Are there any practical applications of pi, aside from calculating the area or circumference of a circle?

pi

Thank you to Linda Johnson from Walnut Creek for the question. Dr. David H. Bailey, co-discoverer of the Bailey–Borwein–Plouffe formula for pi, guest authored this post. Many presume that π is only used for geometric measurements, say to calculate the area of circle or its circumference. While such calculations are done many, many times every day in modern science and engineering, there are numerous other context and applications, some of them quite surprising, where π appears.

One application that will doubtless be familiar to students taking high school statistics classes is the “normal probability distribution”, which governs a wide range of natural phenomena, ranging from rolls of dice and student test scores to measurements of distant supernovas. The normal distribution is given by the formula:

where the μl is the mean or average of the distribution and σ is its standard deviation. π appears prominently in this formula, even though there is no discernible connection to circles or geometry.

Another very important practical application is in the field of "signal processing”. A fundamental operation here is the “Fourier transform”, which converts a signal to a frequency spectrum. Your cell phone does a Fourier transform when it communicates with the local cell tower. Even your ear performs a Fourier transform (although not by digital computation) when it distinguishes sounds of different pitches, or when you recognize a friend’s voice. Mathematically speaking, the Fourier transform is the formula (don’t worry if you don’t understand how it works):

where e is Euler's Number, a famous mathematical constant equal to 2.71828..... Note that π once again appears prominently in this formula. This formula is evaluated digitally in your cell phone, by means of a certain clever algorithm, known as the “fast Fourier transform” or “FFT”, which was discovered by mathematicians in the 1950s (although Gauss may have inadvertently discovered it in the 19th century). It is quite likely that variations of the FFT algorithm are the most widely used numerical algorithm (i.e., computational scheme based on a mathematical formula) performed on computers worldwide. And yes, each of these involves π — in other words, your cell phone or smart phone has the numerical value of π somewhere buried in its computer logic, possibly in more than one place.

π may even be embedded in the fabric of the universe itself. Some formulas of quantum mechanics, which governs the microscopic world of atoms and nuclei, involve π. For example, the Schrodinger equation, when applied to the hydrogen atom, gives the formula (again, don’t worry if you don’t understand it):

Note π in the denominator of the second term on the right-hand side. And π appears prominently in Einstein’s “field equations” of general theory of relativity, which governs the universe on the very largest scales:

In other words, π appears throughout modern science, and is used every day in literally billions of computations, from large-scale supercomputers simulated the earth’s climate to cell phones and key-ring finders. And even more applications would doubtless be found in the future.