 + 3/5 = 1 somehow implies that 2/5 + 3/5 is also a fantasy/pseudo number?
I understand your point but it seems more like this is an issue of symbology rather than mathematics. I’d suggest be careful with this kind of argumentation, because the notion of 0.9… is already very difficult for many students to get their heads around.
0.99... = 1 In this equality, the only number is 1. There doesn't exist any number that would be composed of a whole part equal to 0 and of an inifinite decimal extend with repeating 9. This number doesn't exist because it is the number 1 (whole part = 1 and infinite decimal extension .0000...)
You are confusing a decimal numeral with a number. The integer part of the decimal NUMERAL 0.999... is 0, but that is just because of the way that the "columns" are labelled. The integer part of the NUMBER that the decimal represents is 1.
@@Chris-5318 that's why 0.999... isn't a real number. 1 is a consistant notation for the number it represents : 1. I am very interested by the point of vue of Norman J. Wildberger (He seems to refute all infinite decimal notations included the "proper infinite decimal notation" of fractions. th-cam.com/video/WabHm1QWVCA/w-d-xo.html I am very interested in the sociologic aspect of mathematics too... th-cam.com/video/YBqNDpsO7hY/w-d-xo.html
@@cupiodissolvi9942 Both Mückenheim and Wildberger are cranks. I've exchanged comments with both of them in the past. Mückenheim can't even grasp the fact that the sequence 0.9, 0,99, 0.999, ... doesn't have a last term. He thinks it should end with 0.999... even though it is endless. I read about half of the first paragraph of that pdf and already knew that it was going to be crackpottery for the rest of the document. I glanced at page 346 and that immediately confirmed that view.
I agree with you in general. ".99..." is not 1. I don't believe it is a good representation of 1. I believe it is not a real number. It is better to say it is a "real number space" composed of an infinite amount of numbers that are real numbers with a finite number of the digits, 9 that followi the decimal point in decimal place notation less than 1. (That was a mouthful) Also why mess up decimal notation by defining a integer with digits other than 0 after the decimal. In the late 1500's, infinite numbers in decimal notation became accepted as rational numbers instead of fractions. You pointed out that infinite digits could be obtained by dividing by 9 with a counter example being 9 itself. You could say those, such as ".33...", are number spaces less than the fraction they represent. On a sum of an infinite series of >0 terms, how can it be unique and precise number without someone declaring it to be by formula. (Not by basics) The Archimedean property says we can't define a real number adjacent to 1 as there are always numbers between any 2 real numbers we can conceive of. It is difficult to show that there is no number in base 10 that is closer to 1 than the number space of ".99...". It is always at least 1/10^(n+1) from 1. But in number bases higher than 10, like base 60, 1/60^(n+1) is closer. (Unless you believe 1/n^n^°° equals 0. Then 1/10 and 1/60 represent the same number. [Yes I could have used 1/n^°° but 1/n^n^°° better shows my point, which brings me to...] ) Infinity is incomplete, inconsistent and imprecise. [EDIT Real numbers are unique and precise. ]
So what is ".99...". I liked your comment that 1 and 0 are your favorite numbers. 1 is the basic unit, the starting and controlling counting number. 0 is more exotic. It is a defined non-value number, the additive identity. In the decimal place notation system it has relative value in holding a place value in the notation when that place value is not present. O is a unique and precise non-value. All real numbers have a unique and precise value. Between 0 and 1 all the real numbers are previewed and then repeated from 1 through the infinite real numbers. So what is ".99..."? [An extended real number in the extended real number system.] In decimal place notation. ".99..." is a infinite decimal. A infinite decimal is a notation that represents a sequence of numbers less than, and have a real number limit of, that decimal place notation can't accurately represent such as certain fractions and transcendental numbers. For centuries, rational numbers have been defined to be the ratio of 2 integers. I see no reason to change it. It seams that non-standard (modern?) wants to change this definition to add repeating decimals but do not change the definition in their texts. " 99..." does not meet the definition of a rational number. [Is it a lone example of a repeating decimal not being rational?] Further, math wants to define ".99..." as a integer while for centuries indicating that in decimal place notation there are no non-zero digits to the right of the decimal point in a integer notation. I believe that ".99..." represents a set of real numbers less than 1. An open set between the real number 0.99..9, a indescribably large finite number of 9, and those larger real numbers less than 1 that cant be represented in the decimal place system. The decimal place system has been in use for centuries and modified to be more rigorous. Why mess it up for infinity which is incomplete, inconsistent and imprecise.
bonehead: "So what is ".99...". I liked your comment that 1 and 0 are your favorite numbers." You need to take your meds. bonehead: "1 is the basic unit, the starting and controlling counting number." I guess that you think that that gibberish makes your sound intellectual. LOL. bonehead: "0 is more exotic. It is a defined non-value number, the additive identity." It has the value 0. I doubt that you know what an additive identity is. bonehead: "In the decimal place notation system it has relative value in holding a place value in the notation when that place value is not present." That's almost right. A seven year old could almost be proud of such a sentence. bonehead: "O is a unique and precise non-value." It has the value 0. i.e. it represents the number that we call "zero". bonehead: "All real numbers have a unique and precise value." That twaddle is circular. Values are numbers. bonehead: "Between 0 and 1 all the real numbers are previewed and then repeated from 1 through the infinite real numbers. " LOL. More gibberish. bonehead: "So what is ".99..."? In decimal place notation. ".99..." is a infinite decimal." It's only taken you well over a year to learn that. bonehead: "A infinite decimal is a notation that represents a sequence of numbers ..." No it doesn't. It only represents one number: the sum of the series. bonehead: "... less than, and have a real number limit of, that decimal place notation" Decimals do not have limits. They have sums. You have been told that at least a hundred times. bonehead: "can't accurately represent such as certain fractions and transcendental numbers." Decimals represent fractions (i.e. rational numbers) and transcendental numbers (a subset of the irrational) precisely. You can't have more accuracy than that. bonehead: "For centuries, rational numbers have been defined to be the ratio of 2 integers. I see no reason to change it." We nearly agree about that. Rational numbers can be expressed as the ratio of two integers. bonehead: "It seams [sic] that non-standard (modern?) wants to change this definition to add repeating decimals but do not change the definition in their texts." Nonsense. You are projecting your incompetency and delusions onto the mathematicians. Every repeating decimal can be show to be expressible as the ratio of two integers. Therefore every repeating decimal represents a rational number. The repeating decimal 0.(N) where N is an n-digit string being repeated is equal to N/(10^n - 1) and that is the ratio of two natural numbers, so 0.(N) represents a rational number. bonehead: "That's because they " 99..." does not meet the definition of a rational number." Yes it does. Using 0.(N) = N/(10^n - 1) we have 0.999... = 0.(9) = 9/(10^1 - 1) = 9/9 = 1 and so 0.999... represents a rational number. bonehead: " Further, math wants to define ".99..." as a integer while for centuries indicating that in decimal place notation there are no non-zero digits to the right of the decimal point in a integer notation." Nope. It just happens that 0.999... represents an integer. It has been known for centuries that 0.999... = 1. bonehead: "I believe that ".99..." represents a set of real numbers less than 1." You are unwittingly almost referring to the Dedekind cut definition of the real numbers, except that refers to the set of all the rational numbers less than 0.999.... That is another way to see that that 0.999... = 1. That's because the set of all the rational numbers less than 0.999... is the same set as the set of all the rational numbers less than 1. Of course, it is also the case that the set of all the reals less than 0.999... is the same set as the set of all the reals less than 1. bonehead: "An open set between the real number 0.99..9, a indescribably large finite number of 9, ..." That's vague BS. What is 0.99...9 supposed to be? bonehead: "... and those larger real numbers less than 1 that cant be represented in the decimal place system." BS. All real numbers have at least one decimal representations. Some have two. bonehead: "The decimal place system has been in use for centuries and modified to be more rigorous." What is this BS about being more rigorous? bonehead: "Why mess it up for infinity which is incomplete, inconsistent and imprecise." It's not messed - you are. You are incomplete, inconsistent, imprecise, incompetent and a delusional trolling muppet. You actually believe that you are intellectual, you are that crazy.
OMG I've just seen the last few seconds of your video where you finally admit that 0.999... is just another representation of 1. That's right, that's exactly what it is. So you scored a home goal. LOL.
All you've done, in your nearly 9 minute video, is to inexplicably repeatedly assert that 0.999... isn't 1. You haven't presented any arguments to justify that deIusionaI belief. You even seem to be aware of it, yet you decided to publish the video despite knowing that. That's weird.
Wrong again, Karen. The definition of rational numbers does not state that using the formula to find the integer fraction of a repeating decimal. The definition of a rational number asks for an integer ratio or two integer when divided result in the repeating decimal. 9 divided by 9 results in 1, not ".99...". Learn the basics.
@bonehead, the definition of a rational number simply requires that it can be written as the ratio of two integers. It says nothing about decimals. It happens that if a decimal repeats, then it can be expressed as the ratio of two integers. e.g. 0.123123123... = 123/999 and 0.999... = 9/9 = 1. That is eighth grader math and you suck at it (and everything except being a muppet).
KC You just repeated what I said, and added that 9/9=1, which I agree with. Problem! 9/9 does not equal ".99..." as in the definition of rational numbers. (See Analysis I by Tao) [NOTE: ".99..."/1=".99..." but ".99..." is not a decimal place notation integer. Better find a better formula for a rational number because of this one exception? I have not found any other exception in base 10. Do you think that maybe base 10 notation has a problem representing all real numbers. How about leaving the base 10 place notation system alone, as it is still valuable and commonplace, easier to teach and relatively rigorous in its own right. ".99..." is the closest to 1 in the decimal place notation system]
@@johnlabonte-ch5ul Where is your proof that 0.999... != 1? You've been at least a year and still haven't even tried to prove it. Also, you have seen many proofs that 0.999... = 1, and so far, you haven't shown any faults with any of them. In view of your latest trolling, a relevant one is: 10 * 0.999... = 9.999... (which I have proven to you many times) => 9 * 0.999... + 0.999... = 9 + 0.999... => 9 * 0.999... = 9 => 0.999... = 9/9 = 1 and that directly contradicts your completely baseless claim and it had nothing to do with the definition of rational numbers. OTOH it does happen to satisfy the definition of rational numbers, so it is a rational number. If you change the definition of rational numbers, it won't change the fact that 0.999... = 1. A rose by any other name is still a rose, you muppet. What is the first false statement in that proof? Make sure that you provide a verifiable reason to support your claim - your unsupported delusional opinion is nowhere near good enough. You are off your troIIey. You need medicaI heIp.
You start by assuming 10×".99..." is."9.99..." which implies ".99..." is 1. All algebraic args that show ".99..." is 1 are flawed by using infinite number arithmetic. Of more interest is the Archimedean property of real numbers and the sum of infinite geometric series. The idea that 1/x, when x "reaches" infinity, it's value must be zero. You will find the idea of infinity is dangerous, incomplete, inconsistent and imprecise.
I may have just misunderstood your vid, but my first take on it is that it confuses numbers with our common representation of numbers using decimal digits. That is, "1" is not a number, it is how we represent the abstract concept of the number 1 in our written language. As your vid shows, you can (if you understand limits) also represent the number 1 as a decimal of infinitely repeating nines. (0.9999...) But that's just a quirk of the decimal system we commonly use to represent numbers. Having different representations for the same number also happens with rational numbers. For example, you can write 1 as 1 or 1/1 or 2/2 or 3/3 and so forth. Yes, we refer to all of these representations as simply "a number", but that's just a linguistic short-cut we take for convenience. If you instead always refer to it as, perhaps, "a written representation of the abstract concept of a particular number" it would get very tedious very quickly, no?
You haven't misunderstood the video - it is nonsense.
You didn't prove it doesn't exist. You just proved it's equal to 1 and hence a real number.
He proved that 0.999... = 1, twice, and concluded that means that 1 isn't a real number. I wonder who owned all those math text books that are behind him.
Middleschool math here. You missed an important fact in the video about the algorithm x=".99...". It is circular and designed to give the exact value it started with. Many claim this is the "proof" that ".99..." is 1.
Making a minor change of adding x instead of subtracting, then dividing by 11 instead of 9.
10x="9.99..."
11x="10.99..."
x=".99..." as it should.
What happened.? Can infinite numbers be used in arithmetic?
Adding ".11..." to ".11..." to get ".22..." seems safe. There is a one to one correspondence not affected by infinity. (Math says you can't add to infinity, °°+x=°°)
Adding ".11..." to ".99..."
Let's see. .9+.1=1.0, .09+.01=.10, .009=.010.
So far I've got 1.1100...
I give, can't get rid of the 0 at the end. (Yes, 10 divided by 9 results in "1.11..." but it is incomplete. )
1-".99..."=".00...", where did that 1 go. Again 1.0-.9=.1, .00-.09=-.09, .000-.009=-.009,
So far .1-.099=.001
1/3 step by step. 1.0/3.0= wait for it! WAIT FOR IT!!
UH, No. I can't wait for it, the 3s just won't end. It will always be incomplete and ever changing, unless you say it is finite by ignoring the 3s past some mathical point. (Which is exactly what math is saying, infinity is that point, where ever that is. Where is infinity exactly? How many digits of pi are necessary for the Webb telescope to determine the width of the furthest star or piece of dust in the universe? Do we know? The continuum of Real Numbers keeps us in the game.)
The valuable concept of infinity is dangerous, incomplete, inconsistent and imprecise.
bonehead: "Middleschool [sic] math here. You missed an important fact in the video about the algorithm x=".99...". It is circular ..."
It is not circular. A circular argument requires using the result (or an equivalent result) to prove the result. Nowhere did the proof assume that 0.999... = 1 in order to prove that 0.999... = 1. You have been told this many times, yet you choose to ignore it because you are only here to troII.
bonehead: "... and designed to give the exact value it started with."
Nope. It was designed to show that 0.999... = 1 and it only used junior school arithmetic. You also have had that shown to yoou many times, but you choose to ignore it because you are only here to troII.
bonehead: "Many claim this is the "proof" that ".99..." is 1."
That's right.
bonehead: "Making a minor change of adding x instead of subtracting, then dividing by 11 instead of 9.
10x="9.99..."
11x="10.99..."
x=".99..." as it should."
LOL. You keep on providing a proof that 0.999... = 1. This must be at least the tenth time you have provided this particular proof.
Using your own results: 11x - 10x = 10.999... - 9.999... = 10 + 0.999... - 9 - 0.999... = 1.
bonehead: "What happened.?"
LOL. You happened.
bonehead: "Can infinite numbers be used in arithmetic?"
Infinite decimals have been used in arithmetic for hundreds of years.
bonehead: "Adding ".11..." to ".11..." to get ".22..." seems safe."
That's right, even though you think it can't be.
bonehead: "here is a one to one correspondence not affected by infinity."
What are you blathering on about? What has a one to one correspondence with what?
bonehead: "(Math says you can't add to infinity, °°+x=°°)"
No it doesn't. You even gave an example. However, that has nothing to do with the proof because oo was not being used as a number.
bonehead: "You can "count" to ".99..." without a problem."
??? If you mean as in you can count to 1 without a problem, then I agree. You use terms and phrases so weirdly that it is had to know what you actually mean.
bonehead: "Try adding ".99..." and ".11...". Is the answer 1.00...?"
Off course not. 0.999... + 0.111... = 1.111...
bonehead: "Let's see. .9+.1=1.0, .09+.01=.10, .009=.010.
So far I've got 1.1100..."
If you keep going you will get closer and closer to 1.111..., although you will never actually obtain 1.111...
bonehead: "I give, can't get rid of the 0 at the end."
Nobody can. So what?
bonehead: "(Yes, 10 divided by 9 results in "1.11..." but it is incomplete. )"
What id incomplete? 1.111... = 10/9 exactly.
bonehead: "1-".99..."=".00...", where did that 1 go."
What 1?
bonehead: "Again 1.0-.9=.1, .00-.09=-.09, .000-.009=-.009,
So far .1-.099=.001"
LOL. that was weird.
0.999... - 0.9 = 0.0999..., 0.999... - 0.99 = 0.00999..., 0.999... - 0.999 = 0.000999..., ...
Does that somehow mean that 0.999... != 0.999... in your muppet opinion?
bonehead: "1/3 step by step. 1.0/3.0= wait for it! WAIT FOR IT!!
UH, No. I can't wait for it, the 3s just won't end. It will always be incomplete and ever changing, unless you say it is finite by ignoring the 3s past some mathical point. (Which is exactly what math is saying, infinity is that point, where ever that is. Where is infinity exactly? How many digits of pi are necessary for the Webb telescope to determine the width of the furthest star or piece of dust in the universe? Do we know? The continuum of Real Numbers keeps us in the game.)"
That was weird too. Nobody with a clue says that you can obtain 0.333... that way. You have been told that many times. What has physical measurements go to do with pure mathematics? Math is not constrained by such pedestrian limitations.
What you'll get (using a succinct way of expressing it) is:
1/3 = 0.3 + 1/30 = 0.33 + 1/300 = 0.333 + 1/3000 = ... = ..
The limit is 1/3 = 0.333... + 0 = 0.333..., but you will not get to that limit.
bonehead: "The valuable concept of infinity is dangerous, incomplete, inconsistent and imprecise."
You keep on repeating that mindless mantra. You never attempt to justify it. What are you trying to hide?
I'll revisit your "10x="9.99..."
11x="10.99..."
x=".99...""
10x = 9.999... = 9 + 0.999... = 9 + x => 9x = 9 => x = 1
11x = 10.999... = 10 + 0.999... = 10 + x => 10x = 10 => x = 1
Coupled with 0.999... = x = 11x - 10x = 10.999... - 9.999... = 1
Altogether that means you gave three equations that are equivalent to 0.999... = 1. Yet you somehow think that means you have proved that 0.999... != 1. What it actually proves is that you are a muppet.
PS According to you, a muppet, you had already assumed that 0.999... = 1 in order to say that 10x = 9.999.... According to you that is a circular something or other (algorithm???). How did you use 0.999... = 1 in order to calculate that 10x = 9.999...? Same for 11x = 10.999.... I'll be fascinated to see how you did that. Obviously I won't hold my breath because you never answer straightforward questions because you have something to hide (not your brilliance, that's for sure).
Of interest. Let y=".99..." Now
x=y Multiply by 10
10x=10y add the original
11x=11y divide by 11
x=y then
x=".99..."
@@johnlabonte-ch5ul That definitely would be of interest (to a psychologist).
At about 3:38 you asked for two integers whose ratio is 0.999.... The most "natural" ratio is 9/9. In fact proof wiki shows how to do a delayed long division procedure in order to derive 0.999... from 9/9. Obviously you are trying the good old appeal to personal incredulity fallacious argument there.
Here's an loosely related semi general argument (that you'll hate).
Let 0.(N) represent the n digit string N being repeated. Then:
10^n * 0.(N) = N.(N) = N + 0.(N)
=> (10^n - 1) * 0.(N) = N
=> 0.(N) = N / (10^n - 1)
Note that 10^n - 1 as a decimal is an n digit string of 9s.
Note that N and 10^n - 1 are natural numbers, so 0.(N) represents a rational number.
So 0.(857142) = 857142 / 999999 = 6/7.
I personally like 0.(9) = 9/9 = 0.(99) = 99/99 = 0.(999) = 999/999 = 1/ You'll hate it because you'll see it as blasphemy.
KC, note that it is disputed that your 10^n*N=N.(N). Basically it could be said that n 0s are missing. Using infinite numerals in arithmetic is mostly inaccurate. 10*".99..." does not equal "9.99...". Further the discussion is whether 1=".99..." which is related.
".99..."/1 is not rational and 9/9 is 1. ".99..."/(9/9)=".99..."*9/9=".99..."
bonehead: "KC, note that it is disputed that your 10^n*N=N.(N)."
LOL. You can't even quote me accurately, yet alone reply to my comment in the same thread. Who, other than cIueIess muppets like yourself, is disputing it?
I said, "Let 0.(N) represent the n digit string N being repeated. Then:
10^n * 0.(N) = N.(N) = N + 0.(N)
=> (10^n - 1) * 0.(N) = N
=> 0.(N) = N / (10^n - 1)
Note that the decimal numeral for 10^n - 1 an n digit string of 9s. Also note that N and 10^n - 1 are natural numbers, so 0.(N) represents a rational number."
I'll add that it immediately follows that 0.(9) = 9/9 = 1 and so it represents a rational number.
bonehead: "Basically it could be said that n 0s are missing."
What 0s, you muppet?
bonehead: "Using infinite numerals in arithmetic is mostly inaccurate."
Where's your proof of that ridiculous claim, you deIusionaI muppet?
bonehead: "10*".99..." does not equal ".99...".
LOL. That's right. 10 * 0.999... = 9.999...
bonehead: "Further the discussion is whether 1=".99..." which is related."
What discussion and what is related?
bonehead: "".99..."/1 is not rational ..."
Where's your proof of that false assertion? In over a year you haven't ever tried to prove any of your assertions. I have shown you that 0.999... is rational at least a hundred times.
bonehead: "...and 9/9 is 1."
True, so is 0.999...
bonehead: "".99..."/(9/9)=".99..."*9/9=".99...""
LOL. It's also 1.
----
Yet another rambling post from you. You haven't tried to prove any of your ridiculous assertions. You are only here to troII.
You’re a very brave man, who go against most people’s believe, that 0.9 repeating is equals to 1.
I think most people missed the fact that even 9 over 9 and 10 over 10 are both equal to 1 but they are not the same 1, one is base 9 and the other is base 10 system. The example you used 1 over 9 showed in base-10 system is 0.1 repeating but in base-9 system is 0.1 ( no repeating); 9 over 9 as you said should be 0.9 repeating in base-10 system but since it is base-9 system so it should be 0.9, but again, 0.9 in base-9 system is 1.
"9 over 9 and 10 over 10 are both equal to 1 but they are not the same 1"
LOL Was that satire? If so, then you nailed it.
It's very simple, the sum of a series >>IS
As far as IF you choose ".99..." to represent 1, then it is not a good choice. Basics in decimal based representation of real numbers, ".99..." looks like a real number.
Real number integers in decimal based representation should have no non-zero digits to the right of the decimal point.
Rational numbers are the ratio of 2 integers. Then ".99..."/".99..."=".99..." a rational number?
What about the sum of a infinite geometric series?? The sum of any infinite series with all terms >0 is questionable and is only defined by a limit. The sum of any 2 numbers >0 is > either of the 2 numbers. A infinite series of >0 terms is strictly increasing.
bonehead: "As far as IF you choose ".99..." to represent 1, then it is not a good choice."
Says you, a clueless troIIing muppet. Where's your proof? In fact it is the best choice. However, that choice is not made directly. Rather, it is a consequence of more general choices (and definitions).
bonehead: "Basics in decimal based representation of real numbers, ".99..." looks like a real number."
Your grammar is atrocious. I can barely decipher what you said because of it. What do real numbers look like - bear in mind that they are pure abstract concepts that only exist as ideas.
bonehead: "Real number integers in decimal based representation should have no non-zero digits to the right of the decimal point."
Says you, a clueless troIIing muppet. As -usual- always, you just assert, but never justify.
bonehead: "Rational numbers are the ratio of 2 integers."
That's nearly right. Rational numbers are numbers that can be expressed as the ratio of two integers. You have been told that many times, yet you repeat the same false "definition". That's because you are a clueless troIIing muppet.
bonehead: "Then ".99..."/".99..."=".99..." a rational number?"
Of course it is.
bonehead: "What about the sum of a infinite geometric series??"
What about it?
bonehead: "The sum of any infinite series with all terms >0 is questionable ..."
Yet again, as -usual- always, you haven't justified your ridiculous assertion. If the series is convergent, then it has a sum. There are plenty of ways to test for convergence.
bonehead: "... and is only defined by a limit."
That's right, except YOU don't know what that limit is or how it is defined, despite being told hundreds of times. If you have a point, what is it?
bonehead: "The sum of any 2 numbers >0 is > either of the 2 numbers."
That's right. It must be the most advanced math you know, hence your need to keep on repeating it.
bonehead: "A infinite series of >0 terms is strictly increasing."
Nope. It is strictly not changing. What is increasing is the value of the terms in the sequence of its partial sums as you step through it.
Over a year and you still don't know what sums, limits, series or sequences are.
---
That was yet another of your rambling comments where you fail to make any points, only make a series of (mainly false) assertions, and don't make the slightest attempt at justifying anything you say. That's because you are a clueless troIIing muppet.
LOL. So you proved that 0.999... =1 and then just arbitrarily claimed that 0.999... is not a real number. You didn't make any attempt to justify your claim. You just gave another proof that 0.999... = 1.
You even said something like if 0.999... = 1 then why do we need 0.999... in the first place. So does that mean that 2/5 + 3/5 = 1 somehow implies that 2/5 + 3/5 is also a fantasy/pseudo number?
I understand your point but it seems more like this is an issue of symbology rather than mathematics. I’d suggest be careful with this kind of argumentation, because the notion of 0.9… is already very difficult for many students to get their heads around.
1/1 = .9999999......
0.99... = 1 In this equality, the only number is 1.
There doesn't exist any number that would be composed of a whole part equal to 0 and of an inifinite decimal extend with repeating 9. This number doesn't exist because it is the number 1 (whole part = 1 and infinite decimal extension .0000...)
You are confusing a decimal numeral with a number. The integer part of the decimal NUMERAL 0.999... is 0, but that is just because of the way that the "columns" are labelled. The integer part of the NUMBER that the decimal represents is 1.
@@Chris-5318 that's why 0.999... isn't a real number. 1 is a consistant notation for the number it represents : 1.
I am very interested by the point of vue of Norman J. Wildberger
(He seems to refute all infinite decimal notations included the "proper infinite decimal notation" of fractions.
th-cam.com/video/WabHm1QWVCA/w-d-xo.html
I am very interested in the sociologic aspect of mathematics too...
th-cam.com/video/YBqNDpsO7hY/w-d-xo.html
see p 346 Modern mathematics as religion (by Norman J. Wildberger)
www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity.pdf
@@cupiodissolvi9942 Both Mückenheim and Wildberger are cranks. I've exchanged comments with both of them in the past. Mückenheim can't even grasp the fact that the sequence 0.9, 0,99, 0.999, ... doesn't have a last term. He thinks it should end with 0.999... even though it is endless. I read about half of the first paragraph of that pdf and already knew that it was going to be crackpottery for the rest of the document. I glanced at page 346 and that immediately confirmed that view.
I agree with you in general. ".99..." is not 1. I don't believe it is a good representation of 1. I believe it is not a real number. It is better to say it is a "real number space" composed of an infinite amount of numbers that are real numbers with a finite number of the digits, 9 that followi the decimal point in decimal place notation less than 1.
(That was a mouthful)
Also why mess up decimal notation by defining a integer with digits other than 0 after the decimal.
In the late 1500's, infinite numbers in decimal notation became accepted as rational numbers instead of fractions.
You pointed out that infinite digits could be obtained by dividing by 9 with a counter example being 9 itself.
You could say those, such as ".33...", are number spaces less than the fraction they represent.
On a sum of an infinite series of >0 terms, how can it be unique and precise number without someone declaring it to be by formula. (Not by basics)
The Archimedean property says we can't define a real number adjacent to 1 as there are always numbers between any 2 real numbers we can conceive of. It is difficult to show that there is no number in base 10 that is closer to 1 than the number space of ".99...". It is always at least 1/10^(n+1) from 1. But in number bases higher than 10, like base 60, 1/60^(n+1) is closer. (Unless you believe 1/n^n^°° equals 0. Then 1/10 and 1/60 represent the same number. [Yes I could have used 1/n^°° but 1/n^n^°° better shows my point, which brings me to...] )
Infinity is incomplete, inconsistent and imprecise.
[EDIT Real numbers are unique and precise. ]
Bonehead, it doesn't matter what you believe, like the video poster, you are clueless.
So what is ".99...". I liked your comment that 1 and 0 are your favorite numbers. 1 is the basic unit, the starting and controlling counting number. 0 is more exotic. It is a defined non-value number, the additive identity. In the decimal place notation system it has relative value in holding a place value in the notation when that place value is not present. O is a unique and precise non-value. All real numbers have a unique and precise value.
Between 0 and 1 all the real numbers are previewed and then repeated from 1 through the infinite real numbers.
So what is ".99..."? [An extended real number in the extended real number system.] In decimal place notation. ".99..." is a infinite decimal. A infinite decimal is a notation that represents a sequence of numbers less than, and have a real number limit of, that decimal place notation can't accurately represent such as certain fractions and transcendental numbers.
For centuries, rational numbers have been defined to be the ratio of 2 integers. I see no reason to change it. It seams that non-standard (modern?) wants to change this definition to add repeating decimals but do not change the definition in their texts. " 99..." does not meet the definition of a rational number. [Is it a lone example of a repeating decimal not being rational?] Further, math wants to define ".99..." as a integer while for centuries indicating that in decimal place notation there are no non-zero digits to the right of the decimal point in a integer notation.
I believe that ".99..." represents a set of real numbers less than 1. An open set between the real number 0.99..9, a indescribably large finite number of 9, and those larger real numbers less than 1 that cant be represented in the decimal place system.
The decimal place system has been in use for centuries and modified to be more rigorous. Why mess it up for infinity which is incomplete, inconsistent and imprecise.
bonehead: "So what is ".99...". I liked your comment that 1 and 0 are your favorite numbers."
You need to take your meds.
bonehead: "1 is the basic unit, the starting and controlling counting number."
I guess that you think that that gibberish makes your sound intellectual. LOL.
bonehead: "0 is more exotic. It is a defined non-value number, the additive identity."
It has the value 0. I doubt that you know what an additive identity is.
bonehead: "In the decimal place notation system it has relative value in holding a place value in the notation when that place value is not present."
That's almost right. A seven year old could almost be proud of such a sentence.
bonehead: "O is a unique and precise non-value."
It has the value 0. i.e. it represents the number that we call "zero".
bonehead: "All real numbers have a unique and precise value."
That twaddle is circular. Values are numbers.
bonehead: "Between 0 and 1 all the real numbers are previewed and then repeated from 1 through the infinite real numbers. "
LOL. More gibberish.
bonehead: "So what is ".99..."? In decimal place notation. ".99..." is a infinite decimal."
It's only taken you well over a year to learn that.
bonehead: "A infinite decimal is a notation that represents a sequence of numbers ..."
No it doesn't. It only represents one number: the sum of the series.
bonehead: "... less than, and have a real number limit of, that decimal place notation"
Decimals do not have limits. They have sums. You have been told that at least a hundred times.
bonehead: "can't accurately represent such as certain fractions and transcendental numbers."
Decimals represent fractions (i.e. rational numbers) and transcendental numbers (a subset of the irrational) precisely. You can't have more accuracy than that.
bonehead: "For centuries, rational numbers have been defined to be the ratio of 2 integers. I see no reason to change it."
We nearly agree about that. Rational numbers can be expressed as the ratio of two integers.
bonehead: "It seams [sic] that non-standard (modern?) wants to change this definition to add repeating decimals but do not change the definition in their texts."
Nonsense. You are projecting your incompetency and delusions onto the mathematicians. Every repeating decimal can be show to be expressible as the ratio of two integers. Therefore every repeating decimal represents a rational number. The repeating decimal 0.(N) where N is an n-digit string being repeated is equal to N/(10^n - 1) and that is the ratio of two natural numbers, so 0.(N) represents a rational number.
bonehead: "That's because they " 99..." does not meet the definition of a rational number."
Yes it does. Using 0.(N) = N/(10^n - 1) we have 0.999... = 0.(9) = 9/(10^1 - 1) = 9/9 = 1 and so 0.999... represents a rational number.
bonehead: " Further, math wants to define ".99..." as a integer while for centuries indicating that in decimal place notation there are no non-zero digits to the right of the decimal point in a integer notation."
Nope. It just happens that 0.999... represents an integer. It has been known for centuries that 0.999... = 1.
bonehead: "I believe that ".99..." represents a set of real numbers less than 1."
You are unwittingly almost referring to the Dedekind cut definition of the real numbers, except that refers to the set of all the rational numbers less than 0.999.... That is another way to see that that 0.999... = 1. That's because the set of all the rational numbers less than 0.999... is the same set as the set of all the rational numbers less than 1. Of course, it is also the case that the set of all the reals less than 0.999... is the same set as the set of all the reals less than 1.
bonehead: "An open set between the real number 0.99..9, a indescribably large finite number of 9, ..."
That's vague BS. What is 0.99...9 supposed to be?
bonehead: "... and those larger real numbers less than 1 that cant be represented in the decimal place system."
BS. All real numbers have at least one decimal representations. Some have two.
bonehead: "The decimal place system has been in use for centuries and modified to be more rigorous."
What is this BS about being more rigorous?
bonehead: "Why mess it up for infinity which is incomplete, inconsistent and imprecise."
It's not messed - you are. You are incomplete, inconsistent, imprecise, incompetent and a delusional trolling muppet. You actually believe that you are intellectual, you are that crazy.
OMG I've just seen the last few seconds of your video where you finally admit that 0.999... is just another representation of 1. That's right, that's exactly what it is. So you scored a home goal. LOL.
All you've done, in your nearly 9 minute video, is to inexplicably repeatedly assert that 0.999... isn't 1. You haven't presented any arguments to justify that deIusionaI belief. You even seem to be aware of it, yet you decided to publish the video despite knowing that. That's weird.
Wrong again, Karen. The definition of rational numbers does not state that using the formula to find the integer fraction of a repeating decimal. The definition of a rational number asks for an integer ratio or two integer when divided result in the repeating decimal. 9 divided by 9 results in 1, not ".99...". Learn the basics.
@Bonhead: You are clueless. 0.999... = 1 and so, unlike you, is rational.
@bonehead, the definition of a rational number simply requires that it can be written as the ratio of two integers. It says nothing about decimals.
It happens that if a decimal repeats, then it can be expressed as the ratio of two integers. e.g. 0.123123123... = 123/999 and 0.999... = 9/9 = 1. That is eighth grader math and you suck at it (and everything except being a muppet).
KC You just repeated what I said, and added that 9/9=1, which I agree with.
Problem! 9/9 does not equal ".99..." as in the definition of rational numbers. (See Analysis I by Tao)
[NOTE: ".99..."/1=".99..." but ".99..." is not a decimal place notation integer. Better find a better formula for a rational number because of this one exception? I have not found any other exception in base 10.
Do you think that maybe base 10 notation has a problem representing all real numbers.
How about leaving the base 10 place notation system alone, as it is still valuable and commonplace, easier to teach and relatively rigorous in its own right.
".99..." is the closest to 1 in the decimal place notation system]
@@johnlabonte-ch5ul Where is your proof that 0.999... != 1? You've been at least a year and still haven't even tried to prove it. Also, you have seen many proofs that 0.999... = 1, and so far, you haven't shown any faults with any of them.
In view of your latest trolling, a relevant one is:
10 * 0.999... = 9.999... (which I have proven to you many times)
=> 9 * 0.999... + 0.999... = 9 + 0.999...
=> 9 * 0.999... = 9
=> 0.999... = 9/9 = 1 and that directly contradicts your completely baseless claim and it had nothing to do with the definition of rational numbers. OTOH it does happen to satisfy the definition of rational numbers, so it is a rational number. If you change the definition of rational numbers, it won't change the fact that 0.999... = 1. A rose by any other name is still a rose, you muppet.
What is the first false statement in that proof? Make sure that you provide a verifiable reason to support your claim - your unsupported delusional opinion is nowhere near good enough.
You are off your troIIey. You need medicaI heIp.
You start by assuming 10×".99..." is."9.99..." which implies ".99..." is 1. All algebraic args that show ".99..." is 1 are flawed by using infinite number arithmetic. Of more interest is the Archimedean property of real numbers and the sum of infinite geometric series. The idea that 1/x, when x "reaches" infinity, it's value must be zero.
You will find the idea of infinity is dangerous, incomplete, inconsistent and imprecise.