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Really i'm sorry to my failed attempt to convince my students and also teachers of mathematics of high school of my country that $0.\overline{9}=1$ is a real equality and it is not an approximation , They described me away from the definition of integers ensemble , However I have showed them all standards proofs to approach them the idea of equality but i don't succeed . Then my question here is :

Question: How can i convince students and teachers of high school that $0.\overline{9}=1$ and it is not approximation ?

zeraoulia rafik
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    Ask them what they propose $,1 - 0.\overline{9},$ would be. – dxiv Feb 25 '18 at 21:01
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    If two numbers are different $a<b$, there should be another one in between $a < \frac{a+b}{2}<b$. – rtybase Feb 25 '18 at 21:02
  • @dxiv, if you say them that , you will answer this difference is non zero but closed to 0 – zeraoulia rafik Feb 25 '18 at 21:02
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    @zeraouliarafik It's a real number, so they should be able to produce a value, not just state it's close to 0. – dxiv Feb 25 '18 at 21:03
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    Have you tried telling them that $\frac{1}{3} = 0.\overline{3}$? From here on a childish, but pretty intuitive explanation would be that $3\frac{1}{3} = 1 = 3 0.\overline{3} = 0.\overline{9}$ – Markus Peschl Feb 25 '18 at 21:06
  • If they buy $\frac 13=.\overline 3$ then they should buy this. Or, if they agree that $.\overline 9$ defines a real number $x$ then $10x=x+9$ might be convincing. – lulu Feb 25 '18 at 21:07
  • Just a thought, what about Zeno's paradox, is motion possible ? Do they agree the series converge ? What about the formula for a geometric series ? – Rene Schipperus Feb 25 '18 at 21:08
  • @dxiv they may use Chaitin's constant as a counter example :) – rtybase Feb 25 '18 at 21:11
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    @rtybase Don't give them such ideas ;-) I'd expect their answer would rather be something like $,0.\overline{0}1,$ which may be somewhat easier to argue. – dxiv Feb 25 '18 at 21:14
  • @Pedro, I know all proofs of that , my question is to seek for a simple method to convince students and teachers of my country that 0.99..=1 really it is equality and dosn't approximation , I have edited the question adding some necessary thing – zeraoulia rafik Feb 25 '18 at 21:27
  • @zeraouliarafik Ok, I retracted the vote to close. – Pedro Feb 25 '18 at 21:30
  • Its really frustrating isnt it ? I am surprised that you are having to convince math teachers of this, not a good sign. Its hard to answer such a question since we do not know the details of the situation. I can only say for myself that I try to understand as fully as possible the position of the other person, what do they think an infinite decimal is what does it represent, etc. After you understand their thinking it might be easier to formulate an effective argument. – Rene Schipperus Feb 25 '18 at 21:39
  • Thanks for that , probably i missed to show them the mathematical notion of infinite decimal what does it represent , but their problems if you try to show any thing using limit for infty they w'd juge that is an approach and approximation no more – zeraoulia rafik Feb 25 '18 at 21:43
  • The idea here is that all the finite stages are only approximations but the infinite sum becomes exact. – Rene Schipperus Feb 25 '18 at 21:46
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    @zeraouliarafik as per my first comment, ask them to find a number between $0.\overline{9}$ and $1$, change the research vector so that you won't have to convince them, but rather they will have to convince you. Offer a monetary prise eventually ... – rtybase Feb 25 '18 at 21:53
  • @rtybase, What i understand in your comment that every real number present center of opned interval as the order relation of Real numbers – zeraoulia rafik Feb 25 '18 at 21:56
  • I was having the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ in my mind :) ... but that's not the point. The point is that they have to care about it, not you. – rtybase Feb 25 '18 at 22:00
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    This might be better suited for Math Educators SE. – gen-ℤ ready to perish Feb 25 '18 at 22:03
  • The mathematical content appears to be a duplicate of https://math.stackexchange.com/questions/11/is-it-true-that-0-999999999-dots-1?noredirect=1&lq=1. The distinction with pedagogical content does not seem particularly well-fitted here. I am tempted to close this question as a duplicate. – davidlowryduda Feb 26 '18 at 00:10

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You can use that : let $x=0.\bar{9}$, then $$10x= 9.\bar9= 9+0.\bar{9}=9+x$$ so $$10x-9x=9$$ then $$ 9x=9 $$ and $$ x=1$$ Unless they refuse basics arithmetic operations... (at list they'll have a hard time to counter-attack)

Netchaiev
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  • Yeah, where is the error ? Which step is wrong ? – Rene Schipperus Feb 25 '18 at 21:11
  • @ReneSchipperus: you tell me ... :) – Netchaiev Feb 25 '18 at 21:12
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    @ReneSchipperus How do you know you can multiply both sides of your first equation by 10? The field axioms only give you a FINITE distributive law. In order to prove that you can term-by-term multiply a convergent infinite series, you already need to know what a convergent infinite series is ... by which time you would have no problem believing that .999... = 1. So this is at best a handwavy heuristic "proof" but not really a proof from first principles, since it assumes facts that are more sophisticated than the thing it claims to prove. – user4894 Feb 25 '18 at 21:14
  • @Netchaiev "Unless they refuse basics arithmetic operations." -- What basic arithmetic operation allows you to multiply an infinite series term-by-term? If you check the field axioms you will see that there is no such operation. – user4894 Feb 25 '18 at 21:15
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    @user4894 Multiplication by $10$ shifts the decimal point. A widely accepted rule. – Rene Schipperus Feb 25 '18 at 21:20
  • @ReneSchipperus "Widely accepted" by people who have never seen the real number axioms and who don't understand how much work goes into properly defining the limit of an infinite series. You really don't seem to be tracking this point at all. Go ahead and look up the field axioms and in particular the distributive law. You'll see it only pertains to finite sums. It needs to be proved for infinite sums, after you have rigorously defined the real numbers and infinite sequence and sums of real numbers. – user4894 Feb 25 '18 at 21:25
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    @user4894 And those are exactly the people OP is attempting to convince. – Rene Schipperus Feb 25 '18 at 21:26
  • @user4894 : if the people we are talking about were well-taught enough in mathematics to point out the multiplication of an infinite series, they would sum this $9\times\sum 10^n$ and find out themselves in a blink that it is equal to 1. – Netchaiev Feb 25 '18 at 21:29
  • @Netchaiev I certainly agree that the real numbers are not taught very well until one takes a course in real analysis. Prior to that, people are susceptible to thinking that there's an inherent infinitary distributive law, which of course there isn't until the proper foundation is built. – user4894 Feb 25 '18 at 21:32
  • According to the edit in the question, the OP knows this proof. It seems it is not what the OP is looking for. – Pedro Feb 25 '18 at 21:33
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    @user4894 Yes, indeed. My point was to provide a quick way were a non-mathematician person won't be able to find a solid counter-argument. – Netchaiev Feb 25 '18 at 21:35
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You can tell them decimals are really just a convenient way to represent a series (after all in the decimal 0.123 we call the position of 1 the tenths place, the position of 2 the hundredths place, and so on). Now assure them that the the series $\sum_{n=1}^\infty \frac{9}{10^n} = \frac{9}{1 - 1/10} -9 = 1$.

Jonathan
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The major underlying issues here, IMHO, are the ideas of a limit, infinity, the infinitesimal and students' intuition.

$.\overline 9 = .9 + .09 + .009 + ...$

is an infinite series whose sum is the limit of the sequence of partial sums (as was stated in your link to standard proofs above. Guess you have to get them over that one, too :)

So, when describing limits to students for the first time I tell a story:

"I am standing here in this room a certain distance from the wall and I play a little game. I will cut down the distance to myself and the wall by one-half in successive turns."

[demonstrate]

When I get very close to the wall I ask (naturally) "Will I ever get there?" But as soon as the chorus responds a resounding: "No!" I also ask "does this wall exist?" Maybe even pounding on it a time or two.

And I go on to tell a little bit about the sometimes tragic life and times of Georg Cantor (The natural numbers can be placed into 1-1 correspondence with the integers (et. al.), the mathematical community's reception of transfinite arithmetic, sanitoria ...)

Or, the slope of the tangent line to a curve at a point is the limit of the slopes of the secant lines: "Does the tangent line exist? If I can get as close as I please to it, then I have found it."

"There are more than one encounters with infinity ..." I say, "... and the closer you get to infinity, the close you get to madness."

At some point, mathematical maturity has to kick in. When I was in 8th grade I simply couldn't wrap my head around the idea that an infinite series could have a finite sum. Xeno and I were simpatico. It took a demotion out of the honor's class and a couple of years to come around. Readiness is an issue, as well.

Quoting Cauchy: "When the values successively attributed to a variable approach indefinitely to a fixed number, in a manner so as to end by differing from it by as little as on wishes [my italics], this last is called the limit of all the others." -- from Dunham's The Calculus Gallery.

Hope some of that helped.

estragon
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