To how many significant digits does Pi (and its peers) remain relevant?
Some while ago, I downloaded a file of Pi calculated to one-hundred-thousand digits. A bit later, I downloaded a different file of Pi calculated to one million digits. I thought those were impressive, but just recently I read of a computer calculation of the value of Pi made to an insanely larger number of digits. I can’t find that article again but from memory, the calculation was run to two trillion digits.
The goal wasn’t to seek the value of Pi itself to that level of precision. It was a test of the computer, to see if it could run long enough to do that calculation without some kind of malfunction coming up. It was a test of the computer’s ability to run through very long computational processes without error. In that article, reference was made to NASA depending on the value of Pi to merely fifteen digits. This seeming disparity merited a look-see.
I looked up the definition of a parsec and found its numerical value in light years to a lot of significant digits, fourteen to be truthful. I then set up the geometry on which that number was based (Figure 1).
Figure 1 This graphic provides a visual definition of a parsec.
As the earth moves around the sun, a far-off object is observed for its apparent position in the sky. Because of parallax, there is an angular shift of that apparent position at earth’s two orbital extremes. Knowing the radius of earth’s solar orbit, half of that angular shift is taken as an angle which I call theta for which the distance to that object from the center of the sun may be calculated. The implicit assumptions are that the earth’s orbit is circular and that the sun is at the center of that circle which we know is not exactly so, but we do that anyway.
When the value of theta is one arc second or one degree divided by 3600, the distance D is defined as one parsec. Table 1 derives (with some admitted finagling which I will describe shortly) the distance of one parsec in terms of light years.
Table 1 The calculation detailed here derives parsecs in terms of light years.
The finagling part here is twofold. First, I used a value of Pi to fifteen significant digits, thus mimicking NASA. Secondly, I set the radius of earth’s solar orbit to precisely that value which yields the published value of one parsec that I found online.
That orbital radius looks just about right, but just how precise these numbers really are eludes me. For example, do we really know the earth’s orbital radius to that many significant digits? Earth’s orbit is not really circular. It is slightly elliptic. What precise refinements were made to establish the published value of D to so many significant digits? I have no idea.
Colloquially however, the value of one parsec is usually taken as 3.26 light years, which is good enough for general reading and good enough to satisfy my own curiosity. I’m perfectly happy with that fifteen digit value of Pi.
John Dunn is an electronics consultant and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).
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