Paranormal Numbers and The Ghost of Borel

This post contains a grave error. There will be a flaw in my argument; a fleeting shadow that inflicts a fatal incision.

Most numbers are normal numbers, so says Borel's 1909 theorem. In this post I will introduce the concept of a paranormal number and prove him wrong. Or will I?

Borel was a French mathematician and politician who had a penchant for the infinite; just about the spookiest concept in mathematics. One of his greatest cultural legacies is his thought experiment which has become known as The Infinite Monkey Theorem.

It was the best of times. It was the blurst of times?

Another of his contributions was his coining of the term, normal number. This label seems, at first, to be rather misleading. If I were to ask you to think of a number then I would be very surprised if your answer were normal: 13, 729, 0.34, 0.21938. None of these numbers is normal. No rational number is.

A normal number must have an infinite string of non-periodic decimal places and each digit must appear with equal frequency. If you pick a digit at random, the probability that it is a 7, for example, must be 0.1. However, if this was the only rule needed to be satisfied then the following number would be normal:

0.12345678901234567890123456789012345...

Those with an A* at GCSE, however, will be able to tell me that this is in fact the fraction 1234567890/9999999999, which is rational, and I have already told you that no rational number is normal.

Another rule that a normal number has to satisfy regards each pair of digits. There are 100 possible pairs of digits (00, 01, 02, ..., 98, 99) and each of these has to be appear with equal frequency. For example, if I pick a pair of consecutive digits at random, the probability that it is 73 must be 0.01.

Another rule that a normal number must satisfy is that each of the 1000 possible triples of digits must appear with equal frequency. If I pick three consecutive digits at random, the probability that it is 734 must be 0.001.

These rules continue for 4-digits, 5-digits, 6-digits, ad infinitum. There are infinitely many rules that a number has to satisfy to qualify to be normal. It seems very surprising, therefore, that Borel has proved that most numbers are normal or, as Davar Koshnevisan of the University of Utah put it, "Normal Numbers are Normal". This sounds like a 'Brexit means Brexit' style sound-bite but - Theresa May take note - actually means something owing to the multiple uses in English of the word normal.

You might now expect me to provide you with a list of normal numbers, since there are so many and all.

Things are not that straightforward, however. It turns out to be quite difficult to prove whether a number is or isn't normal. All the usual irrational suspects: Square-root 2, the natural logarithm of 2, e, and pi are all awaiting proof one way or the other. It has been noted that the first 30 million digits of pi seem pretty normal but, when you're dealing with an infinite string of apparently random digits, that is a mere drop in the ocean.

Here's a number, called Champernowne's Constant, which is normal in base 10:

0.1234567891011121314151617181920212223242526272829303132...

It doesn't take a genius to see that this is formed by writing out the positive integers in order after the decimal point. You might, however, be interested to know that joining numbers in this way is known as concatenation; what a very pleasing word! If you're familiar with Benford's Law you might like to think carefully why this number's normality doesn't contradict it.

So there we go. That is just one example of a normal number (even though it's only normal in base 10) so Borel's assertion that most numbers are normal appears to be a little flimsy, especially when I show you that I can make a number that isn't normal from any one of these elusive normal numbers.

Suppose that the following shows the first seventy decimal places of a normal number:

0.1736432859876434117900234566279347758485098723200046253425747352438744......

Let's make its paranormal number by adding one to every decimal place. If we have a nine, however, we'll leave it as a nine. Therefore, the paranormal number begins:

0.2847543969987545228911345677389458869596199834311157364536858463549855......

While the digits still look random, this number is demonstrably not normal. The probability of a randomly selected decimal after the decimal point being zero is now 0. The probability of a randomly selected decimal being 9 has risen from 0.1 to 0.2. Similarly, the probability of a pair of digits being 99 is four times greater than it should be if the number were normal.

Let's think about what we've done here. Most numbers between 0 and 1 are normal, according to Borel's Theorem. We have taken one of these, modified it slightly, and formed a number between 0 and 1 that isn't normal. We could repeat the process for any normal number.

This means that there are just as many non-normal numbers as normal numbers and Borel is wrong!

Or have I made a catastrophic error that is causing Borel to turn in his grave?