First line: "let n = 0.99999999..."

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To the regular visitor of internet bulletin boards it is clear that it's an excellent idea your parents get to choose your real name.

It's just a label we assign to the number denoted by "0.999...", to make the steps in the proof clearer. You could do the same without using a variable.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Dear me. ops: I better go get that second cup of coffee now.

It's always the obvious stuff that trips me up.

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"The oxen are slow, but the earth is patient."

I see nothing wrong with it. Of course, you could say that it "begs the question", in a sense. It assumes that the expression "0.999..." represents a unique real number, and that it is defined in such a way that the usual properties of real numbers that we all know and love remain valid. But then, why define it in any other way?

If we're very picky, then the first proof also assumes some knowledge of calculus, since infinite decimals like 0.333... can only be rigorously defined as limits. However, most people accept such expressions intuitively without protest (at least until we tell them that 0.999...=1 :wink.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

There is NO Rule 6!

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Everything I need to know I learned through Googling.

Perhaps this is where we are going wrong? :wink:

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Brian's mother: He's not the Messiah. He's a very naughty boy!

I don't think it's a rigorous proof. How do we know that 10x0.9999.... = 9.9999.... ? Intuitively it looks right, but how would you actually do the calculation?

10xn means n+n+n+n+n+n+n+n+n+n. So 10x0.9999.... means:

0.9999....
0.9999....
0.9999....
0.9999....
0.9999....
0.9999....
0.9999....
0.9999....
0.9999....
0.9999....
__________
????????????

Since there's no right-hand column in which to start your addition, you can't even begin to do the sum.

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- Learn a lot teaching others.

ToSeek: I beat you!

In the summation part, one must be careful how one simplifies the series:

• 9 * ((1/10) / (1 - 1/10))

• 9 * ((1/10) / (9/10))

• 9 * 1/9

At this point you can either say

• 9/9 = 1

which is cool,

or

• 9 * 0.111111111... = 0.999999999...

and you're right back where you started! ](*,)

Fred

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"For shame, gentlemen, pack your evidence a little better against another time."
-- John Dryden, "The Vindication of The Duke of Guise" 1684

You don't have to do addition from right to left. That's just a convenience. You can go in any order you choose because of the associative property.

Also, I'm not sure that multiplication needs to be defined as a series of additions.

That said, it's not a rigorous proof, just a demonstration.

Eroica, multiplying by ten moves each digit one place to the left (it's one of those nice "properties of the real numbers we all know and love" :wink. [Edit: See Frog march's post below.]

Edited to simplify and correct.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

The 9 in 0.9 is in the tenths column and so represents 9/10(nine tenths) and so when multiplied by 10 becomes 9.

The 9 in 0.09 is in the hundredths column and so represents 9/100(nine hundredths) and so when multiplied by 10 becomes 9/10 ie 0.9

.
.
.
.
.
etc


every thing just moves left by one place.

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Brian's mother: He's not the Messiah. He's a very naughty boy!

But that in itself implies that .999...=1.

What these explanations really show is that .999... must be 1 as long as .111...=1/9, .333...=1/3, etc.

Actually, Nowhere Man, you just cleared it up for me.

Isn't it true that 9 * 1/9 = 1? and isn't 1/9 = .111111...... ?

Well, there you go. I already figured it equalled 1, but sometimes you got that little voice in your head that says : wait, you know its wrong!

Like my test I just took yesterday, how I got the right answer but then thought on it for the remainder of the test time, changed the answer, and ended up getting it wrong.....

kinda sucks.

Edit: ugg, Tomba you beat me to a reply to Fred's comment......

Which is the question that started the whole thing.

It works for me. Some people have trouble with infinity, as in the infinite 9s after the decimal point. I can't picture infinity in my mind, but I can still work with it as a concept or tool.

Fred

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"For shame, gentlemen, pack your evidence a little better against another time."
-- John Dryden, "The Vindication of The Duke of Guise" 1684

Right, and it's exactly what the demonstration was trying to show. I'm saying that what you identified is not a problem, but another way of showing that .999...=1.

Which leads to the trivial conclusion 0.999... = 0.999...

Which proves what we wished to prove.

No problem there. 8)

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Yeah, the first one is definitely a non-rigorous way, but I always thought it was quite elegant and simple. The second one is rooted in more complex mathematics, but is certainly rigorous provided that you accept what calculus has shown about infinite series and things like that.

I don't know if that IS calculus!!!




If you want another problem, try proving whether or not prime numbers ever end, or do they just keep finding higher and higher prime numbers, the more powerful computers get...?




8)



Ok, I'm wrong again....!

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Brian's mother: He's not the Messiah. He's a very naughty boy!

The infinitude of primes has been known for thousands of years. And its an easy proof. In general, for any positive integer n (prime or otherwise), the prime factors of n!+1 are all > n.

Examples:

n=5, n!+1 = 121, Factors are 11,11
n = 6, n!+1 = 721, factors are 7,103
n=10, n!+1 = 39916801, prime
n = 12 , n!+1 = 479001601 , factors are 13,13,2834329


If the primes are finite, then there is a largest one. Call it p. Then p!+1 is not divisible by any number <= p. Therefore the prime factors of p!+1 are all > p, which contradicts the notion that p is the largest prime.

Try to prove or disprove that R and R^2 and C are the same size, were R is the real number line, R^2 is the Plane of all real numbers, and C is the plane of complex numbers. Prove the set R, R^2, and C are all the same size.

Then prove, That for all n,x,y,m size of C^n = size C^x = size R^y = size R^m where n,x,y,m can be any number.

Why?