Of Logarithms and Numbers
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math

More on the digitation theme.

I recently posted a brief conversation about my distaste for digital numbers, and added a bit in another post. Today I want to discuss the general structure of numbers we use.

The core idea behind numbers is to represent quantity as a series of discrete glyphs: 1, 2, 3, and so on. The notion of place value and decimals extend this, but the core idea is this simple series. But this begs the question, what quantities ought adjacent glyphs represent? It begs it so effectively I doubt most of my readers have ever even asked it.

The standard answer is “‍one thing. Get another thing, move to the next number‍”. In other words, to get from one to the next you add a fixed value. But this is not always a good match; for example if X is 150% of Y, then Y is 66⅔% of X, not 50%. In other words, having numbers that are a fixed addition apart isn’t the most natural for ratios and percentages.

Another option is to have adjacent numbers be a fixed multiple of one another. This what we get in logarithmic scales, such as the decibel. Adding 10 decibels is the same as multiplying by 10. This is not good for counting: adding a person to a room of 10dB people yields 10.414…dB people, while adding a person to a room of 20dB people gives 20.0432…dB people. However, for percentages (20dB% = 100%) things are nicely symmetric: if X is 22dB% of Y, then Y is 18dB% of X. We don’t even really need to mess with percentages: X is 2dB more than Y with or without the arbitrary “‍out of a hundred‍” scaling of percentages.

We can keep going on this train if we want, defining a scale were adjacent numbers are a fixed exponent of one another and so on. However, there is rarely a need.

One of the rules of thumb I find applies in my life is that anytime I bump into a number with more than 3 digits (other than money), I’m actually not interested in linear counting numbers. I don’t care if India has 176 million fewer people than China; I do care that they have 13.22% or 0.616dB fewer people. And I’d rather the dB, because I can combine them: Europe having 2.602dB fewer people than China means it has 1.986dB fewer people than India. I can’t do that with simple subtraction using percentages.

I don’t suppose we will ever switch to teaching people logarithmic scales instead of or in addition to linear scales. But I do like them as a tool.