# Magic by Numbers, from Chris Tucker

October 19, 2010 |

Referred to me by the folks at the Museum of Mathematics, there is a great article in NY Times that refers to M. F. M. Osborne's 1962 paper: "Periodic Structure in the Brownian Motion of Stock Prices"

Excerpt from the op-ed article "Magic Numbers" by Daniel Gilbert:

Magic “time numbers” cost a lot, but magic “10 numbers” may cost even more. In 1962, a physicist named M. F. M. Osborne noticed that stock prices tended to cluster around numbers ending in zero and five. Why? Well, on the one hand, most people have five fingers, and on the other hand, most people have five more. It isn’t hard to understand why an animal with 10 fingers would use a base-10 counting system. But according to economic theory, a stock’s price is supposed to be determined by the efficient workings of the free market and not by the phalanges of the people trading it.

And yet, research shows that fingers affect finances. For example, a stock that closed the previous day at \$10.01 will perform about as well as a stock that closed at \$10.03, but it will significantly outperform a stock that closed at \$9.99. If stocks close two pennies apart, then why does it matter which pennies they are? Because for animals that go from thumb to pinkie in four easy steps, 10 is a magic number, and we just can’t help but use it as a magic marker — as a reference point that \$10.01 exceeds and \$9.99 does not. Retailers have known this for centuries, which is why so many prices end in nine and so few in one.

I found this article that says a different numerical bias affects the entire universe under various guises.

Would it be fair to say that "deliberateness", a concept of action, is required for Benford's law to engage in natural environment. A quake, a wind, a measurable force?

## Gary Rogan replies:

I don't think so. I was trying to explain to myself how something so basic yet so powerful can exist and this is the explanation I just came up with (and it's fully consistent with randomness of certain processes).

Imagine yourself shooting projectiles at an infinite log base 10 labeled axis. What are the chance that any number you hit starts with 1 if everything is completely random? My geuss was it's log base 10 of 2, and voila: I just calculated it and it's equal to .301, or the percentage they cite (30.1%). This law must characterize any truly random phenomena where the measurements are distributed over many orders of magnitude. When you don't see this law, this must indicate absence of randomness or a close concentration around some mean.

In the binary number system, all numbers save zero begin with a 1.

## Gary Rogan writes:

The probability of the number starting with a 1 is log of 2 base whatever type number system you have other than of course for the binary system. It's just the ratio of the distance between 1 and 2 on the log axis divided by the distance between 1 and the number equal to the base of the system. There may be even a way to express it so that it works for the binary system since log of 2 base 2 is 0, but not right now.