The problem of ten divided by three

Question

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Does .99999… = 1?

I was thinking about this the other day...

if 10/3 = 3.33333... (series)

why doesn't 3.333... * 3 = 10 it can never be 10 it's always almost 10 or 9.9999... (infinity)

I have read about this but then no one has an answer yet we all accept the fact that it is true when the statement is fundamentally false.

I don't think so, 10=9.999 is an ignorant answer to say i give up thinking about it and it must be this.... If I had a £10 note split it equally in 3 then someone must at some point get a bit extra 1p because for all we know even if you split the 1p in 3 you get a 3.33333 for all we know splitting an atom in 3 may not also be the end of it... — Val, Jan 24 '13 at 21:27
@Val and Indeed, there are theories of mathematics (particularly hyperreal numbers) where 9.999... is INFINITESMALY smaller than 10. Which means, the difference is smaller than any real number. What we mean when we say $10=9.9999...$ is $St(10)=St(9.\overline{9})$ — CBenni, Jan 24 '13 at 21:50
@Val $8.999...=9$. This is true. But why would this make $9.00000=9.1$? — CBenni, Jan 24 '13 at 21:51
@CBenni same reason why 8.8888 would make 8.9 eventually if I can explain it it would be 8.89999... = 8.9 but you can see what I mean if you assume they are the same — Val, Jan 24 '13 at 21:57
NO! Please do read and understand my comments. $8.9-8.8=0.1$, $8.9-8.88=0.02$, $8.9-8.888888888888=0,011111111112$, ... — CBenni, Jan 24 '13 at 22:00
I think, the problem is that not 2 people can agree on anything, thats why the safest option is to say 9.9999... = 10, I was trying to say, 8.899999 = 8.9, just as 9.9999 = 10 — Val, Jan 24 '13 at 22:05

score 4 · Answer 1 · answered Jan 24 '13 at 21:22

4

If $9.9999...$ and $10$ are not the same number, then name a number between them.

answered Jan 24 '13 at 21:22

Thomas Andrews

1

Doesnt mean they are the same ;) (Of course it does, but that would have to be shown at least) – CBenni Jan 24 '13 at 21:24
2

But of course. But almost everybody knows two unequal numbers have a number strictly between them, and those that don't can be shown it fairly quickly using elementary inequalities. Better than trying to explain limits to someone who likely has a pre-calculus level. @CBenni – Thomas Andrews Jan 24 '13 at 22:52

score 3 · Answer 2 · answered Jan 24 '13 at 21:23

3

When we say that $\frac{10}{3} = 3.333\dots$

We have it like this:

$$ \text{Let } p = 3.3333\dots $$

$$10p = 33.33333 \dots$$ or

$$ 9p = 30 $$ or

$$ p = \frac{30}{9} = \frac{10}{3} $$ _{That's how I've always convinced myself of the proof. :|}

answered Jan 24 '13 at 21:23

hjpotter92

I believe since we can't never have truly 0 (nothing), something must give way, e.g: 3.3333 * 3.00000=10 maybe because 0 may not mean absolutely 0 if that makes sense? there must be one point on the serious where something must give way? – Val Jan 24 '13 at 21:32

score 0 · Answer 3 · answered Jan 24 '13 at 21:23

0

$10/3=3.\overline{3}$

$3\cdot 3.\overline{3}=9.\overline{9}=9+\dfrac{9}{9}=10$

This should answer your question pretty much.

answered Jan 24 '13 at 21:23

CBenni

3 Answers3