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**lizard****Member**- Registered: 2006-08-20
- Posts: 2

Hello. I'm new to the boards and I'd be so grateful if someone could answer this question.

My boyf and I were arguing last night about whether 3.3 recurring + 6.6 recurring is 9.9 recurring or 10.

My boyf argued that 9.9recurring = 10 but I do not accept that this is true.

I studied Statistics at a UK university (I'm English) and he studied advance calculus at Berkley!!

Please help. Thank you.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,654

A Big Welcome!

In honour of your Statistics studes we could have a survey:

"Excuse me Sir/Madam do you believe that 9.999... = 10?"

I say Yes (because the "..." implies infinity).

Anyone else?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

yes 9.999.... = 10

The Beginning Of All Things To End.

The End Of All Things To Come.

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**numen****Member**- Registered: 2006-05-03
- Posts: 115

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

Bang postponed. Not big enough. Reboot.

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**lizard****Member**- Registered: 2006-08-20
- Posts: 2

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... )

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

Numen's view is great. You can also say that 3.333... = 3 + 1/3 = 10/3 and 6.666... = 6 + 2/3 = 20/3, and we have

3.333... + 6.666... = 9.999...

but equivalently

10/3 + 20/3 = 30/3 = 10,

so 9.999... = 10.

Another way to see that 9.999... = 10 is to use a geometric series like so:

*Last edited by Zhylliolom (2006-08-21 04:12:54)*

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

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 (2006-08-21 07:22:52)*

IPBLE: Increasing Performance By Lowering Expectations.

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

krassi_holmz wrote:

30/3=3.

*cough* 10

The Beginning Of All Things To End.

The End Of All Things To Come.

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**pi man****Member**- Registered: 2006-07-06
- Posts: 251

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).

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,654

Excellent! So far the survey yields 100% "Yes" votes. But the sample size is too small to draw any conclusions

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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?

**igloo** **myrtilles** **fourmis**

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,654

Ho boy, the conversation is sure gonna take off now.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

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.

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**pi man****Member**- Registered: 2006-07-06
- Posts: 251

The reason that I brought up the whole Base 3 representation is to show it is just a "limitation" of our decimal system. Certain numbers, like 1/3 can't be expressed 100% accurately in the decimal system. But it can be expressed exactly in base 3. .33333... (base 10) = 1/3 = .1 (base 3). So for cases like these, switching to a different base can be helpful.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

If q is rational, then q=10 in base q :):)

IPBLE: Increasing Performance By Lowering Expectations.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

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.

Why did the vector cross the road?

It wanted to be normal.

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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

The closest to 1/3 is 0.333...

And 2/3 = 4/6 = 0.666...

And 3/3 = 6/6 = 1 with no recurring digits.

1/3 expressed as a fraction is different than expressed as a decimal. At least, I think.

So I don't think 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333...

Because 0.333... would equal 0.999... but since there is an infinite number of digits there, 0.999... can't equal 1?

3/3 = 1

3/3 ≠ 0.999...

Because 0.999...is not a whole number. 1, however, is.

*Last edited by Devanté (2006-08-21 21:46:17)*

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,654

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.

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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

Interesting...I've never thought about it that way before.

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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

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 00:06:45)*

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**Shina****Guest**

This is my first time here and i really like the way you guys respond to questions. Am a novice and i'd like you to pls solve this equation for me.

x + y = 5....(1)

x^y + y^x = 17...(2)

i know the answer is 2 and 3 but i just need to know how to arrive at it.

Thanks

**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**Au101****Member**- Registered: 2010-12-01
- Posts: 353

Hi,

Just about every way to show that 3.333... + 6.6... = 10 has been given, although I think that bobbym's suggestion of analysis is a good one. I believe, however, that it will yield the same answer. I wonder if it is also possible to think of it in terms of limits:

I'm fairly sure that that's true, however, I'm not sure if that's the best and most illustrative way of using limits, but somebody more knowledgeable might be able to help. Perhaps:

Would be better?

What I can point out, though, is that multiplication by ten doesn't mean that there will be a zero on the end of the number, e.g. 33.3 times 10 = 333. Since we use base ten, multiplication by ten simply means that we move everything up one place value - one ten. If, then, we are dealing with an integer, then everything gets moved along one place value and since the last digit is in the 1s column, we move it to the 10s and add a 0. If we have a decimal, however, then we move everything along, e.g. from the hundredths up, which will move our last digit into the tenths - no need for a zero. Since 3.3... is a non-terminating decimal we will never get to the end and so we will never have to worry about putting a zero there.

Thanks

*Last edited by Au101 (2011-09-22 00:01:49)*

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**Calligar****Member**- Registered: 2011-09-24
- Posts: 272

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.

There are always other variables. -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -Aristotle

Everything makes sense, one only needs to figure out how. -[unknown]

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