【1 paradox】Why 0.999... is not equal to 1? 【1 paradox】Why 0.999... is not equal to 1?Written in 2012 The current mathematic theory tells us, 1>0.9, 1>0.99, 1>0.999, ..., but at last it says 1=0.999..., a negation of itself. So it is totally a paradox, name it as 【1 paradox】. You see this is a mathematic problem at first, actually it is a philosophic problem. Then we can resolve it. Because math is a incomplete theory, only philosophy could be a complete one. The answer is that 0.999... is not equal to 1. Because of these reasons: 1. The infinite world and finite world.We live in one world but made up of two parts: the infinite part and the finite part. But we develop our mathematic system based on the finite part, because we never entered into the infinite part. Your attention, God is in it.0.999... is a number in the infinite world, but 1 is a number in the finite world. For example, 1 represents an apple. But then 0.999...? We don't know. That is to say, we can't use a number in the infinite world to plus a number in the finite world. For example, an apple plus an apple, we say it is 1+1=2, we get two apples, but if it is an apple plus a banana, we only can say we get two fruits. The key problem is we don't know what is 0.999..., we can get nothing. So we can't  say 9+0.999...=9.999... or 1, etc.We can use "infinite world" and "finite world" to resolve some of zeno's paradox, too. 2. lim0.999...=1, not 0.999...=1.

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ouppp I forgot the .... at several places

 

Let x be 0.999999...

10x=9.9999999...

now subtract x on both side

10x-x=9.9999999...-0.9999999...

9x=9

There is only one solution to the equation: x=1.

Therefore 0.9999999...=1

 

 

Another question: if 0.9999.... were not 1, there would be a number between those two number. Can you find one ? I don't think so... :-)

 

 

 

 

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Another proof of the fact that 0.9999999... is equal to 1.

Let x be 0.999999...

10x=9.9999999

now subtract x on both side

10x-x=9.9999999-0.9999999

9x-9

There is only one solution to the equation: x=1.

Therfore 0.9999999=1

 

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This is the world of real numbers.  What you describe with your apples is the realm of integers, and mathematically, these are different things.

What you are describing with the repeating fraction is an asymptotic limit, where you will get closer and closer to convergence, but will never actually reach it.

 Adding more 9's to the repeating decimal only makes the number closer to one; as more are added, the closer the fraction is in value to 1.  It really only ever approximately equals 1, but as an infinite number of 9's are added, in practical terms, it does equal 1, as the minute differences between an infinite number of 9's and an infinite number of 9's minus a single 9 is moot.

 

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Also see this discussion-- http://www.enotes.com/math/discuss/1-0-999-114836.

.999... exactly equals 1. To say otherwise is to posit the existence of a number smaller than any given number -- namely 1-.999.... But there cannot be a smallest number, as you can always divide a number by 2.

There is no number between .9 repeating and 1, so they are the same number.

There are a number of arguments for this: limits, sums of infinite series, etc... If you argue that these arguments all involve infinity, your belief that .9 repeating does not equal 1 involves an infinitesimal -- trading a large infinity for the small.

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In simple and practical terms, the number 0.99999 rounds to 1.   However rounding is just for simplification.  It is still not equal.  The more nines after the decimal point, the closer it gets to 1 until, for all intents and purposes, it basically is 1, but it is not EQUAL to 1.

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