Hi,
Today i had decided to take an older book of mine out of the shelf about the number 'pi'. One of the first chapters (it is a highschool leveled mathbook) devotes itself on Archimedes method for finding the digits of pi. For the ones may not know, he did this by setting exact upper(QN)- and lower(pN)- barriers on the value of pi. The uppervalue is defined by the length of a polygon that circumscribes the unit circle (radius = 1/2) and the lowervalue is defined by a polygon that inscribes the unit circle. What makes the idea of Archimedes so smart is that by increasing the amount of angles these polygons have, the length of these are going to aproximate te length of the unit circle (wich is pi! ). I hope the image I have copied from the internet is visible:
Now I will adress my actual question
: So I am very keen on doing my calculations only whit pen and paper, which means I try to do them whitout a calculator when I can. Well, Archimedes had found a way to find exact mathematical expressions for the QN and PN by a method of iterating (iss that how you say it?) two formulas alternatingly (which can be relatively easy derived from the trigonometric identities from the unit circle). Archimedes had to iterate these, each two formulas, 5 times, using roman(!) numerals. I offcourse use modern notation but I still found these really large algebraic expressions, from my third iteration (stopped here as my brains were getting a bit tired). I can only imagine how complicated these expressions will become after five iterations, let alone notated in roman numerics. To specify, the answers you get out of the aproximation consist of a summation of fractures whit a whole load of non-interdevidable large numbers, square roots of 3 and square roots of square roots upon square roots of things. And than... he had to simplify these huge expressions to the aproximations of pi he actually found, wich were: 3.1408 < pi < 3.1429 .
Although I realise I may be asking a stupid question: but how did he manage to do these calculations whitout a calculator? Are there methods available today to make this kind of 'complicated' arithmatic easier or is simplifying large expressions always a laborious proces?
I am really interested to read the anwers this post may get.
thanks!
Today i had decided to take an older book of mine out of the shelf about the number 'pi'. One of the first chapters (it is a highschool leveled mathbook) devotes itself on Archimedes method for finding the digits of pi. For the ones may not know, he did this by setting exact upper(QN)- and lower(pN)- barriers on the value of pi. The uppervalue is defined by the length of a polygon that circumscribes the unit circle (radius = 1/2) and the lowervalue is defined by a polygon that inscribes the unit circle. What makes the idea of Archimedes so smart is that by increasing the amount of angles these polygons have, the length of these are going to aproximate te length of the unit circle (wich is pi! ). I hope the image I have copied from the internet is visible:
Now I will adress my actual question
: So I am very keen on doing my calculations only whit pen and paper, which means I try to do them whitout a calculator when I can. Well, Archimedes had found a way to find exact mathematical expressions for the QN and PN by a method of iterating (iss that how you say it?) two formulas alternatingly (which can be relatively easy derived from the trigonometric identities from the unit circle). Archimedes had to iterate these, each two formulas, 5 times, using roman(!) numerals. I offcourse use modern notation but I still found these really large algebraic expressions, from my third iteration (stopped here as my brains were getting a bit tired). I can only imagine how complicated these expressions will become after five iterations, let alone notated in roman numerics. To specify, the answers you get out of the aproximation consist of a summation of fractures whit a whole load of non-interdevidable large numbers, square roots of 3 and square roots of square roots upon square roots of things. And than... he had to simplify these huge expressions to the aproximations of pi he actually found, wich were: 3.1408 < pi < 3.1429 .Although I realise I may be asking a stupid question: but how did he manage to do these calculations whitout a calculator? Are there methods available today to make this kind of 'complicated' arithmatic easier or is simplifying large expressions always a laborious proces?
I am really interested to read the anwers this post may get.
thanks!
