We all know about pi and its common numerical equivalent for practical purposes (3.14). But where did pi come from, and why? The development of pi over the centuries is an example of mathematical innovation. As you know, today is Pi Day. If you’re interested in links to help students engage in pi for Pi Day, skip to the bottom. If you want to know a bit more about pi … keep reading.
First, pi is the ratio of a circle’s perimeter to its diameter.
The short history is pi was first approximated by the area of polygons with n=6 sides (a hexagon) that is inscribed (inside a unit circle with diameter = 1) and circumscribed (around the same circle). Pi was approximated to be between 3 and 4. Over a long stretch of time the polygon was bisected and bisected and bisected … and you get the point. The last time this method was used to approximate pi is in 1630 using a polygon with n=1040. At this time, pi had 38 digits to the right of the decimal, but it took over 25 years to make this approximation.
That’s a long time to simply calculate a very precise numerical representation of pi. Here is where the innovation happens. In 1666 Sir Isaac Newton, while recovering from Bubonic plague (what discoveries may come from the conditions Covid-19 placed on society?) decides there has to be a better way. Through widespread understanding of Pascal’s triangle and the development of the binomial theorem, Newton “breaks” the rules of algebra, and applies calculus to the geometric theories of circles, and discovers the irrational nature of pi allowing anyone with a computer to calculate pi to whatever level of precision they desire. I’m sure everyone can find pi on a calculator.
This isn’t about how many digits of pi we can calculate, but rather to recognize that something as mundane as pi has a rich history of innovation that allows anyone to use it in its most simplistic form (3.14) in the discovery, creation and invention of both simple and complex ideas. Newton provides an example of how the following three aspects of mathematical innovation come together for a discovery that is now used by every student across the globe.
Arte Scienza – development of balance between science and art, logic and imagination. [Pascal’s triangle and the binomial theorem]
Conneccione – everything is connected to everything else. [pi is an amalgamation of algebra, geometry and calculus]
Curiosita – the curiosity to find the connections. [Newton was curious about how to calculate pi without the arduous task of bisecting polygons]
This line of thinking allows mathematical and scientific innovation to propagate into other disciplines. Here is a brief timeline of pi and it’s uses in other innovations.
Archimedes uses the geometry of a circle to introduce the concept of pi …
Keplar’s laws of planetary motion …
Galileo’s pendulum …
Euler’s use of algebra, trigonometry and geometry to develop Euler’s constant and what some consider the most beautiful formula in mathematics using 5 important constants:
Gauss’ normal distribution …
Einstein’s theory of relativity!
To hear more about the history of pi and it’s discovery check out the Veritasium youtube channel.
For resources to connect innovations in mathematics with your students check out MoMath .
If you’re looking for activities for Pi Day, What We Do All Day offers some great projects, and Jo Boaler just updated her youcubed website with this Finding Pi activity.
CSD Innovation Blog 