Let's look at some definitions:
1. The 16th letter of the Greek alphabet:
2. The ratio of the circumference of a circle divided by the diameter: C/d
3. The area of a unit circle: A=r^2 ,where r=1
4. A trancendental number, approximately 3.14159
5. The first zero of the sine function
Where did it come from? Here are a few highlights of a long and colorful history:
1. The ancient Hebrews knew about the relationship between diameter and circumference: See the Biblical references:
I Kings 7:23 and II Chronicles 4:2.
2. The Rhind papyrus, (ca. 1650 B.C.) gives 3.1604 as the quadrature of the circle.
3. Archimedes (ca. 240 B.C.) was the first known to attempt to compute pi in a scientific way. He found this
approximation: 3 and 10/17 < pi < 3 amd 10/70.
4. A Chinese astronomer thought that pi= 355/113. (ca. 480 AD)
5. Ludolph van Ceulen (1540-1610), a German mathematician, calculated pi to 35 decimal places:
3.14159265358979323846264338327950288. This approximation was engraved on his tombstone!
6. Leonhard Euler adopted the symbol in 1737 and caused its wide usage.
7. In 1761, Johann Lambert, an Alsatian mathematician, proved that is an irrational
8. David and Gregory Chudnovsky calculated over a billion digits of pi in 1989.
Using a graphing calculator or a mathematics program such as Matlab or Algebra Xpressor , we can discover pi for ourselves accurate to as many places as we like. We'll use Algebra Xpressor here. Using the definition of pi as the first zero of the sine function, we'll graph that function.
Notice that the first zero of the function is between 3 and 4. Now we'll zoom in and look a little closer.
Can you see the line crossing the x-axis between 3.1 and 3.2? Zoom in further.
Now we can see that it is just a little more than 3.14. Let's keep zooming in.
It seems the line is between 3.1415 and 3.1420. Keep zooming.
Now the line is very close to 3.1416. Let's try one more zoom.
From this graph we can see that pi is a little more than 3.14159. So we have found pi to five decimal places. This process can be continued to obtain as many decimal places as desired.