From: 3blue1brown
The way mathematicians approach problem-solving can be illuminated by examining the work of historical figures like Leonhard Euler. Rather than adhering rigidly to conventions, a flexible and context-dependent approach to mathematical concepts can lead to profound insights and extensive output.
Euler’s Flexible Use of Pi
The ongoing “pi vs. tau” debate highlights how different fundamental circle constants could be chosen. While pi (π) is commonly defined as the ratio of a circle’s circumference to its diameter (approximately 3.14) [00:00:14], some advocate for tau (τ), which is the ratio of circumference to radius (approximately 6.28) [00:00:11] [00:00:19].
Historical documents, including the notes and letters of Leonhard Euler, reveal a fluid approach to this constant [00:00:57]. Surprisingly, Euler was found to define “pi” as the circumference of a circle with radius 1, equating it to the 6.28 constant now called tau [00:01:18] [00:01:22]. He likely used the Greek letter pi as a “p” for perimeter [00:01:28].
Despite this, Euler was also responsible for establishing the 3.14 constant as the standard. His 1748 calculus book defined “pi” as the semi-circumference of a unit circle [00:01:45] [00:01:53] [00:02:09], a definition that spread throughout Europe and the world [00:02:16]. Furthermore, Euler occasionally used “pi” to represent a quarter turn of a circle (what is now known as pi halves or tau fourths) [00:02:33] [00:02:37].
Euler’s use of the symbol “pi” was more akin to the modern use of the Greek letter theta (θ), which represents an angle that can vary depending on the context [00:02:41] [00:02:44]. He allowed “pi” to represent whatever circle constant best suited the problem at hand [00:02:58]. While he typically framed things in terms of unit circles, the 3.14 constant was consistently understood as the ratio of a circle’s semi-circumference to its radius, rather than its circumference to its diameter [00:03:06] [00:03:08].
Euler’s Focus on Problem Solving
Euler’s approach stemmed from his relentless focus on solving problems. He was incredibly prolific, writing over 500 books and papers during his lifetime, equating to about 800 dense math pages per year [00:03:37] [00:03:41] [00:03:47]. After his death, another 400 publications surfaced [00:03:47]. His mind was not preoccupied with which circle constant was “fundamental,” but rather with the specific task in front of him [00:03:58] [00:04:02]. For some puzzles, the quarter circle constant was most natural; for others, the full or half circle constant was more appropriate [00:04:09] [00:04:13] [00:04:15].
A General Lesson for Math Education
Euler’s pragmatic use of notation offers a valuable lesson for mathematical education and problem-solving. Often, math education focuses on which of several competing views is “right” [00:04:21] [00:04:24]. However, questions about the “correctness” of definitions or rules, when isolated, are not as important as the focus on specific problems and puzzles [00:04:41] [00:04:45] [00:04:47].
When standards arise, they should be evaluated within a specific context [00:04:53] [00:04:55]. Different contexts will naturally lead to different answers about what seems most appropriate [00:04:59] [00:05:01]. Euler’s incredible output and transformative insights were correlated with his flexibility towards conventions, rather than a rigid insistence on what standards were objectively “right” [00:05:06] [00:05:09] [00:05:13]. This suggests a general approach where adaptability to the problem’s demands takes precedence over adherence to a single, universal convention.