5.6 Simhan, Bandi. The Fermi Estimate
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The Fermi Estimate By Jay Simhan ’23 and Kush Bandi ’22 Abstract
Many questions have been proposed which are physically unanswerable by our current technology. Often, scientists and mathematicians have no basis or starting point on how to proceed in answering a certain question. However, having the ability to make educated and justified guesses is extremely significant in nearly all fields involving numbers. These are Fermi Problems and their solutions are counter-intuitive. The more estimates one makes regarding a certain proposition, the further they approach the true answer. The solutions to Fermi Problems involve several calculus-based concepts. Specifically, functions and equations that fluctuate around a certain value, which then converge to that value, exemplify the premise behind why Fermi Problems are equatable. Through our analysis of converging functions representing error in estimations, we exhibit how the nature of educated estimations concentrates to a single value as extra guesses are added. Though most Fermi Problems are generally not provable by current data, those which have been mathematically computed result in values extremely close to the estimates made in solving with minimal real data. These types of accurate estimations can help further humanity’s knowledge on abstract concepts we have yet to understand, and can potentially allow us to progress further in all regards.
Figure 1: Enrico Fermi [4] The reason Fermi’s estimations work in these problems is because the approximations of individual terms are generally close to correct, and the overestimates and underestimates help cancel each other out. Broadly, if bias is excluded, a Fermi calculation that involves calculations on several levels of different approximations will continuously become more accurate, and eventually will be more accurate than first supposed.
This paper will discuss the theory behind Fermi Problems, famous examples and solutions to these problems, and finally, the mathematical notation and calculaIntroduction tions proving why infinite Fermi estimates result in accuHave you ever wondered how many piano rate determinations of seemingly unanswerable questions. tuners there are in Chicago? Probably not - but how would one go about answering such a proposition? An ap- Methods/problems proximation could be severely incorrect, acutely precise, The most noted example of Fermi’s Estimate or anywhere in between. These types of indeterminable comes from the question posed above – how many piano problems which can’t be determined by mathematical or tuners are there in Chicago [1]? To take on this broad yet scientific deduction are known as Fermi Problems [3]. oddly specific inquiry, let us start with something a little bit easier to grasp. How many people are in Chicago? We don’t know the exact number, but we could assume Enrico Fermi, whom these questions are named somewhere between 2 and 3 million since it is probably after, was known for having the ability to make extremely less than one-third of the New York population (Which approximate calculations with minimal data [3]. He is somewhere around 8.5 million). Lets assume there are solved these problems by making numerous justified and 2.5 million people in Chicago. An average household has educated guesses about quantities. about 4 people, so we can say there are about 625,000 households in Chicago. Of course not all households have pianos, so let us assume 15 houses has a piano. This gives us about 125,000 pianos in Chicago. Pianos need to be tuned about once a year, so we need to find how many people it takes to service all 125,000 pianos per year. Consider a piano tuner who works full time. They
