Calculating Pi... By Hand!
Entry posted by Drew Espinosa
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Today, we know π to well over the trillionth decimal place. It is a number that has shown up throughout mathematics, so it is no surprise that mathematicians have been dealing with the number for millennia.
The mathematicians of the Ancient world observed that the circumference of a circle was just over three times the diameter. So, they set out to estimate the value of π… and began a centuries long journey to take π from being “just over 3” to “3.14159265... etc.”
Throughout the world, various methods were devised to approximate π, however, the one I’ll be focusing on is the method developed by Archimedes.
He knew that the perimeter of an inscribed polygon would be less than the circumference, and the perimeter of a circumscribed polygon would be greater than the circumference.
So, he used this principle to work out a way in approximating π. He inscribed a hexagon on a circle, calculated the lengths of the hexagon’s sides, and then calculated its perimeter. He then doubled the sides of that hexagon, to get a dodecagon. And repeated the process of calculating its side length and perimeter.
And he repeated that process for a 24-gon, then a 48-gon, and finally a 96-gon.
Ah, but wait! He wasn’t done, because he then circumscribed a hexagon on the circle, and repeated the entire process. Archimedes was able to calculate the perimeters as 223/71 for the inscribed 96-gon and 22/7 for the circumscribed 96-gon.
Meaning, π was somewhere in between those two numbers, or in mathematical notation:
223/71 < π < 22/7
And since both fractions begin as 3.14 in decimal form, Archimedes found the first three digits of π.
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While Archimedes, was the first to use this method, he certainly wasn’t the last. Which brings us to Ludolph van Ceulen, a Dutch mathematician from the 16th Century.
This man gave the most accurate approximation of π up to that point, all thanks to Archimedes’ method.
The only difference was that he started with a square, instead of a hexagon.
The following is to give you an idea of what he did:
First, consider the circle above. We will be inscribing and circumscribing polygons on it.
The perimeter of the inscribed square is about 2.83, and that of the circumscribed square is 4. Which gives us the first inequality:
2.83 < π < 4
The perimeter of the inscribed octagon is about 3.06, and that of the circumscribed octagon is about 3.31. Which gives us the second inequality:
3.06 < π < 3.31
The perimeter of the inscribed hexadecagon (16-gon) is about 3.12, and that of the circumscribed 16-gon is about 3.18. Which gives us the third inequality:
3.12 < π < 3.18
As you can see, these inequalities hone in on π, eventually getting the digits of π as you continue doubling the sides of the inscribed and circumscribed polygons.
As for van Ceulen, he doubled the sides until he had 262-gons. And with that, he found π to the first 35 decimal places:
3.14159265358979323846264338327950288
This was an impressive achievement. So impressive in fact, that for a time, π was called the Ludolphine Number in Germany.
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Of course, this record would be surpassed over the next couple centuries, when mathematicians began using infinite series to find π (where before they could only find tens of digits, they could now find hundreds).
And those records were broken during the last several decades, thanks to computers, and we now know π to well over a trillion decimal places.
We have certainly come a long way.
If you have any questions, I’ll be happy to answer them. 
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