The Smallest Boring Number

Oct. 2004

By Ofer Faigon


People use numbers every day. Some numbers, such as 10 or 2, are very useful. I bet we all mention them daily. Other numbers, like 64 or 35, are less useful. We probably mention these a few times a year. On the other extreme, there are numbers no one will ever think of, such as 3,735,926,102,393,...

Being somewhat lazy, people have always managed to come up with short comfortable names (or descriptions) for the useful numbers. If ever there were a useful number with a "long" name, people would gradually reduce it to or replace it by a shorter name.

Since these useful numbers are in a way "interesting", let's call the no-one-will-ever-think-of numbers "boring". In order to develop some valuable reasoning about the "boring" numbers, we shall give them a more precise definition:

A "boring" number is a natural (whole and positive) number that cannot be described in less than six English words.

Now it is clear from this definition that "ten" and "thirty-five" are not boring. It is also clear that "ten thousand and seven, squared" is not boring. This last example is a bit far from our intuitive notion of interesting or boring, but this is not important for what we are about to do next.

Which numbers are boring, according to our definition?

Let's start with one. "One" is not boring. Nor are "two", "three" and so on up to "twenty", as these can all be described using a single word. Let's move on. "Twenty-one" requires two words, as do the following numbers up to "twenty nine". Counting up to "hundred" we encounter only one- and two-word numbers. From here on we enter the realm of the three- and four-word numbers, which includes the vast majority of numbers up to "nine hundred ninety nine" (there is no need to add the word "and" in the middle, as its presence does not contribute to the identity of the number.)

We can go on like this and every now and again encounter numbers that require more and more words to name them. Some of these numbers can be described by a short formula, such as "nine to the seventh power", but as the numbers grow larger, we are sure to come across a number that cannot be described in less than six words. It is easy to see this if we use a simple counting argument: obviously, there is a finite number of words in the English language, and therefore, a finite (though quite sizeable) number of five-word combinations; on the other hand, there is an infinite number of natural numbers.

Let's get back to the first number that did not have a short description. By our definition, this is a boring number. Not only that, it is the smallest boring number; after all, we scanned every number from 1 upwards and saw only 'interesting' numbers before reaching this particular one. But wait, doesn't that immediately rob it of its title? "The smallest boring number" is a four-word description that uniquely identifies that number, so by our definition, it is not boring!

Let's look at this a bit differently. Clearly, the numbers "one" through "ten" are not boring. It is also pretty clear that most seven or eight digit numbers cannot be described in less than six words, which means they are boring. We can try it with an example, say, "three million four hundred seventy one thousand six hundred and fifteen" (3,471,615). I count 11 words here, and I am almost certain there isn't a short description for this number (if there is, then a. I apologize, and b. you can pick a different number of the same magnitude and try again). So, evidently, there are boring numbers out there, and they are not even overwhelmingly large. Now, somewhere between 10 and 3,471,615 there must be a boring number that is smaller than all the others. It immediately follows that we can speak of "the smallest boring number". However, being described in only four words, this number cannot be boring. A paradox.

What just happened here?

In order to understand this, let's go back a few hundred years. In medieval times, philosophers and theologists pondered about many interesting questions, one of which was roughly this:

What will happen if a cannon that can break any wall fires at a fort whose walls are so strong that no cannon can break?

Since the cannon can break any wall, it will break the walls of this fort too. If these walls are so strong that they are cannon- resistant, they will not break. A paradox.

Have we discovered some new phenomenon that should be investigated by physicists? Of course not. There is no such thing as a cannon that can break any wall, just as there is no wall that can withstand any cannon ball. We cannot change reality just by pretending it is different or by playing with words. This game only demonstrates how powerful our language is, so much so that it can describe non-existent and even contradicting concepts. (Goedel's incompleteness theorem comes to mind.)

This same paradox appears in many shapes and forms. If we try to distill it by removing all the ornaments and verbal noise, we finally reach its purest form, namely:

This sentence is false.

If this sentence is true, then what it claims - that it is false - is true, which renders it a false sentence. On the other hand, if it is false, then what it claims is not true, which means the opposite is true, and therefore the sentence is true. In short: If it is true then it is false, and if it is false then it is true.

All three examples (the sentence, the cannon and the wall, the smallest boring number) share a common theme: we haven't broken any physical law; we were just playing with words.