Although being a fundamental concept in pure mathematics, infinity is quite a challenging one to get one’s head around. I do not aspire to an exhaustive discussion on the topic, but merely to illustrate some recurrent issues that arise when exploring themes where infinity creeps in. In this process we will also meet in passing other mathematical concepts and problems that we often take for granted and are usually the subject matter of the philosophy of mathematics.

There is a very simple example, which I often like to present and discuss with my students, that reveals the puzzling nature of infinity. This example falls within the topic of converting recurring decimals into fractions, and the exposition adheres to the common method that we follow. However, we will see two more proofs of the same statement.

The question is: prove that 0.999… = 1.

Let’s start with our first step, which is the creation of the simple equation:

x = 0.999…, that is, setting a variable for the recurring decimal number.

Our aim is to somehow eliminate the recurring part after the decimal point of the number, and end up with another equation for x.

A second equation is obtained when we multiply the first by 10:

10x = 9.999…

Now we observe that the recurring parts agree in both equations, so we can subtract the first from the second giving:

10x = 9.999…

x = 0.999…

10x – x = 9.999… - 0.999…

9x = 9

x = 1

But is this not a contradiction? We started by letting x = 0.999… and end up with x = 1; can the variable x be two different numbers at the same time? So what did just happen?

Our intuition urges us towards the claim that the two numbers are infinitesimally close to each other (they are not completely random), and the math obtains an even stronger relation, that of equality.

The question then addressed to my students is: do you think there is a fundamental issue with the above proof? Did we overstep or misuse our well-defined mathematical principles and operations? What kind of numbers do we have at hand?

There are various reasons why this short and simple result appears as a paradox. First, there’s the assumption that the two numbers are obviously different, instead of being considered as the same number but with two different representations (similar to when we consider, say, fractional, decimal, and percentage forms as equivalent). Although 1 is a quite easy, familiar, and non-problematic number, a number that can also be immediately experienced in our empirical lives, 0.999... does not seem to admit a similar response. What if we think of 1 as 1.000..., represented clearly as another infinite number, and assume that the two are in fact different? Now since they are different there should be at least one number that would lie in between 0.999... and 1.000.... But could we think of such a number; does it exist?

Secondly, it is quite demanding to grasp the idea of subtracting recurring decimal numbers. Although many students do not initially recognize any problem with it and intuitively agree straight away with the subtraction, once the recurring decimal is understood as a fixed mathematical object, instead of resolving the issue it becomes worse. Think of the following finite differences,

1 – 0.99 = 0.01

1 – 0.999 = 0.001

1 – 0.9999 = 0.0001

So at some point,

1 - 0.99...9 = 0.00...01, which means that as long as 0.99...9 is finite, then there will always be a difference between the two. Yet, for the subtraction of infinite numbers we have that: 1 – 0.999... = 0.00...00... = 0.

Since I have infinite 9s when subtracting there will be infinite 0s after the decimal point, thus the difference will be 0. This also responds to our previous question of having a number fit in between them. Therefore, we conclude again that 1 = 0.999...

Thirdly, students often comprehend, or try to do so, a recurring decimal by mentally adding more decimal digits at the ‘end’ of the number all the way to infinity. Thus, while students easily grasp the number 1 as a fixed mathematical object, the same does not hold with the recurring decimal 0.999…, which is instead thought as a flowing, changing number with trailing 9s continuously added along. The problem lies exactly in that conceptualization of the recurring decimal as a mentally ‘finite’ process of adding infinite 9s at the ‘end’. This appears non-sensical since there is *no end* to an infinite number, the whole idea of it is as such. Similar conceptual problems surround many other mathematical areas where infinity ‘shows up’.

Fourthly, the proof above incorporates the notion of limit, which is also troubling for some students. In the particular example, one may think that 0.999… is still a sequence (of repeating 9s) and not a limit in itself, which should be a fixed number – another unique mathematical object. Students tend to think of 0.999... as “approximating” 1, as “getting arbitrarily close” to 1, and yet never equalling 1. However, such expressions are mistaken since they express a number – something specific – as changing, moving towards another number, which makes no sense. The recurring decimal 0.999... is an *infinite* number and as such it does not have a lot or many but infinite 9s, we cannot count them, so it does not help to think of it in terms of how many ‘9’ digits there are; and that’s exactly how we should conceive and represent the particular number.

A sequence, on the other hand, is changing and could perhaps approximate a particular number when it is convergent. So let’s define a convergent sequence that would both serve as a proof of 0.999... = 1, and illuminate the idea of limit as “getting arbitrarily close” to a number.

For this purpose, we will define a geometric series, thus asserting 0.999... as the limit of the sequence of partial sums.

The infinite decimal number 0.999... can be expanded as an infinite series:

0.999... = 0.9 + 0.09 + 0.009 + ...

If we think of the partial sums of this series, we have that:

S_{1 }= 0.9, S_{2} = 0.99, S_{3} = 0.999, …, S_{n} = 0.99...9

Thus,

lim_{n→∞} (S_{n}) = 0.9 + 0.09 + 0.009 + ...

Now, when factorizing we observe that we end up with an infinite sum of a geometric sequence inside the bracket,

0.999... = 0.9 + 0.09 + 0.009 + ...

= 9/10 + 9/100 + 9/1000 + ...

= 9/10 * (1 + 1/10+ 1/10^{2} + 1/10^{3}...)

Taking the limit of the infinite geometric series of {1/10^{n}}, which exists since 1/10 < 1, we have that:

0.999... = 9/10 * [ 1/(1 – 1/10)]

= 9/10 * [1/(9/10)]

= 1.

Now, for the sequence {S_{n}} n ≥1, it makes sense to state that it “approximates” or “gets arbitrarily close” to 0.999..., since as it progresses, the difference of each term from the limit decreases, and I can always find another term that has a smaller difference from 1.

We have thus reflected on the concept of infinity through a very common and, at first glance, trivial example. We have touched upon the nature, conceptualization and understanding of real numbers. But the ultimate point of this ‘exercise’ was to understand that mathematics requires a deeper understanding of and an inquisitive attitude towards concepts and principles, and it also demands that one doubts inner, ‘obvious’ intuitions and given assumptions. Approach maths as a philologist approaches a poem.