A Friendly Chat About Whether 0.999... = 1

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121

Somehow this discussion has some similarity or relation to the equation:
1/0 = infinity.

Of course it has been settled to be UNDEFINED because it is absurd to come up with the result that 0 x infinity =1 . How is it possible to have so many nothings(0) and come up with one? That is only true in the world of fantasy and magic!

Same thing with infinitely repeating decimals when converting fractions. No matter how many times you repeat it you would never arrive at the true exact value of the original fraction the decimal was derived from. Infinity by definition cannot be arrived at and it cannot reach its end.
0.9999… therefore can never reach the value of 1.

122

I’m sorry, I just realized that through my haste I made a fatal false assumption in my previous “simple proof”.

However, I’m still not buying the claim that 1=.999…
Infinitesimally close maybe, but never equal to 1.

I need to make sure next time that I don’t post after I drink my beer. :slight_smile:

…

123

Earlier I argued why repeating decimals cannot “end” with a “different number”. For example, there can’t be such a number as .0…1, the presumed “nonzero difference” between .9… and 1.

I feel like my specific argument wasn’t adequately countered. I’d be very suprised if there were a number system in which one-twelfth is not equal to one-third of one-quarter, and that the difference between the two amounted to “0.0…25” or some such.

I’d like to re-iterate the point in a new form: Are certain rational numbers only expressible in certain bases? Mathematicians say no: If it can be expressed in base 10, it can be expressed in any other base, from 2 upwards – so long as repeating decimals are permitted.

However, if .9… does not equal 1, then this becomes false, at which point major issues arise. Ordinarily, we might say that .9… is equal to “one minus the smallest rational number”, which in turn means “one minus zero”, or simply “one”. But if we accept the “unequal” argument, then instead we are saying something like “one minus the smallest rational number greater than zero and expressible in base ten”. Which leads to more madness.

In binary, the number 0.1… is ordinarily equal to 1. If it’s not, then it can’t be equal to 0.9… either; in fact, no binary number would be equal to .9…. This is especially strange because ordinarily a “binary number” isn’t actually a kind of number but a way of describing a number, rather like a language.

Ultimately, if you reject the equality, you reject repeating decimals in general as meaningful or useful. That’s something you have every right to do (they’re merely a tool for which fractions can usually substitute). What you can’t do is say that 0.9… is a coherent, actual number and that it’s not equal to one. It just doesn’t work.

124

@Lenoxus on June 20, 2013 at 5:16 pm:
What do you mean by “finitely-numbered step” and “infinitely-numbered step”? As far as I can tell, there are only finitely-numbered steps on this list, and none of these infinitely many finitely-numbered steps is a shift to position [-1, 0].

125

Hi Conor, thanks for the note! Good point, I’ll see if I can clarify the article. Correct, in our current decimal system 0.999… = 1 (because we cannot represent infinitesimal differences), but the deeper response is that we could represent it in another system. A little like how the Romans didn’t have a symbol for zero, but it doesn’t mean the concept is inherently impossible :). I’ll need to see if I can make this subtlety more clear though!

126

But it’s not just a matter of notation.

0.999… necessarily means “Nine-tenths plus nine-hundredths plus nine-thousands, etc, forever.” There’s no disagreement with this, right?

Well, mathematicians find that the sum of these fractions is exactly equal to one. There’s no point in the process where our means of representation come into play. We don’t say “Well, there’s an infinitesimal part that we lack the means to express, so we’ll round it away.” We simply find that the number “one” is equal to that infinite sum.

127

If I am mixing 9 liters of red paint and 1 liter of green paint, the mixture ratio is 9:1. If I am mixing 99 liters of red paint and 1 liter of green paint, the percentage of red paint is 99%. If I am not stopping increasing the percentage of red paint, the percentage of red paint is 99.999…% (even if this means that the whole infinite universe is filled with red paint). But a universe filled with red paint and 1 liter of green paint is different to a universe solely filled with red paint, and no green paint in it. How do we point out this difference using standard decimal notation?

128

An excellent question, netzwelter. My understanding is this: It depends on what you want to express. If you’re asking for the actual amount of each type of paint, we would say there are “aleph naught” (or “aleph null”) liters of red paint and one liter of green (or no liters of green). There isn’t exactly a decimal representation of aleph naught, it’s just the word for the number of integers there are (and also the number of rational numbers).

If you’re asking about the ratio of red paint to all paint, that would be 1:1, despite the fact that not all the paint is red paint. I know it makes no sense, but I believe that’s the mathematical answer. Likewise, the ratio of green paint to red paint (or of green paint to all paint) is 0:1, even though the actual amount of green paint is not zero. This also seems strange, but it’s the way it has to work. In fact, the ratio of any finite amount of green paint to an infinite amount of red will be 0:1.

All that said, I think there are mathematics in which you could make that distinction better, by dealing with infinitesimals and similar concepts. The math of infinitesimals doesn’t actually change the equality of .9… and 1, but it does deal with other distinctions that standard arithmetic does not.

129

Hi, thanks a lot for this post. My mathematical philosophy course was talking about the 0.999=1 idea and I couldn’t believe how just because something was infinitesimally small meant that all the sudden it jumped to becoming another number. But you showed me the idea of hyperreals. Also, depending on which base is chosen, then the number that goes to its equivalent “1” is different. And since this number is different for each base, then it just seemed arbitrary to me… And that didn’t seem right. Seems like math still has a long way to grow.

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Hi Khalid. I very much like your site. It’s great to see somebody explaining maths ideas simply.

When it comes to this question, I agree with you that we usually give the wrong answer. It’s much better to discuss how we deal with infinities than to assert that 0.999… = 1 and then blind the other party in the conversation with a barrage of analysis.
I think it would have been nice if you could have concluded that 0.999… is 1 in the standard decimal system that people learn in school at the end, only because it’s a little annoying when people come up to you and try to tell you that 0.999… isn’t 1!

131

Mike on March 2, 2013 at 3:14 pm said:
“Also, the problem with the graph with an infinitesimal difference between the .999… graph and the 1 graph, is that you didn’t go on to infinity. You stopped at a finite value. This is analogous to saying 1/2+1/4+1/8… ≠ 1 because you’ll never reach one. Clearly this is false.”

t = 0: I am shifting the line [0, 1] to position [-0.5, 0.5]
t = 0.5: I am shifting the line from position [-0.5, 0.5] to position [-0.75, 0.25]
t = 0.75: I am shifting the line from position [-0.75, 0.25] to position [-0.875, 0.125]
…

This is a list of infinitely many steps (I am not stopping at a finite value). What is the position of the line after execution of all of the steps on this list, at t = 1? Is there a step on this list shifting the line to position [-1, 0]?

132

I’m replying to a comment that appears in the email alert but has yet to show up here. netzwelter asked about the position of xir line after executing “all” of an infinite number of steps, “not stopping at a finite value”. The position is indeed -1, 0], not just “close to” -1. Xe also asked “is there a step on this list shifting the line to position [-1,0]?” The answer is that it depends on what is meant by “a step”. A finitely-numbered step, no. All the infinitely-numbered steps, yes.

This gets at the core strangeness of infinity. There is no finite number of steps for which these principles hold, but they do hold for an infinite number.

And nearly all objections to 0.999… miss this. They say, in effect, “But we’re not there yet!” when this whole concept of “yet” hides an assumption of finiteness. In short, the sum of a truly infinite set of numbers of the form 0.9, 0.09, 0.009, etc is exactly 1.

133

[…] There’s no difference between the number 1 and the seemingly smaller number 0.9999… (with 9 repeating endlessly). I used to think that there must be some tiny but key difference, but it turns out there just isn’t any. They seem different and are written differently, but they are the same. […]

134

@Netzweltler, If I understand what you are asking correctly, then this is my answer :slight_smile:

If the amount of green paint amongst the red paint is measurable, then you can measure it and write down the value. 0.000000000001 perecent is fine to write down if you can measure it as such.

If though you are the point where the amount of green paint is no longer measurable, you can not assign a number to it, it is equivelent to zero. In other words, if the amount of green paint is so small you can not measure it, how do you know it is not infact orange paint? Or unicorns? If you can’t measure it, it can be anything, inlcuding nothing. So we repreent that as zero: 0. If you CAN measure it, then you represent it withthe measure number: 0.00000001 percent. Hope that helps :slight_smile:

135

@D-Physicist on September 13, 2013 at 12:27 am:
My question on June 20, 2013 at 2:46 pm (#119) might be a better fit to your answer than the red paint to green paint question (since we are dealing with one liter of green paint - which isn’t immeasurably small and which is always one liter of green paint).

This list does not contain a step shifting the line to position [-1, 0]:

t = 0: I am shifting the line [0, 1] to position [-0.5, 0.5]
t = 0.5: I am shifting the line from position [-0.5, 0.5] to position [-0.75, 0.25]
t = 0.75: I am shifting the line from position [-0.75, 0.25] to position [-0.875, 0.125]
…
So, we are missing the target by an immeasurably small segment, which is 0, if we are trying to use standard decimal notation to describe the size of the segment. Nevertheless, these infinitely many steps don’t have the same effect as moving the line from [0, 1] to [-1, 0] in one step.

136

The question of whether .999… =1 can be easily addressed without recourse to philosophical nit-picking.

.999… is short hand for the series 9/10 +9/100+9/1000…, which everyone agrees has the limit 1. The question people are asking is whether the series really adds up to 1 or whether or not it just barely misses it by some infinitesimal result.

The problem is that addition is defined for finite inputs. The axioms behind addition don’t allow you to calculate an infinite number of additions. In order to make a sense of an infinite series, you have to define what the sum of an infinite series is, and that definition cannot be derived from the base principles of addition alone.

For convergent series, we define the sum to be the limit of the partial sums. Doing so has useful properties, it preserves linearity and other properties that we associate with addition. But, it is merely a definition. You can’t prove that the infinite series actually does , when carried out completely, add up to the limit, because the sum of an infinite series has no meaning unless you give it a definition.

You are free to not to define the limit of the partial sums of a convergent series as its sum. If you can find an interesting alternate definition, feel free to explore the mathematical consequences. But, if you reject that .999… = 1, then you must reject the use of the limit as the sum of any convergent series, which wipes out a large portion of very useful and interesting math.

I wonder how many people who doubt whether .999… = 1 also doubt whether 1+1/2+1/4+1/8… = 2, or whether e = 1 + 1/2! +1/3!..

137

With regards to Peter post #70

I’ve long regarded (and still do) ‘numbers’ like 1/3 to be a process and not a ‘number’. Even in denoting 1/3 we have not denoted a number, we have denoted 2 integers and a process (division). Still, we like to conceptualize a platonic concept of a number, n and set it equal to 1/3 then ask ourselves “What is n? Is it a process, or the result at the end of a process?”

For me it is similar to ‘looking at’ a quantum particle and asking what is it? Is the particle a quantum wave function (a process)? Or is the particle the result at the end of a process? E.g. does this marble on the desk have a position, or can I simply apply the position operator to the wave function describing the marble? Gary Zhukav wrote a fascinating book in 1979 entitled ‘The Dancing Wu Li Masters’ addressing this issue of perception quite eloquently. He often says ‘the dance is the dancer, the particle is the wave function’. It may sound odd at first, but consider the two following recursions:

  • does the dance exist if there is no dancer to perform it?
    -can the dancer be so called if there is no dance to perform?

-Is 1/3 a number without a process to use as an approach vector?
-Is 1/3 a process if there is not a target to point to?

138

The ole Euler proof for infinite series is based on mathematical induction for finite partial sums, then , with a grand leap of faith, POOF! We can conclude something for infinity!

Sorry to burst this bubble, but Euler incorrectly subtracted the 2 “numbers” in order to attain his desired result, fooling many of the mathematical community.

The Limit concept prevents this equality in the first place (as a sum). That is besides the point for now. Let’s show how the proof is flawed, shall we?

First, let’s try and apply the “proof” using infinity:

$$S = a + ar + ar² + … + arⁿ + … ar^∞$$
$$rS = ar + ar² + … + arⁿ + … ar^(∞+1)$$

Now what usually happens here is the magical INCORRECT subtracting of infinite series terms. You see, here is the correct way of subtracting infinite converging series: $$ Σ { a_i - b_i } = Σ a_i - Σ b_i $$
But you see in the proof, they subtract like this: $$a_i+1 - b_i:$$
$$S - rS = (a - 0) + (ar - ar) + (ar² - ar²) + … $$
This forces the results to be what is desired along with asinine things such as 2 different decimal numbers being the same decimal number, ie, 0.(9) = 1
The particular detail here is that rS has one more term in the sequence at all times since you are subtracting the “next term” in S from the “current term” in rS.
However, when done correctly, you get consistent results:
$$S = Σ a_i$$
$$rS = Σ r x a_i$$
$$S - rS = Σ {ra_i - a_i}$$

Apply to 9.(9) where a = 9, r = 1/10:
$$S = 9 + 9/10 + 9/100 + …$$
$$rS = (1/10) x S = 9/10 + 9/100 + …$$
$$S - rS = (9 - 9/10) + (9/10 - 9/100) + … = 8.1 + 0.81 + 0.081 + … = 8.999…$$

You are led to believe that S - rS = a, but clearly above:
$$S - rS = (a - ar) + (ar - ar²) + …$$

If a = 9, and r = 1/10, then S - rS = 9
But if we do it correctly, S - rS = 8.999…

As we can see, the proof is flawed using incorrect infinite series subtractions to achieve the results that were desired. The proof was not a proof at all since it is invalid… unless you assume 9 = 8.999… which is completely circular, that is, assuming $$S - rS = a$$ which is the basis of the entire “proof” of 0.(9) = 1.

139

Lets try that without the latex:

Let’s try and apply the “proof” using infinity:

S = a + ar + ar² + … + arⁿ + … ar^∞
rS = ar + ar² + … + arⁿ + … ar^(∞+1)

Now what usually happens here is the magical INCORRECT subtracting of infinite series terms. You see, here is the correct way of subtracting infinite converging series: Σ { a_i - b_i } = Σ a_i - Σ b_i
But you see in the proof, they subtract like this: a_i+1 - b_i:
S - rS = (a - 0) + (ar - ar) + (ar² - ar²) + …
This forces the results to be what is desired along with asinine things such as 2 different decimal numbers being the same decimal number, ie, 0.(9) = 1
The particular detail here is that rS has one more term in the sequence at all times since you are subtracting the “next term” in S from the “current term” in rS.
However, when done correctly, you get consistent results:
S = Σ a_i
rS = Σ r x a_i
S - rS = Σ {ra_i - a_i}

Apply to 9.(9) where a = 9, r = 1/10:
S = 9 + 9/10 + 9/100 + …
rS = (1/10) x S = 9/10 + 9/100 + …
S - rS = (9 - 9/10) + (9/10 - 9/100) + … = 8.1 + 0.81 + 0.081 + … = 8.999…

You are led to believe that S - rS = a, but clearly above:
S - rS = (a - ar) + (ar - ar²) + …

If a = 9, and r = 1/10, then S - rS = 9
But if we do it correctly, S - rS = 8.999…

As we can see, the proof is flawed using incorrect infinite series subtractions to achieve the results that were desired. The proof was not a proof at all since it is invalid… unless you assume 9 = 8.999… which is completely circular, that is, assuming S - rS = a which is the basis of the entire “proof” of 0.(9) = 1.

140

Let x= 0.99999…
Hence 10x = 9.999999…
Therefore,
10x - x = 9.9999… - 0.9999999
Therefore 9x = 9
so, x=1,
But x=0.9999…
Therefore, 1 = 0.99999999…
this ways 10 = 9.999999…
but 9.9999… is completely divisible by three ( the quotient is 3.33333… )
Therefore, 10 is completely divisible by 3 and it leaves the remainder 0 when divided by 3