The mathematical constant pi is calculated by taking the circumference of a circle and dividing it by its diameter. This value is accepted to be 3.1415926 approximately.
But is it? Prove it! Measure a circle and see if you can calculate that number using that method.
I’ve been in my new house for almost a year now and have barely unpacked let alone decorated. So I’ve been thinking a lot lately about what I want to do and what kind of pictures I want to put on the walls. As I was drifting off to sleep last night, an image came to me of a circle with pi inscribed around the circle out to a hundred digits or so.
It is easy enough to look up this number. Mathematicians claim to have calculated pi to over three trillion digits. The first hundred digits should be readily available.
But then it hit me. Given the standard definition that pi is equal to a circle’s circumference divided by its diameter, it is impossible to calculate pi to more than a few digits.
The precision to which you can calculate pi is dependent upon the precision of your measurement of the circumference and diameter. Taking measurements by hand will only be possible to within about a millimeter. If you are working with a large circle, where you are using units of meters, a precision of millimeters means that you are working with three digits of precision.
The second problem is that you need a perfect circle. Try drawing a large circle that is exactly the same radius, at every point, to within a millimeter. Given enough time and proper tools I could probably manage it. But again, using units of meters and a precision of millimeters yields only three digits of precision. I am certain that I could never achieve any better.
Is it possible to draw a circle perfectly enough and make measurements accurately enough to calculate pi to more than a few digits? You may manage slightly better but I would question the possibility of more than three or four digits, let alone anything beyond that.
No matter how perfectly you draw the circle, it is bound to have imperfections that would alter the outcome if you are trying to calculate pi to any significant precision. Similarly, measurements are severely limited in how accurately they can be made. Beyond a certain point even the measurement tools themselves become inaccurate due to the inherent shrinking and expanding that wood and metal will undergo due to changes in heat and humidity.
In short, I do not believe that it is possible to calculate pi directly with any precision beyond a few digits.
There are numerous simple calculations that will approximate pi: 22/7, 28/5, 333/106, 355/113, 52163/16604, (16/9)^2, sqrt(10), etc.
Although useful in some situations, these approximations are not pi and will cause problems in calculations requiring more precision.
Archimedes (circa 250 BCE) estimated pi using polygons. He created two polygons, one interior to a circle with the points of the polygon having the same radius as the circle, and the second one exterior to the circle with the sides of the polygon having the same radius as the circle. It is easy to calculate the perimeter of the two polygons, with the circumference of the circle then somewhere between these two values. The greater the number of sides of the polygons, the narrower the difference between the two and the more precisely you can estimate the circumference of the circle (and therefore the more precisely you can calculate pi).
Unfortunately, it requires polygons with over 12,000 sides to estimate pi to just seven digits. That is a long shot from the insanely accurate precision claimed by current day mathematicians.
It is, however, the most accurate method with a clear, easily-understandable proof that the result is, in fact, a true representation of pi.
Infinite Series Approximations
There are a great many infinite series calculations that have been developed to approximate pi. Some were created an unimaginably long time ago, while better and faster methods are constantly being discovered. With the recent availability of enormous computing power, such calculations have been used to approximate pi to over THREE TRILLION digits!
I have to assume that greater minds than mine have verified that these numbers actually do represent pi. However, I have been unable to track down the associated proofs. The infinite series equations and iterative algorithms that I have seen do not intuitively present any relationship to circumference over diameter. (hereafter are just a few)
Three terms of calculus, taken for fun decades ago, is no match for mathematicians who live in the world of numbers on a daily basis. Therefore, I have to accept their findings. But still, I’d very much like to understand the relationship myself. The equations are straightforward enough, but I just don’t see how they relate to the problem at hand. Where, in the equations above, are circumference and diameter represented? Without them, where is the proof that these equations actually describe pi?
Regardless of their accuracy, they are derived through approximation. It is a perfectly valid approach but it is not direct. For some reason, I find it disconcerting to be faced with the sudden realization (in the midst of thinking about home decorations and trying to fall asleep) that there is in fact no accurate method of directly calculating pi with any significant precision.