Apparently 0.999… = 1

Let me just say, wow. I haven't had my brain rattled like this in a while!

A fellow over at another blog made an intriguing post. He argues and claims to "prove" that 0.99999… repeated in fact equals 1. If you are anything like me, your first reaction is that of utter rage. 😛 How can that be? Have my years in math class gone completely to waste? Why was I never informed of this. So I decided to dig a little deeper and see what the guy had to say.

Guess what? I still disagree! He does has some interesting "proofs" though.

Proof 1:
Let x = 0.99999…

10 * x = 9.99999...
-    x = 0.99999...
--------------------
 9 * x = 9
     x = 1

Now as far as this proof is concerned it sure is scary to look at. Here are my thoughts about it.

When you multiply a number by 10 all you're doing is shifting the decimal place, correct? So for example 1.234 * 10 = 12.34. At first there were 3 digits after the decimal, after the multiplication there are only 2…so one less. What is so different here?

0.99999…(infinite number of nines) * 10 = 9.99999…(infinite number of nines – 1 nine)

Now when you go to subtract the original x = 0.99999… from the not multiplied x value, what you have is 10*x has one less nine after the decimal than the original nine.

10 * x = 9.99999...
-    x = 0.99999...
--------------------
 9 * x = 9.00000...9
     x = 1.00000...1

Of course none of this makes sense because I am assuming that even though the number of nines after the decimals is infinite, it is a fixed infinite… The problem is that infinity is a concept and 0.99999…,1 are numbers. In which case we either consider both of our arguments valid or invalid. Because I do not understand how we can be sure that 0.99999… cleanly cancels out with the decimal portion of 9.99999…!

Proof 2:

1/3 = 0.33333...
+
2/3 = 0.66666...
----------------
3/3 = 0.99999...
1 = 0.99999...

Now consider 2.99999…/3, which interestingly enough equals 0.99999… . So what does that mean? If 0.99999… is assumed to be equal to 1, then 2.99999…/3 = 3/3. That in turn means that any number which has an infinite number of trailing nines after the decimal can in fact be always rounded up? In other words, every number has a infinite decimal representation…

Proof 3:

The last proof worth mentioning is that if 0.99999… to be not equal to 1, then we need to find a number which is in between the two. What is between 0.99999… and 1? The answer is seems to be obviously 0.99999… Because you can always find a number that is greater than 0.99999… but less than 1. For example:

0.9   < x < 1, x is 0.99
0.99  < x < 1, x is 0.999
0.999 < x < 1, x is 0.9999

And so on…so what is greater than 0.99999… and less than 1? Well 0.99999… of course.

In reality of course this has nothing to do with the reality. In reality 0.99999… is NOT equal to 1. However 0.99999… behaves like 1 when current math rules and math assumptions are applied.

Now going a step further, if 0.99999… is interchangeable with 1.0 when doing mathematical calculations using Real numbers, then what is the number that can be interchanged with 0.333333… or 0.66666… or 0. What I'm saying is how come non-repeating numbers have substitutes of repeating numbers but not vice-versa.

I find that a post by Michael VanDeMar on a continuation of the original thread follows my logic well.

Saying that .999… is infinitely close to one is indeed saying that it is at the same time discreet from it. This is not just semantics. If you were to follow your argument through, you would then say that below .999… there is also a number infinitely close to it, which is equal therefore equal to it, and continued on infinitely, you would then end up with an infinite series proving 1 equals 0.

Next thing that bothers me, is 0.99999… equal to 0.99999…? They are as concepts but not as numbers. Because how can we possibly have a finite definition of infinity, meaning that 0.99999… cannot directly equal 0.99999… since infinity does not equal infinity and we don't know the exact number of trailing nines, there are infinite… Or maybe I'm far off the field to make such statements.

Anyways my brain already feels twisted up enough, so here are some links of interest:

http://en.wikipedia.org/wiki/Proof_that_0.999…_equals_1
http://mathforum.org/dr.math/faq/faq.0.9999.html
http://www.newton.dep.anl.gov/askasci/math99/math99167.htm
http://qntm.org/pointnine
http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/
http://polymathematics.typepad.com/polymath/2006/06/the_saga_contin.html

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10 Responses to “Apparently 0.999… = 1”

  1. Mindless » Blog Archive » The Monty Hall Problem Says:

    […] A place to write « Apparently 0.999… = 1 The Monty Hall Problem June 21st,2006 […]

    • Jenelie C. Alamban Says:

      For me it’s yes, because 0.9999..can be 1 through rounding off the numbers. If 0.99999… is assumed to be equal to 1, then 2.99999…/3 = 3/3. That in turn means that any number which has an infinite number of trailing nines after the decimal can in fact be always rounded up? In other words, every number has a infinite decimal representation.When the number is less than 5 the preceding number is unchanged and when the the number is more 5 one is added to the preceding number.If a calculation has several steps, it’s generally best to round off at the end after all the steps have been carried out.

      • Bill Says:

        @Jenelle – Your argument is invalid from beginning to end “For me it’s yes, because 0.9999..can be 1 through rounding off the numbers. If 0.99999… is assumed to be equal to 1” If it’s rounded, you can’t use the equals sign -_-

  2. catmint Says:

    Actually it is equal to 1. I don’t like it but it is true. There isn’t a number that is a smudge below 0.9 recurring as that would be 0.999…..(for infinity)…8. It doesn’t exist as a number, you can’t stick something on the end of infinity as infinity has no end.
    This has of course been done to death on various forums, and in the real world maths teachers, professors etc know its true. Its just accepted.
    However I dislike alot of people who share my view as they are crap uni students who think the sun shines in interesting places because they know a couple of proofs and can talk about set theory.

    • Bill Says:

      If it has no end, then how could it, in the end, be 1? it would still just be 0.9 forever, why would one round it up to 1? And Michael VanDeMar also said, If there was an infinitely close number to 1, then there is also an indinetely close number to 0.9 repeated, and thus it goes down from there. 0=1

  3. dave Says:

    “Next thing that bothers me, is 0.99999… equal to 0.99999…? They are as concepts but not as numbers. Because how can we possibly have a finite definition of infinity, meaning that 0.99999… cannot directly equal 0.99999… since infinity does not equal infinity and we don’t know the exact number of trailing nines, there are infinite… Or maybe I’m far off the field to make such statements.”

    If you cannot even trust 0.999… = 0.999… , then you must doubt that 1.000… = 1.000… and you must then doubt for any real number x: x=x. If cannot accept something equals itself than you are incapable of agreeing to anything proven to you because you are both mathematicaly and logicaly lost.

  4. Thule Says:

    Think about thy pi-problem: pi has infinit numbers, right. so, a value, which has an “ever – growing” number is and must be overdimensional.

    example:

    0,1 > 0,13 > 0,136 > 0,1364

    and so on.

    iit’s weight must grow constantly. imagine it as a reversed cylinder.

    so: 0,99999 can NOT be equal to 1 because it’s a whole other type of number.

  5. Thule Says:

    add: (… in behavior.

  6. Random geometry student Says:

    I’m no math major, but ive been studying the field of logic for over 8 years, and it is obvious that 0.9999… MUST equal 1. All of the listed proofs are logical and true. Why shouldn’t 0.9999… equal 1? You must not understand what infinity means, otherwise you would agree.
    @Thule- uhhh i guess 1/3 cant be equal 0.3333… and 14/25 is no longer equal to 56%. What shame, it looks like countless middle school math classes have been taught lies. Oh well, i guess we’ll just have a generation of math retards /sarcasm

  7. Ben P Says:

    Hi,

    Interesting post, thank you.

    I can tell you’re not a mathematician, as you put your idea of ‘reality’ above mathematical proof.
    What mathematicians are interested in is ‘mathematical reality’.

    “Now consider 2.99999…/3, which interestingly enough equals 0.99999…”
    This is of course a true statement, as it is equivalent to 3/3 = 1.
    But I wonder how you did this ‘calculation’?

    Have you thought about the fact that eg 7.0 = 7.000…?

    If I described a car as ‘blue and shiny’ and you described it as ‘having four wheels’ we are not necessarily talking about different objects.


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