Best Answer:
Hey there... I like these questions... although its been asked a zillion times on here already.

In fact there are many proofs... I will try to show some of them.

Hold on...

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Proof 1 - Sum of fractions

1/3 = 0.3333~

1/3 = 0.3333~

1/3 = 0.3333~

------------------

3/3 = 1 = 0.9999~

This works for other fractions

1/3 = 0.3333~

2/3 = 0.6666~

------------------

3/3 = 1 = 0.9999~

1/9 = 0.1111~

8/9 = 0.8888~

------------------

9/9 = 1 = 0.9999~

Of course, you have to first agree that the fraction and its decimal equivalent are equal. Does 1/3 equal 0.3333~? Thats the same question as the header, isnt it?

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Proof 2 - Conversion to fraction

n = 0.9999~

10•n = 9.9999~

10•n - n = 9•n

10•n - n = 9.9999~ - 0.9999~ = 9.0000~

9•n = 9

n = 1

∴ n = 1 = 0.9999~

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10•n ≠ 9.999~...0

There is no zero affixed to the end, not really. I suppose it is there... but only after the infinitely long, never ending string of nines. So, its meaningless.

As a result, 9.999~ - 0.9999~ does not equal 8.999~...1

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Proof 3 - Infinite Geometric Series

0.9999~ = 0.9 + 0.09 + 0.009 ... =

9•10^-1 + 9•10^-2 + 9•10^-3 + 9•10^-4 ... =

∞

∑ 9 • (1/10)^n

n=1

Theorem - Infinite geometric series can be evaluated:

∞

∑ c • (r)^n = c•r / (1 - r)

n=1

Therefore:

9•(1/10) / (1 - 1/10) = (9/10) / (9/10) = 1

The infinite series that represents 0.9999~ can be simplified to 1

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Proof 4 - Argument from Philosophy - The definition of the real numbers as a continuum

"If two numbers, x and z, are not equal such that x < z, there exists a third number, y, such that x < y < z"

This means that if two numbers are different, there is a number in between them.

If 0.9999~ does not equal 1, what number could possibly exist between them?

No number exists that is greater than 0.9999~ but less than 1.

∴ the two are equal.

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Proof 5 - From Averages

The average of two numbers, m and n, is found by adding them and dividing by 2.

[m + n] / 2

The average, A, is greater than the lesser number and smaller than the greater number

m < A < n

Unless the average is made between a number and itself:

A = a = [ a + a ] / 2

Assume that 0.9999~ < 1

Find the average of 1 and 0.9999~

[ 1 + 0.9999~ ] / 2 =

[ 1.9999~ ] / 2 =

0.9999~

The average is equal to the smaller of the two numbers, A = m. Therefore A = n, also, and thusly m = n.

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Proof 6 - By Arithmetic

Any number, n, minus itself, n - n, will equal zero.

Take 1 and subtract 0.9999~

1 - 0.9999~ = 0.0000~...1 = 0

Therefore, 1 = 0.9999~

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Interesting that 0.000~...1 equals 0, isnt it?

0.000~...1 is as arbitrarily close to 0 as 0.999~ is to 1.

After that never ending infinite string of zeros, there is a one. Meaningless! Of course, this proof relies on you believing that 0.000~...1 = 0

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Proof 7 - As a Limit.

Infinite sums cannot be evaluated as easily as the limit of a finite sum approaching infinity.

0.9999~ = 1 - 0.0000~...1

lim [ 1 - 0.0000000.....001 ] =

n→∞ ...... |__n digits__|

lim [ 1 - 10^-n ] =

n→∞

lim [ 1 ] - lim [ 10^-n ] =

n→∞ ....... n→∞

1 - 0 = 1

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Proof 8 - Comparing Limits

The limit of two functions are equal only if they evaluate to an equal value at the limit, if the value exists

If the limits are equal:

lim [ f(n) ] = lim [ g(n) ]

n→c .......... n→c

Then the values are equal, if they exist:

f(c) = g(c)

The limit of 1 is equal to the limit of the decimal expansion of 0.9999~. As more 9's are affixed, the limit approaches 1

0.9

0.99

0.999

0.9999

...

0.9999...999

...

0.999~

(0.1, 0.01, 0.001, 0.0001, 0.00001, ...., 0.000.....0001)

Because the limits approach the same value, they have the same value:

1 = 0.9999~

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And to answer the second part of your question... about contradictory proofs

Proof 9 - A Fallacious Proof - Lack of contrary evidence

There is absolutely no proof or demonstration that concludes 0.9999~ is less than 1, unless you count shear intuition

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There are in fact many more proofs. And there is absolutely no evidence or proof to the contrary. Any contrary proof Ive ever seen has been fallacious in some way.

When we look at 0.999~ is appears, intuitively, to be less than 1. But that is only because of our conceptualization of the decimal point and the nines that follow it.

I do agree with David H though... I still cant grasp it myself

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Anonymous
· 1 decade ago

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