When Isaac Newton computed pi to 15 decimal places in the 17th century, he wrote, “I am ashamed to tell you to how many figures I carried these computations, having no other business at the time.”
Interestingly enough, that is about as accurate as anyone needs to get for practical purposes, as NASA’s Jet Propulsion Laboratory uses exactly 15 decimal places for pi, 3.141592653589793, in all of its “highest accuracy calculations, which are for interplanetary navigation.” Thanks to modern computing, we have now calculated pi to over 62.8 trillion digits and counting, and if we want to know just a specific digit of pi, we can find them out to the quadrillions.
But where does the number pi come from? Represented by the Greek symbol π, pi is a ratio that represents the diameter of a circle against that circle’s circumference. It is an irrational number, meaning it cannot be expressed as a whole fraction, has a never-ending sequence of digits, and will never fall into a repeating pattern of digits. Specifically, π = C/d.
If you were to take a circle’s diameter and multiply it by pi, you would have the circumference—which is indeed the same as the formula 2πr. If you’re having trouble holding all that in your head, this visualization should help clear up the relationship between diameter, circumference, and pi:
But pi corresponds to a number of different measurements as well, which is what makes it so useful in a wide group of calculations. For example, pi can be measured according to radians, the standard unit for angular measurements. A radian (abbreviated “rad”) is the angle created at the center of a circle when the arc is equal to the same length of the circle’s radius. There are exactly π radians in half a circle, and 2π radians in a full circle. Again, it helps to see it visualized:
Pi also shows up when measuring circles and sine waves, which are graphed curves that represent smooth and repetitive oscillations. When you plot the positions of points around a circle, you get a sine curve, and the curve crosses the x-axis, representing a vertical value of 0, at π, 2π, and so on.
As you can see, the ratio of a circle’s diameter to its circumference pops up just about everywhere. Of course you need pi to calculate circumference and area of a circle, but you also need pi for things like the cosmological constant, Einstein’s field equation of general relativity, Coulomb’s law for the electric force, and the period of a pendulum.
Without pi, we never would have built complex infrastructure, advanced our knowledge of mathematics, or visited other planets. This is why every March 14, we celebrate pi with its very own day.
This story was published in 2018. It’s been updated for Pi Day.
Jay Bennett is the associate editor of PopularMechanics.com. He has also written for Smithsonian, Popular Science and Outside Magazine.
