Core Concepts
In this article we will discover the origin of pi, discuss what exactly it is, and explore some of its uses.
What is pi?
Pi is the ratio of the circumference and diameter of the same circle. Circumference and diameter are important measurements of a circle, and we can use pi to relate them. The circumference of a circle is the measure of the entire distance around the circle. Contrarily, the diameter is the measure of the distance across the circle.
Using the 16th letter of the Greek alphabet, pi is most often illustrated as π. The numerical representation of pi comes with a bit more nuance.
Pi is classified as an irrational number, or a number that cannot be expressed via a fraction. Irrational numbers are most frequently seen as a non-terminating, non-repeating decimal. As a result of pi’s non-terminating nature, it is commonly abbreviated in decimal form to 3.14. However, depending on the purpose, including more digits of pi may improve calculation accuracy. For example, at the Jet Propulsion Laboratory (JPL), NASA uses about 15 digits. They cite the notion that 15 digits give them a high enough accuracy that, even if a mathematical error happens, the error would only be miscalculated by a distance no more than your finger!
Occasionally, we approximate pi via a fraction — placing emphasis on “approximated,” because we cannot approximate it in fraction form. That being said, the fraction 22/7 is a common approximation. However, mathematicians generally don’t use the fraction form in calculations, although it serves as an impressive example of how closely we can approximate pi. Significant study has been done with pi and, specifically, how to approximate it. Let’s shift our discussion to the discovery of π, and what exactly makes it such a unique number.
The Discovery of Pi
The history of π is expansive, and not as easily followed through the past as other mathematical discoveries. Many records track the existence of π to the Babylonians many centuries ago! Archaeologists discovered a tablet where π was recorded as three. This is thought to reference how Babylonians found the area of a circle by taking three times the square of the circle’s radius.
We can compare and contrast the Babylonian approach with those of other civilizations throughout history. For example, an ancient Egyptian document referred to as the Rhind Mathematical Papyrus, uncovered in 1865 by Alexander Henry, contains various mathematical techniques, formulas, and strategies. Among the many approaches addressed within this papyrus is a method to find the area of a circle.
In the papyrus, work is done on a circle with a diameter of 9 units. Translations state roughly “Take away 1/9 of its diameter, namely 1. The remainder is 8. Multiply 8 times, making 64. Therefore the area is 64 setjat.” (A setjat was a unit of measurement in ancient Egypt.) This formula was the guide for how Egyptians calculated the area of a circle.
Much later in time, this fraction was found to be a very close approximation to pi: 3.16, versus the often-cited 3.14. The formula also closely resembled the modern-day area of a circle equation: A =πr2, where A is the circle’s area and r is its radius.
Calculating Pi
Archimedes is often recognized as the calculator of pi. Born in Sicily, Italy, Archimedes was a mathematician and inventor famously known for his work with circles and spheres!
While the concept of pi had been posited before Archimedes’s time, his work was revolutionary. It offered us a theoretical approximation of pi, rather than a measurement as we discussed above. In other words, he found a way to describe pi for all circles with his work!
Archimedes’s method involved a polygon of n sides within a circle. To simplify, let’s refer to the unit circle around a square, where n = 4:
To calculate the sides, Archimedes took advantage of the double angle identify for sine, which allows one to find the sine of twice the angle! Archimedes performed what we today call a “doubling” of the sides, where we multiply the current value of n by 2. With the new n value, we then calculate the new area and sides for the polygon. See below for a simplified version of the work!
If you were to continue working out the area of the polygon as the sides double, you would notice an important observation. As the number of sides increases, the area within the circle gets closer and closer to pi! Visually, the polygon also appears more circular as its number of sides increase.
Archimedes’s work uncovered the boundaries of pi via this methodology, which at that point had not been done before! His work holds up to this day, despite his lack of access to modern algebra and trigonometry, instead relying on geometric truths. Thanks to his work, we are now able to use pi across different subjects. Let’s explore what it looks like to use it in these diverse applications.
How is pi used?
In math, as we briefly discussed above, pi represents the relationship between a circle’s diameter and its circumference. As such, it is a key player found throughout geometry. We see it featured in formulas that calculate the area, volume, and circumference of not only circles, but spheres and cylinders as well!
Pi’s discovery also opened up the door for the polar coordinate system. Polar coordinates are unique because they refer to distance and angles, rather than referring to points on the (x, y) plane. In the polar coordinate system, the distance is also known as the radius, and we notate the angles in radians rather than degrees. A radian is the angle formed when the radius is wrapped around the circle, as shown below.
Radians are given in terms of pi, due to the interconnected nature of pi and a circle. The polar coordinate system is popular with mathematicians because of its simple basis in the radius of a circle.
As we can see, pi is not just a mathematical topic. It has applicability anywhere there is a curve! For instance, in space research, pi is critical in studying the orbital patterns of planets and trajectory of spacecrafts. Engineers who manage flaps on aircraft use pi to calculate how fast the motors need to spin. In nature, pi can describe the motion of the ocean, light waves, and sound waves due to pi’s relationship to circles! Meteorology uses the meandering ratio, which describes jet patterns using pi. Even statisticians can use pi to calculate population dynamics, a complex analysis of how populations ebb and flow over time.
Conclusion
Pi is found in many fields, either explicitly or implicitly, and every day serves as a means to daily life. As approximations for pi improve, we are able to more accurately explain the world around us! This concept is pivotal in allowing many industries to interact with curves and circles on an incredible scale. It also brought many students into fields like math with its popularity! March 14th is widely recognized as Pi Day because its date matches a common rounded form of pi, 3.14. This “holiday” serves a way for passionate people to share their love and appreciation for this special number over the homophonous treat, pie! As pi studies continue to advance, there seems to be no end to its uses and promise.
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