Fortunes and Geometric Means
I saw a post on X recently that said
Bill Gates is closer to you in wealth than he is to Elon Musk. Mind blown.
For round numbers, let’s say Elon Musk’s net worth is 800 billion and Bill Gates’ net worth is 100 billion. So if your net worth is less 450 billion, the statement in the post is true.
The reason the statement above is mind blowing is that in this context you naturally think on a logarithmic scale, even if you don’t know what a logarithm is.
Or to put it another way, we think in terms of orders of magnitude. Musk’s net worth is an order of magnitude greater than Gates’, and Gates’ net worth would be an order of magnitude greater than that of someone worth 10 billion. Musk is a notch above Gates, and Gates is a notch above someone with a net worth around 10 billion, where a “notch” is an order of magnitude.
To put it another way, we think in terms of geometric mean √ab rather than arithmetic mean (a + b)/2 in this context. 100 billion is the geometric mean between 12.5 billion and 800 billion. Geometric mean corresponds to the arithmetic mean on a log scale. And on this scale, Gates is closer to Musk than you are, unless you’re worth more than 12.5 billion.
Here are three more examples of geometric means.
The size of Jupiter is about midway between that of earth and the sun; it’s the geometric mean. On a linear scale Jupiter is much closer to the size of the earth than the sun, but on a logarithmic scale it’s about in the middle. More on that here.
The tritone (augmented fourth) is half an octave. So, for example, an F# is in the middle between a C and the C an octave higher. Its frequency is the geometric mean of the frequencies of the two C’s. More here.
Finally, the humans body is a middle-sized object in the universe. From Kevin Kelly:
Our body size is, weirdly, almost exactly in the middle of the size of the universe. The smallest things we know about are approximately 30 orders of magnitude smaller than we are, and the largest structures in the universe are about 30 orders of magnitude bigger.
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