Decimal expansion question

Al-Allo · 2014-06-24 08:47:16

I was reading something which I found really special. It goes like this : Imagine we have a line with unity division (0,1,2,etc.) Now, we have a point on this line. The point can be on a point of division or it can be contained between two points of division. If it isn't on a point of division, then we can continue cutting the line into 10 equal pieces. (The first piece being 0, the second 1, the third 2,...) So, if my point was between 1 and 2, I'll have 1.0,1.1,etc. We continue this cutting until our point arrives at a point of division.If this arrives, we have two choices : We can pick the interval on the left or the interval on the right. And we can continue this way at infinity.

We also have the following formulae to express our numbers : g+a/10+a2/100+a3/1000+... g being an integer and a being the number of the piece which contains our point.

Now, my question concerns this :

http://i.imgur.com/RlKuUG9.jpg

How can it be possible that we can have 1/4=0.2499... and 1/4=0.25000... Also, for this idea to be valid, must it necessarily be an infinite decimal expansion or I could have the following 1/4=0.249 and 1/4=0.250 ?

Thank you !

anonimnystefy · 2014-06-24 08:58:50

That is a know fact. The usual discussion is whether 1=0.999... The answer is yes and there are much simpler proofs than the one given in that image. The proof in that image is, also, not a very rigorous proof, but it does provide a nice way to visualise both of the decimal expansions.

Al-Allo · 2014-06-24 09:05:33

Well, I gave a resume of the first part, if you want, I can give you the whole.

Al-Allo · 2014-06-24 09:06:55

Also, must the decimal expansion of 0.249 be an infinite one so that we can say that it is equal to 0.25 ??? Couldn't we have 0.249=0.25

anonimnystefy · 2014-06-24 09:07:51

0.249 is not equal to 0.25. 0.24999... is.

Al-Allo · 2014-06-24 09:13:58

For a certain reason, I find it weird. So, we can't see that 0.249..=0.25 with the example of the line subdivision because each subdivision point represents a finite decimal. Or am I wrong ? For example, If I have 0.24999 and 0.25000 on my line, I couldn't say that they're equal.... I would need to imagine an infinite process of subdivision of a line to establish the true equality ?

anonimnystefy · 2014-06-24 09:16:23

Well, basically, yes. You need infinite "cuts" to "reach" 0.24999...

For a nice proof of that fact, look at Joel's answer.

Al-Allo · 2014-06-24 09:18:49

Joel's answer ?What answer ? Who's Joel?

Al-Allo · 2014-06-24 09:19:50

Oh you mean on stack exchange, right ???

anonimnystefy · 2014-06-24 09:20:51

Yes. The first proof he wrote is commonly used to show the equality. The second proof is a bit more rigorous.

Last edited by anonimnystefy (2014-06-24 09:21:21)

Math Is Fun Forum

#1 2014-06-24 08:47:16

Decimal expansion question

#2 2014-06-24 08:58:50

Re: Decimal expansion question

#3 2014-06-24 09:05:33

Re: Decimal expansion question

#4 2014-06-24 09:06:55

Re: Decimal expansion question

#5 2014-06-24 09:07:51

Re: Decimal expansion question

#6 2014-06-24 09:13:58

Re: Decimal expansion question

#7 2014-06-24 09:16:23

Re: Decimal expansion question

#8 2014-06-24 09:18:49

Re: Decimal expansion question

#9 2014-06-24 09:19:50

Re: Decimal expansion question

#10 2014-06-24 09:20:51

Re: Decimal expansion question

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