I was walking to the tennis courts in Battersea Park a few years back, when I heard something on my Walkman radio. It stuck with me for years, and until tonight I haven’t followed up on it, read about it or written about it. Though I have told everyone at my work, which has resulted, as usual, in groans about how nerdy I am (and genuine amazement at how I could spend valuable time pondering these things).

What I heard was a very short anecdote about someone who wrote a little regarded paper in the 1940’s (see ref below) in which he made an attempt to define a ‘measure’ for information. Although I never read any more about it (until today), what I heard was enough to set me thinking…

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Now, if you know lots about this subject then bear with me. Those readers who don’t know what he came up with: I challenge you to this question:

- what contains more information, a phone-number, a ringtone or a photo?

Are they even comparable?

**Bits & Bytes…**

In this computer age, we already have some clues. We know that text doesn’t use up much disk space, and that photos & video can fill up the memory stick much quicker.

But what about ZIP files? These are a hint that file-size is not a very accurate measure of information content.

So what is a megabyte? Is it just so many transistors on a microchip? Happily, its not, its something much more intuitive and satisfying.

### Information: what is it?

If you go to Wikipedia and try to look up Information Theory, within a few seconds you are overrun with jargon and difficult concepts like Entropy; I hope to avoid that.

Let’s rather think about 20 questions. 20 Questions is the game where you have 20 questions to home in on the ‘secret’ word/phrase/person/etc. The key, however, is that the questions need to elicit a yes/no response.

To define information simply: the more questions you need in order to identify a ‘piece of information’, the more information content is embodied in that piece of information (and its context).

This helps us to answer questions like: “How much information is in my telephone number?”

Let’s play 20 questions on this one. How would you design your questions? (Let’s assume we know it has 7 digits)

You could attack it digit by digit: “is the first digit ‘0’? Is the first digit ‘1’? Then changing to the next digit when you get a yes. If the number is 7 digits long, this may take up 70 questions (though in fact if you think a little you will never need more than 9 per digit, and on average you’ll only need about 5 per digit – averaging ~35 in total).

But can you do better? What is the optimum strategy?

Well let’s break down the problem. How many questions do we really need per digit?

We know that there are 10 choices. You could take pot luck, and you could get the right number first time, or you might get it the 9th time (if you get it wrong 9 times, you don’t need a 10th question). However, this strategy will need on average 5 questions.

What about the divide and conquer method? Is it less than 5? If yes, you have halved the options from 10 to 5. Is it less than three? Now you have either 2 or 3 options left. So you will need 3 or 4 questions, depending on your luck, to ID the number.

Aside for nerds: Note now that if your number system only allowed 8 options (the so-called *octal* system), you would always be able to get to the answer in 3. If you had 16 options (*hexadecimal*), you would always need 4.

For the decimal system, you could do a few hundred random digits, and find out that you need, on average 3.3219… questions. This is the same as asking “how many times do you need to halve the options until no more than one option remains?’

Aside 2 for nerds : The mathematicians amongst you will have spotted that 2^{3.3219} = 10

Now, we could use 4 questions (I don’t know how to ask 0.32 questions) on each of the 7 digits, and get the phone number, and we will have improved from 35 questions (though variable) to a certain 28 questions.

But we could take the entire number with the divide and conquer method. There are 10^{7} (100 million) options (assuming you can have any number of leading zeroes). How many times would you need to halve that?

1. 50 00o 000

2. 25 000 000

3. ….

…

22. 2.38…

23. 1.19…

24. 0.59…

So we only needed 24 questions. Note that calculators (and MS Excel) have a shortcut to calculate this sort of thing: log_{2}(10^{7}) = ~23.25…

OK, so we have played 20 questions. Why? How is the number of questions significant? *Because it is actually the accepted measure of information content!* This is the famous ‘*bit*‘ of information. Your 7 digit number contains about 24 bits of information!

### Epilogue

As you play with concept, you will quickly see that the amount of information in a number (say the number *42*), depends hugely on the number of possible numbers the number *could* have been. If it could have been literally any number (an infinite set) then, technically speaking, it contains infinite information (see, I’ve proven the number *42* is all-knowing!).

But the numbers we use daily all have context, without context they have no practical use. Any system that may, as part of its working, require ‘any’ number from an infinite set would be unworkable, so this doesn’t crop up often.

Computer programmers are constantly under pressure to ‘dimension’ their variables to the smallest size they can get away with. And once a variable is dimensioned, the number of bits available for its storage is set, and it doesn’t matter what number you store in that variable, it will always require all those bits, because it is the number of possibilities that define the information content of a number, not the size of the number itself.

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I hope that was of interest! Please let me know if I’ve made any errors in my analysis – I do tend to write very late at night 😉

References:

1. Claude Shannon, “A Mathematical Theory of Communication” 1948