Raymond-Bergman wrote:Ok we all know that Pi is 3.142.....
But how did early maths bods know that the ratio of circumference to diameter was the same for all circles?
Its something thats puzzled me for a long time, anyone have a clue?
This is no proof that will give you the value of Pi. There is, however, proofs that will show that Pi is a so-called transcendental number, meaning that it is a number without end and which never repeats. As such, it can never be represented by a ratio of integers. Only transcendental numbers are the square root of 2, Euler's number, e, and a fascinating number called omega, or Chaitin's constant.
The proofs are often elegent and quite invovlved but often surprisingly simple as well. For instance, showing that the square root of 2 is transcendental take no more than a handful of lines.
As for calculating the values of any transcendental, this is a computational task, and hence requires large amounts of computer processing time to achieve high percission of accuracy and large decimal place values.
As for the early mathematical inquisitive minds who "calculated" the value of Pi, it was essentially done experimentally. The ancient Egyptians, for example, literally would take varying lengths of rope, which they fastened to a pole and trace out a perfect circle, whose circumfrance they they measured with yet another rope. They then determned the ratips of the circumfrance to the diameter and found that that they were getting the "same" number, with one perplexing finding: as they increased the length of their rope to trace the circles, they found that they got ever more changing, trailing number. They simply didn't understand that Pi was a number that could never be calculated, even though it is a constant. Although its value is fixed, only the next decimal place can be obtained, one decimale place at a time, computationally. The "computational" method was ropes, men, sand, slates, and brain power.
It was the Greeks that forwent the physical experimenting with math and sat time and came up with what would become what we call today "proofs:" the showing of the truth of a statement in a particular context that often illuminates a larger generality or underlying struture or truth. In this case, the Greek proved that all circles, regardless of the length of their radius, are associated with the same constant, and moreoever that this constant had the strange property of never ending with any repetition and hence never able to be determined in totality, as their is an infinite number of decimal values.
Pi has a wonderful history and you should go to Wikipedia or get hold of the wonderful book The Story of Numbers by John Mcleish.
Incidentally, it was not the ancient Greeks that assigned the Greek symbol of Pi to the constant, but rather the 18th century English mathematician William Jones.
Wonderful questions you asked! I hope your interest in math burgeon, as math is a wonderful, worthwhile pursuit that has many benefits regardless of your career choice.
All the best.
