In Praise of Logarithms

25 10 2008

It occurs to me at this time that powers (or logarithms) are an equally justifiable numbering system of their own, indeed they may in fact be more meaningful and representative of ‘reality’ than the linear numbering systems we use so often. What I am referring to is a numbering system where consecutive ‘numbers’ are not simply the last number +1, but the last number multiplied by some factor. So: 1 2 3 4 may be used to represent 1 10 100 1000 in the case of a base 10 log (or exp), or 1 2 4 8 in the case of a base 2 system. (You can see that the numbers are simply obtained by raising the base (2 or 10) to the power of the number in question – so these really are just the logarithmic version of normal numbering)

But wait! The notable thing here is that this system has no apparent negative numbers: -2 -1 0 1 2 3 becomes 0.01 0.1 1 10 100 1000 for base 10.

Aside 1: you will note that addition in this system is ‘altered’. Addition and multiplication are mixed up! 2*3=5 while 6*9=15 (true for all bases!) and on the other hand 2+2=3 (in base 2) while 2+2=2.3010… (in base 10).

Aside 2: Negative numbers? The concept of negative numbers in this context has a strong (and genuine) relationship with imaginary numbers in conventional numbering systems. The only way to obtain negative numbers is to raise your base to the power of an imaginary number: e^(i*pi) = -1 being the famous example of this.

There is however much benefit to had to stop thinking of these numbers as powers but rather think of them as numbers in their own right – a numbering system, extending from -infinity (representing the infinitely small) to +infinity (representing the infinitely large). This is much more in keeping with reality – in which negative numbers don’t really exist! In fact we are so used to them that we have forgotten that they are just as weird as imaginary numbers (they were the imaginary numbers of their time). They, just like imaginary numbers, are so darn useful and sensible that we forget that really don’t have any basis in reality. They are firmly stuck in the platonic world.

So what of reality then? Imagine two points in open space. How far apart are they? A yard? A mile? One cannot say as the space has no reference measure besides the two points. The only definition we might attach is to say the distance is “1”. I.e. we define all distance in that world to be the distance between the two points. If you added more points the distances between them could then be expressed as multiples of the length AB. Using (for example) a base 2 system – because a base two system gives us the case of doubling (or splitting in half) with each increment. So if CD where the AB doubled 10 times then its length would be 10. If EF where AB halved ten times then its length would be -10.

So what’s the point? This ‘numbering’ system allows a better basis for attacking the big cosmological question: What is the nature of space?

Aside 3: “Information” has been shown to be binary (each bit of info halves the unknowns). If you have two boxes, 1 bit will tell you which one has the prize – 4 boxes will need 2 bits – 32 boxes – 5 bits. There is no such concept as negative bits. This numbering system linearises information content.

Aside 4: Which base? Well I am not tied down on this yet. 2 is good and e has a strong case. 10 is probably not as useful as we think (Q: why is 10 ‘special’? A: Its not.)

Please, o blogosphere, dispense thine thoughts!




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