Happy Pi day

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Pi is significant in any computation involving curves. March 14th is 3/14 and Pi is 3.14. Well, more accurately it is 3.14168, but that's still rounded. Currently, there are more than 22.4 trillion known digits, which show no hint of ending or repeating. In computations, Pi is always rounded; always inexact. But does that make any difference to us?

A quote from an article at

https://www.vox.com/science-and-health/ ... i-day-2019

"If you were to draw a circle with a diameter of 25 billion miles, using 15 digits of pi, you’d only arrive at a measurement of the circumference that’s off by 1.5 inches, NASA’s Marc Rayman explained in a post on NASA’s JPL website. And that’s good enough. If you wanted to calculate the circumference of the known universe, he explained, you’d only need 40 decimal places to be accurate within a range the size of a single atom of hydrogen (the smallest element)."

So, on the face of it, it seems Ok, if we modulate number of decimal places to suit our application of the Pi value. After all, we don't really have to deal with distances of 25 billion miles for anything but astronomical computations. But, I think that in iterative systems like computer programs, the accuracy of PI might become quite significant. Einstein observed that the most fascinating thing that he ever learned was the miracle of interest compounding. So, say we cube Pi = 3.14^3 = 30.959... But if we cube the more acculturate value of 3.14168^3 = 31.0088. The accuracy of the value of Pi that we use is dictated by the requirements of the particular problem that we are working on. So 1. How do you determine the decimal places of accuracy required of Pi, for your particular application? 2. Error keeps compounding like interest, so If we take 3.14^16 = 89304710.96, and 3.14168^16 = 90072280.17. So will we ever run into a problem with iterative systems which keep building up error, due to the initial rounding of Pi? No answers here .

Happy Pi Day to you, also.

For those who don't know, it's not just that we have calculated trillions of digits of pi without reaching a stopping place or finding any repeats; we can actually prove mathematically that the decimal representation of pi neither ends nor repeats.

happy π day