Exploring the Beauty of .99 Repeating Equals 1

[raven1]

i accept that .99 repeating equals 1. I read a lot of posts as why it is true and I accept them. ii had a discussion with some people about this and their response is there must be a difference. the basic reason is based on what happens when you divide 1 by 3. they claim 1/3 dose not equal .33 repeating, that even if u go out to infinity there was always be something left over, and with any repeating decimal there will always be something left over . I am not sure how to respond. Part of the problem is they never went that far into math

 

[arildno]

because they chose never to go far into math, you should tell them not to bother about stuff they don't have the competence to discuss.

Alternatively, tell them that in the constuction of the real numbers (for example with the use of the equivalence class concept) it is a trivial, almost definitionally true statement that 0.99..=1

 

[matt grime]

Ask them to define what Real Numbers are. When they can't blind them with the correct definition:

the completion of the rationals

Note there are other equivalent definitions. In anyone of them those two *different symbols* represent the same Real Number. I am capitalizing the 'r' of real to indicate that Real Numbers refers to a specific set of objects, not their hand wavy (understandably) ill informed idea. I don't understand how anyone can actually accept that there is such a thing as 0.9.. in the first place yet not accept this. I suspect it is because they do not get what infinite sums are.

Also ask them if they accept that symbols are just representations of these objects and that it is perfectly possible for two different symbols to represent the same thing, like 1/2 and 3/6.

 

[mathman]

In high school algebra, at some point, the concept of infinite series is introduced. If you understand it, then the idea that .999...=1 or that .333...=1/3 is trivial. Otherwise you can run into fundamental difficulties in undertanding.

 

[ramsey2879]

In high school algebra, at some point, the concept of infinite series is introduced. If you understand it, then the idea that .999...=1 or that .333...=1/3 is trivial. Otherwise you can run into fundamental difficulties in undertanding.

Yeah , let x = .999... then 10 x = 9.999... Since the decimal part is the same infinite series for both x and 10x, it cancels by subtracting. Thus we have 9x = 9 which gives x = 1.
Similarly x = .333... gives 9x = 3 showing that x = 1/3.

Another example is the infinite geometric series
$$\frac{1}{2} \quad \frac{1}{4} \quad \frac{1}{8} \dots$$ which can be shown to equal 1 by letting $$x$$ equal the sum and subtracting $$x$$ from $$2x$$ It is indeed trival if you understand infinite series.

 

[HallsofIvy]

Of course, people who claim 0.999... is not equal to 1 also will not accept that you can do arithmetic on infinite digits like that. There's probably no way of arguing with them- they think of numbers as being their decimal representations rather than just represented by them: 0.999... is a different representation than 1, therefore a different number!

 

[arildno]

I think that the lack of mathematical skill hinders most people from regarding "infinity" as anything else than a whopping big number.

That is, without mathematical expertise, the ideas of infinity (and many others) remain rough and naive (and useless).

Not a particularly revolutionary insight, I guess..

 

[Alkatran]

i accept that .99 repeating equals 1. I read a lot of posts as why it is true and I accept them. ii had a discussion with some people about this and their response is there must be a difference. the basic reason is based on what happens when you divide 1 by 3. they claim 1/3 dose not equal .33 repeating, that even if u go out to infinity there was always be something left over, and with any repeating decimal there will always be something left over . I am not sure how to respond. Part of the problem is they never went that far into math

As soon as they say there's always something left you whip out "Oh ya, well one of the proofs is showing there's nothing left!" like so

$$1 - .999... = 0.0000...1$$
Now, I'm aware of how bad an idea it is to put "1" after an infinite series. But they'll accept this. You really don't want to use the summation sign going from 0 to infinity with non-math people. Anyways, continuing:

$$0.000...1 = 1*10^{-\infty}$$
It helps if you show that .1 = 10^-1, .01 = 10^-2 and so forth to backup this step.

Then explain how X^-Y = 1/x^y and do this:
$$1*10^{-\infty} = \frac{1}{10^\infty}$$

and how 10^infinity = infinity:
$$\frac{1}{10^\infty} = \frac{1}{\infty}$$

and anything divided by infinity is 0:
$$\frac{1}{\infty} = 0$$

so that means:
$$1 - .999... = 0$$

Then adding .999... to both sides you get:
$$1 = .999...$$

 

[moose]

Alkatran, people who do not believe .9999=1 will not believe that 1/infinity = 0.

 

[Alkatran]

Alkatran, people who do not believe .9999=1 will not believe that 1/infinity = 0.

Well then there's no hope.

 

[ramsey2879]

Alkatran, people who do not believe .9999=1 will not believe that 1/infinity = 0.

Then tell them how 1/9 = .111...11
10/9 =1.11...10 implies that
10/9 - 1/9 = 9/9 = 1 = 1.00...(-1) = 1 - 1/infinity and thus 0 = 1/infinity!

If they still don't acept that, say too bad since anyone having success in math accepts this and until you have faith that this is true you will not succeed in math.

 

[Hurkyl]

There's nothing wrong with these latest posts... I just don't like necromancy of .999~ = 1 threads.