yes 9.999.... = 10

numen wrote:

If 9.999... is not equal 10, you should be able to find a number between the two, but you can't.

That is an excellent way for me to understand it. Thank you so much.
(Only thing now is I have to admit I was wrong and he was right... )

Another way:
if
c=3.333...
then
10c=33.333...=30+c
So 10c = 30+c
9c=30;
c=30/9=10/3;
Let d=6.666...
Then
10d=66.66...= 60+d
9d=60
d=60/9=20/3;
Then c+d=10/3+20/3=30/3=10.
[edited]

Last edited by krassi_holmz (2006-08-22 05:22:52)

You could also use a system other than decimal.   

3.333.. = 3 1/3 (base 10) = 10.1 (base 3)
6.666.. = 6 2/3 (base 10) = 20.2 (base 3)

In base 3:
10.1
20.2
====
101.0

101 (base 3) = 10 (base 10).

I think they are different.
Also I think 3.333... times ten is not 33.333... because there is a zero on the end when you times by 10, so it's not infinite 3's anymore.
So I think 9.999... is probably not 10, but perhaps it is, but I doubt it.
Because infinity is not really infinity, do you think?

John, if that's the case, then tell me, what is ∞ - 1? Because instead of ∞ 3's after the decimal place in 3.333... × 10 there will be "only" ∞ - 1, in your view of the problem. Krassi's display may leave room for ambiguity, but Numen and I's posts have rather clear methods/reasons which leave little to be debated.

Certain numbers, like 1/3 can't be expressed 100% accurately in the decimal system.

...with a finite number of digits.

Actually, any rational number can be expressed with a finite number of decimal digits in some base.  The proof is fairly straightforward, anyone want to try it?

John E. Franklin wrote:

there is a zero on the end

The flaw there is that as there are infinite 3's, there is no end.

I vote with the majority, 9.999... = 10.

This is what I like about this forum ... you can express a different idea and people will discuss it nicely.

As it should be, really, because wise people know that ideas are our servants, not our masters.

And there's also the other example:

x = 0.999...
10x = 9.999...
9x = 9
x = 1

Which means that 9x + x = 9 + x, which is true, then concluding that 9x = 9.

I still say that 3.333... + 6.666... = 9.999..., though.

Last edited by Devanté (2006-08-22 22:06:45)

Hi Shina;

That set contains a non linear equation. Probably there are no algebraic methods to solve it. There is a much wider branch of math for solving such systems than algebra. It is called numerical analysis and requires a computer to do.

The first method would be the graphical method.

I'd have to disagree with most of what people are saying here, reason being is because to me it sounds like most people are assuming 3.3 recurring = 3 1/3, however, what if 3.3 recurring were just that?  I'll agree with what most people are saying if you meant that 3.3 recurring was SUPPOSED to equal 3 1/3, although I can't because it doesn't.  3 1/3 expressed as a decimal equals 3.3 recurring because you can never get to the end, because ultimately it can not be accurately defined as a decimal.  So in turn, I would argue that using 3.3 recurring + 6.6 recurring = 9.9 recurring not 10, unless 3.3 recurring was meant to be 3 1/3 in the beginning.  The problem I'm having is what you originally meant, because if you meant 3.3 recurring as 3 1/3, then what I'm saying here is wrong, but looking at 3.3 recurring by itself, I'd disagree with what most people are saying.