Philosophy Discussion Forums

I have read a proof that 0.99 recurring is equal to 1.

I do not know enough mathematics to judge whether the proof is valid or not but, if valid, it appears to generate a paradox which, so far as I know, has not been commented on in the forums.

Given that the conclusion 0.999… = 1 is valid, it is easy to prove, in like fashion, that an infinitely small fraction must be equal to zero, thus:

1 - 0.99… = 0.0…1

(where the number of zeroes between the decimal point and the ‘1’ is assumed to be infinite).

But 1 - 0.99… = 1 - 1 = 0; therefore,

0.0…1 = 0

We can see immediately that if 0.0…1 = 0, then -0.0…1 must also be equal to zero. We thus have three values (so far) which, although logically distinguishable, are equally entitled to the name of ‘0’.

Another moment’s thought will suggest that the set of such numbers must be infinite. Since 0.0…2 differs from 0.0…1 by the same magnitude which differentiates the latter from zero, then it too must be equal to zero. And so on ad infinitum.

The set of numbers which are equal to zero is thus infinite, and every member of the set is a logically unique value.

I suppose the paradox does not arise if it can be shown that 0.0…1 (an infinity of zeroes, bounded at one extreme by a decimal point, and at the other by the number 1) is not an allowable number – not a ‘legal’ number, as Microsoft would say. Is this in fact the case?

This paradox is bothering me. Can anyone help me resolve it?

This has a fundamental equivalence to the Sorites paradox (the paradox of the heap), and characteristics of Zeno's paradox (the paradox of half distances). In the paradox of the heap, at what point does the heap become a non-heap as grains of sand are removed? The same as: at what point does the 0.000....1 value become a true 0 or a 1 value in an additive or subtractive method? My answer would be when it actually becomes that value. I think that when we are dealing with numbers, we must take an absolute view of what they actually represent, not what we can attribute to them theoretically.

Thanks for your answer, Spiral Out! I've posted this question on a couple of forums, with no response, I was beginning to think I was talking a foreign language... thanks for throwing me a rope. I'm not conversant with all the elements in your reply (the heap thing, for example) so I will need to research them before replying more fully.

But you raise the issue of the additive/subtractive procedure. That is the red-hot iron which I was reluctant to touch. Is it possible to get from 0.0...01 to 1 by a purely additive process? Georg Cantor might say yes; he was the first to suggest, I think, that 'infinity' is equal to the largest possible natural number; if that is the case, it would be logically possible to get from infinity to 1 by normal arithmetic process.

If it is possible to get from 0.0...01 to 1 by a process of natural addition, then it follows from my argument that all of the natural numbers are equal to zero, which is obviously absurd. Thus 0.99... = 1 is proven false.

I think it's instinctual for Humans to disregard such insignificant measures of quantity (and quality) for reasons of economy. We like to take complex numbers and round them to more comfortable and simpler numbers to better handle the functions of such numbers. If we consider the actual numbers of pi and phi, 3.1415 and 1.618 respectively, and draw them out to 1000 decimal places then what actual functional use does that serve for us? To say that 0.999999999999999999999999 is equal to 1 is accurate enough for functional purposes but not for exact measure or technicality. We could take the value of phi and say it is equal to 1.6(0000) for economical handling, but if used for specific calculations, it would give us an incorrect result.

If you take an "infinitely" insignificant measure and compound it "infinitely" upon itself then you end up with a significant measure. If you stacked 1 micron thick plates atop each other enough times, you would eventually end up with a stack that would reach the moon. A long journey starts with one step.

As "Spiral Out" says, this is essentially Zeno's paradox. It arises because we're dealing with infinities and infinitesimals. It led Newton/Leibniz to invent the Theory of Limits and, as a consequence, differential and integral calculus. Basically, in mathematics, you can talk about something "tending towards" infinity or the sum of an infinite series "tending towards" a particular number. But you can never actually reach infinity.

Sorry. Not very well explained!

The human problem is seeing things, including minuscule, as something separate. Is it a disorder of sorts or practical skill? In Universe nothing is a tiny separate part, like 0.001, because it is part of something bigger and something smaller is attached to it as well. Being marked by humans as something of almost zero, does not make it stop being together with the rest of things and the whole world. I can even say that size is spatial characteristic, quantitative, but it can be qualitative. Now there are two infinities, one for quantity and one for quality, and then one on relationship between quantity and quality, then on longevity of those relationships, etc? 0.001 of human DNA molecule can mean it is zero of a person. While 0.001 of a harmful bacteria can mean full blown epidemic. Why do people have problems with concepts they invented? Because those concepts are not perfect. It got me thinking about bottlenecking and the founder effect and how those change which theories or concepts proliferate in human population. Are we victims of narrowed genetic pools, so we are stuck now on a concept of infinity which is product of our mathematical innovations?

I think 0.0000...infinite....1 is basically equal to zero. The one is after infinite zeroes. That is, 0.000...the zeroes never end, infinite is infinite. A one after never ending zeroes? can such a number exist?

It appears to mean that everything is nothing, and nothing is everything.

Or, to put it another way, infinity minus infinity equals .... whatever you want it to be. The concept of infinity is not used in sciences like physics. In physics, they say that things "tend to infinity" or can be "arbitrarily large", which is not the same thing. If something is "arbitrarily large", it means that, for any given value, it's always possible to make it bigger. That's not the same as being infinite.

Infinity is a mathematical concept. There is no law which says all mathematical concepts correspond to something in the physical world. The fact that some do is useful.

After thinking about it a bit more, I realised that whether “0.0…01” is a genuine or permissible number is actually irrelevant, a mere problem of finding the appropriate terminology, because the fraction to which I'm referring is surely a real one.

The set of decimal fractions bounded by 0 and 1 is an infinite set, the members of which are individually unique and evenly distributed, the interval between any adjacent pair of values being the same everywhere in the set.

Thus there must exist a fraction which bears the same relation to 0 as 0.99… bears to 1, since the beginning of the set is, from the logical point of view, just a mirror image of its end. I called this fraction 0.0…01, but if that offends, we could just as easily call it f; it makes no difference to the original question.

Anyway, thank you all for some interesting and enlightening comments! I’m not a mathematician, but I can understand the magic attraction it has for those who love the subject, and know how to do it well.

Both 0.999999....... and 0.00000.........1 are unreal

Alan: Maths is interesting but it can drive you mad!

If you're interested in these kinds of ideas, like the infinite number of fractions between 0 and 1, and all that, you should look up the work of Georg Cantor and his ideas about the different levels of inifinity that he denotes with the "Aleph" symbol. The basic idea is that some infinities (bizzarely) are bigger than others. And you can say that two infinite sets are of the same size - the same level of infiniteness - if it's possible to establish a one-to-one correspondance between all of their members.

The reason why 0.999....... = 1 is that there is no number between them. Hence they can be considered to be the same number. (1 - 0.99....... = 0)

There is no such number as 0.00.....1 in mathematics, nor for that matter is there 0.0.......2 , so any arguments using such 'numbers' are necassarily not a part of mathematics.

Alan Masterman wrote:I have read a proof that 0.99 recurring is equal to 1.

I do not know enough mathematics to judge whether the proof is valid or not but, if valid, it appears to generate a paradox which, so far as I know, has not been commented on in the forums.

Given that the conclusion 0.999… = 1 is valid, it is easy to prove, in like fashion, that an infinitely small fraction must be equal to zero, thus:

1 - 0.99… = 0.0…1

(where the number of zeroes between the decimal point and the ‘1’ is assumed to be infinite).

There is no such number as "0.0...1". The "..." never ends, so you can't put a "1" at the end because it doesn't exist.

1 - 0.999... = 0

But 1 - 0.99… = 1 - 1 = 0; therefore,

0.0…1 = 0

If there is a "1" at the end, the above equality is not true.

We can see immediately that if 0.0…1 = 0, then -0.0…1 must also be equal to zero. We thus have three values (so far) which, although logically distinguishable, are equally entitled to the name of ‘0’.

You are simply wrong.

Another moment’s thought will suggest that the set of such numbers must be infinite. Since 0.0…2 differs from 0.0…1 by the same magnitude which differentiates the latter from zero, then it too must be equal to zero. And so on ad infinitum.

The set of numbers which are equal to zero is thus infinite, and every member of the set is a logically unique value.

I think the problem is related to the lack of that "moment".

I suppose the paradox does not arise if it can be shown that 0.0…1 (an infinity of zeroes, bounded at one extreme by a decimal point, and at the other by the number 1) is not an allowable number – not a ‘legal’ number, as Microsoft would say. Is this in fact the case?

So, what is an infinite string with a "1" at the end? It's a contradiction. I do not know or care what Micro$oft think.

This paradox is bothering me. Can anyone help me resolve it?

There is no paradox.

Actually there is a simple solution, to your so called paradox. There is no such thing as 0,0......(ad infinity)01, but even if there was, there can be infinitely many ways to label a number and there is no paradox in it. You can say that 3=2.9999999999...., but also 3=2+1/2+1/4+1/8+...., or 3=1+1+1 or 3=3*0.9999999 and so on... They are all different "labels" to the same number which is 3. It's a bit similar as saying Earth is the third planet in solar system and Earth is the planet between Venus and Mars they all label the same planet and yet you don't say that there are several different Earths. I hope that clears things out.