Numbers are everywhere, like it or not. We deal with them daily, from the simple speed of sound to the more complex speed of light. Managing larger numbers can be tricky, but scientific notation makes it easier. For instance, instead of writing 299,792,458 meters per second, we can shorten it to 3.0 times 10 to the eighth meters per second. In this format, the first number must be greater than one but less than ten, followed by the power of ten.
Scientific notation isn’t just for convenience. It helps in making quick estimates when exact values aren’t needed. Take the diameter of an atom, roughly 10 to the power of negative 12 meters, or the Earth’s diameter, about 10 to the power of seven meters. Using these quick estimations, often called Fermi problems, is a handy skill in both math and science. Named after physicist Enrico Fermi, these problems involve making rapid, rough estimates with limited data.
Fermi, who worked on the Manhattan Project, was famous for this. During the first test explosion at the Trinity site in 1945, he estimated the blast’s strength by observing pieces of paper he dropped. He guessed it to be 10 kilotons of TNT, impressively close to the actual 20 kilotons.
A classic Fermi problem involves estimating how many piano tuners are in Chicago without precise data. Start with the city’s population. Chicago is large, but exact numbers can be unclear, so we estimate it around 10 to the sixth, approximating the actual three million people. Next, guess the number of pianos. If one in every hundred people has a piano, this gives us about 10,000 pianos or 10 to the fourth.
Then, consider how many pianos a tuner can handle yearly. Estimating each tuner manages about 100 or 10 to the second pianos, we find around 100 piano tuners in Chicago, or 10 to the second. This method balances over and underestimates, often landing us within one order of magnitude of the real number. Checking the phone book reveals there are actually 81 piano tuners listed. Not bad for a rough guess, right? That’s the power of using the power of 10.
