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Is this proof too hard to understand?
There is some doubt on whether 1=0.999... (continues). I would like to hear people's opinions about the following proof. If 1>0.9999..., then 1-0.99... is a number greater than zero, let
a=1-0.999..
Then we see that a must be smaller than 1/10, because 1/10= 0.1 and
0.999... + 0.1=1.09999 >1. Similarly a<1/100 as
0.999...+0.01=1.00999..>1,
and so on... So we see that a<1/10, a<1/100, a<1/1000,.... . On the other hand we also see that as a is non-zero we can perfectly talk about 1/a which must be greater than 10, 100, 1000,... (just by inverting) and therefore any whole number. But there is no such real number which is greater than any other number, hence we get a contradiction. Therefore a can't be anything other than zero and so 0.9999.... has to be equal to 1 although it looks funny and different.
BTW I am not wondering if this proof or the statement is correct or not. There is no real controversy among mathematicians about this at all. Just wondering how difficult it is.
rshaunders: You answered my question in the negative sense. As it appears when you read the proof you somehow got the impression that I claimed the existence of a number which is greater than any other. That is not the case please re-read. As for your limit concern, 0.9999...=1 is to say that the limit of the sequence 0.9, 0.99,0.999 is one. The limit of that sequence and 0.9999... are the same.
Tom: You are still claiming that
1>0.9,
1>0.99,
1>0.999,
...
being true and 1>0.999... being true are the same thing. Sorry but that bird won't fly. Anyway this question is not for you, it is for people who are willing to read a proof and try to understand it. You already have your religion.
Tom: You are avoiding the gap in your proof very conveniently for the millionth time. You can't show 1>0.9999...,
by showing
1>0.9,
1>0.99,
1>0.999,
...
and so on. This is the 4th time I am telling you this. Go read any elementary logic book of your liking if you don't believe me.
Well this is the last. For the sake of your beloved "significant figure" method you are willing to add numbers which are greater than all the integers in your real number system, as well as those which are smaller than 1/n for any n. Then you lecture about infinity not being attainable and everything being an approximation and so on. You even deny convergence. Sorry to upset you but academics are neither stupid nor in a conspiracy to make people believe in nonsense. And talking to you is no different than talking to a religious fundemantalist who would repeat his religion's teachings like a parrot.
Liar? If there were a slightest bit of controversy wouldn't there be a single established mathematician who thinks like you? Are they all morons? Are you the single smart person on earth? Even Evolution, Relativity, Big Bang and etc. are more controversial than this. I dare you to find me a single established mathematician who thinks like you.
4 Answers
- Anonymous1 decade agoFavorite Answer
Your proof is valid, and fairly easy to understand. Here's how I would write it:
Let a=1-0.999... and so 0.999...+a=1. So a<1/10, a<1/100, a<1/1000,... and in general a<(1/10)^n for all natural numbers n, and hence a<=0. Similarly a>(-1/10)^n for all natural numbers n, and so a>=0. Therefore a=0 and 0.999...=1.
The 1/a stuff is of course true, but it's silly to talk about it - and you're adding in unnecessary trouble with 1/0 cases.
- Anonymous1 decade ago
The limit of the sequence 0.9, 0.99, 0.999, ... is 1 - I don't have a problem with this either. I have a problem when you try to say that the sum of 9/10+9/100+9/1000+... is equal to 1. It is in fact never equal to 1 but always LESS. Thus 0.999... is ALWAYS LESS than 1. The limit of the sum 9/10+9/100+9/1000+... is 1. The actual sum is ALWAYS LESS.
Oh and yes, I am still claiming that 1 > 0.999...
I don't know what made you think otherwise?!
As for religion, the following is meant for the readers, not you firat:
Academics will tell you that the limit of an infinite sum is the value of the sum. However, they are at a complete loss when you ask them why they do not compare all numbers in the same way. As an example: If we compare 3.145 with some more exact value of pi, then we start with the most significant digit and compare until we find a smaller digit. From this we deduce which is larger. When you ask academics why they don't do the same for 0.999... and 1, they have no answer. Please, if you will not compare pi, e and other irrational numbers in the same way, then don't tell me that 0.999... = 1! The method of comparison does not matter - you are comparing partial sums in the case of pi and 3.145 but in the case of 0.999... and 1, you are comparing the limit of a series with the number 1. How inconsistent. firatc, I don't need to convince anyone. You are the one who needs to do the convincing.
You say there is no controversy? What a liar!
Just search and see how successful your teaching has been. All students question this lie every time it is presented because it is not only intuitively incorrect but it is mathematically incorrect!
Firatc says: "For the sake of your beloved "significant figure" method you are willing to add numbers which are greater than all the integers in your real number system, as well as those which are smaller than 1/n for any n." My response: he must be smoking pot.
You are trying to use the Archimedean principle incorrectly. You can only use the Archimedean principle when you know what kind of number you are dealing with. In other words you need to know the full extent of a given number. You don't know what 0.999... is.
Check the Wiki archives to see how all there is no contradiction with the Archimedean property.
"Then you lecture about infinity not being attainable and everything being an approximation and so on." You got this one right!
"You even deny convergence." I have not denied convergence anywhere! More pot for you?
"Sorry to upset you but academics are neither stupid nor in a conspiracy to make people believe in nonsense." - I don't ever recall using the word 'conspiracy' but as for the word 'stupid' - oh YES! And I stand by it. Academics today are stupid beyond belief!
"And talking to you is no different than talking to a religious fundemantalist who would repeat his religion's teachings like a parrot." - I am sorry you think this way but truth be told, I am not surprised. In fact I hate every religion with every fibre of my being. As for a parrot? Hmmm, makes me think of how you have been taught...
I thought that comment would be your last? There are many mathematicians who think like me. The morons are the ones who argue the way you do. Cheers!
- Anonymous1 decade ago
Does not follow. Although there is no finite number which is greater than any other number, you are at liberty to pick any number you please and I can give you back a bigger one by simply adding 1 to it. The bottom line on this is:
0.999999.... < 1.000000, but it approaches it as a limit.
- Anonymous1 decade ago
Zero isn't just a number... some people can be zero's too.