A clever way to estimate enormous numbers - Michael Mitchell

Whether you like it or not, we use numbers every day. Some numbers, such as the speed of sound, are small and easy to work with. Other numbers, such as the speed of light, are much larger and cumbersome to work with. We can use scientific notation to express these large numbers in a much more manageable format.

So we can write 299,792,458 meters per second as 3.0 times 10 to the eighth meters per second. Correct scientific notation requires that the first term range in value so that it is greater than one but less than 10, and the second term represents the power of 10 or order of magnitude by which we multiply the first term. We can use the power of 10 as a tool in making quick estimations when we do not need or care for the exact value of a number.

For example, the diameter of an atom is approximately 10 to the power of negative 12 meters. The height of a tree is approximately 10 to the power of one meter. The diameter of the Earth is approximately 10 to the power of seven meters. The ability to use the power of 10 as an estimation tool can come in handy every now and again, like when you're trying to guess the number of M&M's in a jar, but is also an essential skill in math and science, especially when dealing with what are known as Fermi problems.

Fermi problems are named after the physicist Enrico Fermi, who's famous for making rapid order-of-magnitude estimations, or rapid estimations, with seemingly little available data. Fermi worked on the Manhattan Project in developing the atomic bomb, and when it was tested at the Trinity site in 1945, Fermi dropped a few pieces of paper during the blast and used the distance they traveled backwards as they fell to estimate the strength of the explosion as 10 kilotons of TNT, which is on the same order of magnitude as the actual value of 20 kilotons.

One example of the classic Fermi estimation problems is to determine how many piano tuners there are in the city of Chicago, Illinois. At first, there seem to be so many unknowns that the problem appears to be unsolvable. That is the perfect application for a power-of-10 estimation, as we don't need an exact answer - an estimation will work.

We can start by determining how many people live in the city of Chicago. We know that it is a large city, but we may be unsure about exactly how many people live in the city. Are there one million people? Five million people? This is the point in the problem where many people become frustrated with the uncertainty, but we can easily get through this by using the power of 10.

We can estimate the magnitude of the population of Chicago as 10 to the power of six. While this doesn't tell us exactly how many people live there, it serves as an accurate estimation for the actual population of just under three million people. So, if there are approximately 10 to the sixth people in Chicago, how many pianos are there?

If we want to continue dealing with orders of magnitude, we can either say that one out of 10 or one out of one hundred people own a piano. Given that our estimate of the population includes children and adults, we'll go with the latter estimate, which estimates that there are approximately 10 to the fourth, or 10,000 pianos, in Chicago.

With this many pianos, how many piano tuners are there? We could begin the process of thinking about how often the pianos are tuned, how many pianos are tuned in one day, or how many days a piano tuner works, but that's not the point of rapid estimation. We instead think in orders of magnitude, and say that a piano tuner tunes roughly 10 to the second pianos in a given year, which is approximately a few hundred pianos.

Given our previous estimate of 10 to the fourth pianos in Chicago, and the estimate that each piano tuner can tune 10 to the second pianos each year, we can say that there are approximately 10 to the second piano tuners in Chicago. Now, I know what you must be thinking: How can all of these estimates produce a reasonable answer?

Well, it's rather simple. In any Fermi problem, it is assumed that the overestimates and underestimates balance each other out, and produce an estimation that is usually within one order of magnitude of the actual answer. In our case, we can confirm this by looking in the phone book for the number of piano tuners listed in Chicago.

What do we find? 81. Pretty incredible, given our order-of-magnitude estimation. But, hey - that's the power of 10.