【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...

【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.

Expert Answers
enotechris eNotes educator| Certified Educator

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.


embizze eNotes educator| Certified Educator

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.

cosinusix eNotes educator| Certified Educator

ouppp I forgot the .... at several places


Let x be 0.999999...


now subtract x on both side



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... :-)





cosinusix eNotes educator| Certified Educator

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

Let x be 0.999999...


now subtract x on both side



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

Therfore 0.9999999=1


litteacher8 eNotes educator| Certified Educator

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.

mentch | Student

my math teacher proved 1 = 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

mitodoteira | Student

1.  Most people agree that 1/3 = 0.3333333...

2.  Everyone agrees that 3*1/3 = 1

3.  Everyone agrees that 3*0.333... = 0.99999

4.  Therfore, 3*1/3=3*0.333...=0.999...=1

richon | Student

I think that 0.999... is equal to 1.

One third is equal to 0.333..., two thirds is equal to 2 x 0.333... which is then equal to 0.666..., and three thirds is equal to 3 x 0.333... which equals 0.999...

However, three thirds is also known as one as any number divided by itself is 1 except 0.

Therefore, 0.999... must equal to 1.

science321 | Student

O.. Wait.. I juz realise u guys r teachers.. LOL!:)

science321 | Student

LOL. You guys explain it so complicatedly. All I can say is, 0.999 IS 0.999. It can never be equal to 1.

elekzy | Student

I don't get this, can someone explain it again please.

selyrou | Student
In reply to #1: 3.The indeterminate principle. Because of the indeterminate principle, 1/9 is not equal to 0.111.... For example, cut an apple into nine equal parts, then every part of it is 1/9. But if you use different measure tools to measure the volume of every part, it is indeterminate. That is to say, you may find the volume could not exactly be 0.111..., but it would be 0.123, 0.1142, or 0.11425, etc. Now we end a biggest mathematical crisis. But most important is this standpoint tells us, our world is only a sample from a sample space. When you realized this, and that the current probability theory is wrong, when you find the Meta-sample-space, you would be able to create a real AI-system. It will indicate that there must be one God-system in the system, which is the controller. Look our world, there must be one God, as for us, only some robots. Maybe we are in a God's game, WHO KNOWS? More info, three other download points(written in Chinese): yourfilelink.com/get.php?fid=780934 d01.megashares.com/dl/0LZix2o/the end of the world.rar localhostr.com/file/3LtuSLb/the%20end%20of%20the%20world.rar