Jan 092016
Are we being shown a numeric illusion?

Is “0.999… = 1” a numeric illusion?

About the sting

A ‘sting’ is a deceptive operation designed to catch people out. For those of us who do not accept 0.999… =1, it can feel as if members of the mathematics establishment have conspired to perform a sting operation on the rest of the world.

This 0.9 recurring sting is an elaborate mathematical con trick that’s been around for centuries. It is not just one trick; it is a series of deceptions that work together to reinforce the illusion that 0.9 recurring equals 1.

The convincer: the one-third deception

One of the key stages of a good con is called ‘the convincer’. With financial cons, this involves an actual or apparent pay-off such as a small amount of money or a fake stock dividend document. The purpose of the pay-off is to convince the victim to part with bigger sums of money.

In our case, the convincer comes in two parts. These two tricks create the illusion that infinity is a valid concept in mathematics. Part 1 claims we can create an infinite object from a finite object, and part 2 claims we can convert it back again. These ‘convincer’ arguments can persuade us to accept a huge body of theory built on the apparent validity of mathematical infinity.

Parts 1 and 2 are described below. But beware, both parts are sneaky deceptions.


Part 1: to infinity…

  1. If we try to convert 1/3 into a decimal, our algorithm produces an endless stream of threes, 0.333…
  2. So we can say 1/3 equals 0.3 recurring.

We should first consider if it is possible to represent 1/3 as a decimal. One third can be represented in its entirety as a fraction or in a base that has 3 as a factor, such as base 12, but can it be represented in base 10?

We can use short division to attempt to divide 1 by 3, but this technique has no defined end point; it cannot complete. For each digit processed the decimal result gets closer to one third but no finite number of digits will achieve equality with 1/3. After each ‘3’ digit we produce by our decimal division algorithm, there will always be a positive non-zero remainder part. In order to attain equality with 1/3, any remainder part must be processed in its entirety, but our algorithm cannot achieve this.

We are told that after ‘infinitely many’ digits the decimal will equate to one third exactly, there will be no remainder part, and there will be no more digits after infinitely many digits.  But they cannot demonstrate how the algorithm can achieve infinitely many iterations. They cannot explain how the algorithm will then end. They cannot explain why there are no more terms after ‘infinitely many’ and they cannot explain why this does not form a contradiction with the statement that the decimal is endless. Importantly, they cannot show how the result can equate to 1/3.

In order to accept 1/3 becomes 0.333… in base 10 we do not need to see all the digits, but we do need to know how the conversion algorithm can end and how this will make the result equate to one third.

The first trick they use here is to assume ‘infinitely many’ is a valid concept. Then with infinitely many decimal places, it follows that any finite value in any other base can be entirely represented in base 10. But there still remains the problem that we cannot construct an infinite decimal.

So the second trick they use is to claim that proof of construction is not required. Instead of constructing the infinite decimal in its entirety, the repeating pattern is said to imply infinitely many digits, and this must equal 1/3 because it cannot equal any other number.

We can think of 0.333… as being the series 3/10 + 3/100 + 3/1000 + … and it is true we can always choose a series length to prove the sum does not equal any number that is not equal to 1/3. But it is also true that whatever length we choose, the sum cannot equal 1/3. So by the same argument that shows 0.333… cannot equal any number that is not 1/3, we can argue 0.333… cannot equal 1/3. This does make sense because the series has endless non-zero terms; hence it has no fixed value which means it is not a number at all.

There is no proof that 1/3 can be represented as a decimal, it is just assumed it can.


Part 2: and back…

We can easily convert our infinite recurring decimal back into one third:

  1. Let x = 0.333…
  2. So, 10x = 3.333…
  3. Subtracting (a) from (b) we get 9x = 3, hence x = 3/9 = 1/3

As an extra convincer, we don’t even need to multiply by 10; other powers of 10 like 100, 1000 (and so on) all work just as well as 10.

This is the same flawed logic that is behind the most famous proof that 0.999… equals 1. It involves multiplying it by 10 and subtracting the original series, and this supposedly removes the endless part. The problem with this approach is demonstrated in the first of the two videos near the bottom of this page (from about 6 minutes in to 08:45).

Note that we can apply the same logic to the series 1+2+4+8+…

By simply multiplying it by 2 and subtracting the starting series we can supposedly remove the endless part giving the result 1+2+4+8+… = -1.

And so by exactly the same logic that gives 0.999… = 1, we can say 1+2+4+8+… = -1.

In both cases the logic is flawed because we are not lining up the 1st terms of the series when we do the subtraction. In both cases above we are comparing the first (n+1) terms of the multiplied series with the first n terms of the starting series in order to create the illusion that the trailing terms all cancel out. We are simply choosing to line up the terms in a way that will appear to remove the trailing part of the series.

The con men will claim there are different ‘rules’ for diverging series. They expect us to blindly accept a made-up law that stipulates we cannot apply the same logic as we use for converging series.

One point to note, however, is that 1 is related to 0.999… in the same way that -1 is related to 1+2+4+8+…, it is the ‘fixed part’ of the expression for the sum to the nth term (or the fixed part of the ‘partial sum expression’ if you prefer, but beware of using the word ‘partial’ as it implies there may be a full sum).


The sting: ‘infinitely many’ and ‘convergence’

With the convincers expertly delivered, it is easy to be taken in by the main deception. Again this is not just one trick, it is several highly dubious notions supposedly backing each other up.

The argument that 0.999… equals 1 is same as the argument that 0.333… equals one third. And so if we have been swayed by the convincers then the main con trick is very close to completion.

They start by saying since 0.333… equals 1/3, we can simply multiply both sides by 3 to get 0.999… = 1. Next they perform the 10x – x ‘proof’, exactly as described for 1/3 in ‘Part 2’ above.

But we might still have some niggling doubts at the back of our minds… how and why does the algorithm end when it reaches ‘infinitely many’? How can it reach ‘infinitely many’? How can we get a fixed value without a last term? How can there be no more terms after ‘infinitely many’ when the series is supposedly endless?

They won’t say the sum mysteriously ends and inexplicably becomes equal to a fixed value after a magic number of terms, because this sounds ridiculous and farcical.

Instead they will dress-up this very same argument in a story using better sounding words. They will explain that if we examine how the partial sum varies as n increases, we can see the trend where the sum ‘converges to’ a ‘limit’ as n approaches infinity. A nice touch is that they simply define the sum to be equal to the limit, then this is true by definition (who needs proof when we can simply define things to be true!).

In reality, to examine how the partial sum varies we should find an expression for the sum to the nth term. This expression will have a fixed-part (a constant) and a variable-part (containing ‘n’). Note that the ‘fixed part’ equals the value defined by mathematicians to be the ‘limit’ of the series.

Then we are being asked to accept that if the nth sum diminishes as n increases, the sum somehow equates to the fixed-part of the expression (or the ‘limit’) after ‘infinitely many’ terms. And since there are no more terms (after ‘infinitely many’), the sum can be said to be a ‘real number’.

But if the nth sum does not diminish, then we are supposed to accept it doesn’t equal the fixed-part, and the sum is not a real number.

If we still have some doubts the con men will use put-down tactics.

They will tell us to not worry about such things; we should put them to the back of our minds. Better still, we should just forget about them completely. When we become clever enough to understand infinity we will realise there are no contradictions here; everything is completely rigorous and well-defined. After all, we are not talking about finite objects, these are infinite objects, so our concerns are simply not relevant.


The sucker punch: the epsilon-delta ‘proof’

The epsilon-delta formulation provides the standard definition of convergence. The con men can now use this in another way to deliver a sucker punch, just to make sure they have us down and out.

In the case of 0.999… and its ‘limit’ of 1 (or of 0.333… and its ‘limit’ of 1/3) the argument goes like this:

  1. If we consider the sum of the first n terms of the series (called the ‘partial sum’), then we can make this sum as close to the limit of the series as we like.
  2. It follows that there is no number that can be placed between the value of the series and the value of the limit, therefore they must be the same value.

The problem with point (1) is that ultimately we would like the sum to equal to the limit, and this cannot be achieved.

The problem with point (2) is that it assumes the series can somehow exist in its entirety, in which case it would admittedly have a sum that equals a fixed value. It ignores the problem of constructing a so-called infinite decimal because it assumes infinite decimals are well-defined. But if they were well-defined, their definition would describe how the terms can form a completed series, and their construction would no longer be a mystery.

For the epsilon-delta argument to make sense, you have to accept that decimals can have ‘infinitely many’ digits and that these ‘infinite decimals’ do have a static/fixed values. But if a series has endless non-zero terms then it cannot have a fixed sum. It does not have a fixed value and so it is not a number. To claim it does have a fixed value forms a contradiction with the claim it has endless non-zero terms. The con men cannot resolve this contradiction so they will simply deny it exists. No clear explanation of why this is not a contradiction will ever be provided.


If 0.999… is not a real number, what is it?

Like 0.333…, 0.999… is an endless series that cannot equate to any fixed value because the series does not end; it has no last term. An endless series should be treated as an endless series, not a fixed value. Like it or not, mathematics can handle such objects without having to accept the notion of ‘infinitely many’.

The two videos near the end of this article show how such objects can be handled in mathematics, without accepting infinity.


Sum to the nth term (where n is indeterminable)

All of mathematics is procedural. We always get from one place to another via a series of steps.

If we try to add up the natural numbers, 1 + 2 + 3 + 4 + …, we follow a process. Similarly, if we attempt to add up 3/10 + 3/100 + 3/1000 +…, we follow a process.

After n steps of the process we will have added up the first n terms. Here ‘n’ is an unknown that does not have a fixed value. Once we appreciate that n is a indeterminable variable, we can appreciate that any ‘endless sum’ cannot equal a fixed value, but it can be completely described by an expression for the sum to the nth term (where n is indeterminable).

Moreover, an endless series (summation) is completely and uniquely described by the expression for the sum to the nth term (where n is indeterminable).

Mathematics can easily work with these objects as shown in the two videos found further down this page.


How to check for equality between 1 and 0.999…

As we said previously, an endless series is uniquely defined by the expression for the nth sum. In the case of 0.999…, the nth sum is:

1 – 1/10n

This has a fixed part ( 1) and a variable part ( – 1/10n).

The correct way to compare a number with an endless series is to express the number in the same format as an endless series. And so we could write 1 as:

1 + 0.n     {i.e. one plus (zero times n)}

By elevating 1 to the same structure as 0.999… we can clearly see these are not equal.

It would not make sense to try to convert the endless series to a number in order to do the comparison, as we would then be comparing just a part of the endless series, not the whole object.


The car salesman con

Infinity bumper sticker

Infinity bumper sticker

The unsuspecting car buyer handed over a large amount of cash in order to purchase a particular vehicle.

After gleefully accepting the money, the salesman handed a bumper sticker to the man. “Here’s something exactly equivalent to the car you wanted, I hope you enjoy it.” he said.

The salesman pointed out to the customer that the same bumper sticker was on the car he had intended to purchase. Everything could be explained in terms of convergence and the simple matter of defining one thing to be something completely different.

What had happened here was that salesman had compared the two objects:

Object 1:          [Bumper sticker]

Object 2:          [Bumper sticker] + [Rest of the car]

He could have elevated the first object into a format that made it easy to compare without losing any information about either object, like this:

Object 1:          [Bumper sticker] + Zero.[Rest of the car]

Object 2:          [Bumper sticker] + [Rest of the car]

Then it would have been obvious that the two objects were not identical. But the car salesman adopted a different approach. He decided to ‘define’ object 2 to be equal to [Bumper sticker].

He could even ‘prove’ this was true by taking the bumper sticker off and holding it alongside the view of the car as the car drove off into the distance. The car appeared to get smaller and smaller in comparison to the the bumper sticker.

This shows that [Bumper sticker]+[Rest of the car] converges to [Bumper sticker].

So they must be exactly the same thing! (just as 0.999… converges to, and therefore is equal to, 1).


Videos about 0.999…

Below are two 10 minute videos about why 0.999… does not equal 1.


Why 0.999… does NOT equal 1 (Part 1: The Problem)


Why 0.999… does NOT equal 1 (Part 2: The Solution)



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