Suppose you’re looking for sources of random numbers. You choose numbers from a newspaper. How random will the numbers be? Answer: not very. This surprising result is called Benford’s Law. This law says that real-world numbers (measurements, populations, stock prices and so forth) are distributed logarithmically. More specifically, the leading digit of a given real-world value is 1 about 30% of the time (instead of 10% of the time, as you might expect), and leading digit is 9 less than than 5% of the time (again, not the expected 10%).
There are many fascinating things about Benford’s Law. For instance, even though the law was discovered in the late 1800s, it was only proven only 10 years ago. More interestingly, the IRS uses Benford’s Law to detect fraudulent tax returns.
Eric Lippert makes the case in his blog that Benford’s Law means that binary numbers are actually the best way the do accurate floating-point math calculations on real-world numbers.
I find the intersection between mathematics and the real-world to be absolutely fascinating.