How can I convince students and teachers of my country that $0.\overline{9}=1$ and it is not approximation?

Question

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 ?

Answer 1

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)

Answer 2

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 $\sum_{n=1}^\infty \frac{9}{10^n} = \frac{9}{1 - 1/10} -9 = 1$.

Answer 3

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.