Transcendental Nature of Pi
Whether or not the 10 to the tenth power to the tenth power to the tenth power decimal digit of pi exists, your editorial posing the question is unnecessarily confusing.
It reminded me of the day when my 3-year-old, proudly demonstrating his newly acquired counting skills, answered my question, “How many fingers do you have?” with the prompt and sure reply, “Nine, Dad.”
When asked to explain, he very obligingly but somewhat condescendingly counted out loud on his digits, “One, two, three, pi, four” and continued on the other hand, “five, six, etc.,” thus clearly demonstrating the correctness of his answer.
While his method was unimpeachable, and his fund of knowledge surprising (the subject of pi and circles had come up casually months before), the subtleties of integers and cardinal numbers, to say nothing of numbers irrational, imaginary and transcendental, awaited his grasp.
So, while we know by context what you meant to say, still, “Ten to the second power” does not have “100 digits”--it has three (you know, the digit before pi), nor does 10 to the sixth power have a million digits, it has seven.
And so, your question “Does it (the 10 to the 1,000th power decimal digit of pi) exist?” suffers from similar problems of imprecision. The fact that you can and did write the number so simply in one notational system (although my son prefers “the googol to the tenth power decimal digit”) answers your question self obviously. The inability of one notational system to express a number or concept or thing may merely reflect the shortcomings of that system.
When put more precisely, your question becomes somewhat less metaphysical: “Will it ever be possible for mankind to correctly express as an integral function of the null power of 10 the 10 to the one-thousandth decimal digit of pi?” The problem can then be seen merely as a question of expressibility within a limited system of expression, which endeavor, unfortunately, your writer appears overly fond of pursuing.
C. MICHAEL GANSCHOW
Santa Barbara