I was just wondering if you're really serious about your proof of the rationality of pi or whether you did it just so people like me would write to you to ask if you were really serious about your proof of the rationality of pi.
— <jezek [AT] jupiter.scs.uiuc.edu>
Also featured on http://www.ihatecalculus.com/
It all started when I offered my original proof that Pi was rational, using induction on the number of significant digits!
A fourteen-year-old (back then) genius gives us this proof based on the fact that there are no infinities in the physical universe!
Ben Guaraldi discovers a use for this new and wonderful branch of mathematics.
Jacob Walker offers us his observations on why infinite sums of rational numbers are rational. I'm keeping his old comments here for historical reasons.
Proofs are in the indented text; text which falls beyond the indentation is commentary by me or someone else.
Other Goofy Stuff
[Thanks go to Joe Hoffman, firstname.lastname@example.org, for an improvement.]
We will prove that pi is, in fact, a rational number, by induction on the number of decimal places, N, to which it is approximated. For small values of N, say 0, 1, 2, 3, and 4, this is the case as 3, 3.1, 3.14, 3.142, and 3.1416 are, in fact, rational numbers.
To prove the rationality of pi by induction, assume that an N-digit approximation of pi is rational. This number can be expressed as the fraction
M / 10N. Multiplying our approximation to pi, with Ndigits to the right of the decimal place, by 10Nyields the integer M. Adding the next significant digit to pi can be said to involve multiplying both numerator and denominator by 10 and adding a number between between −5 and +5 (approximation) to the numerator. Since both 10N + 1and 10M + Afor A between −5 and 5 are integers, the (N + 1)-digit approximation of pi is also rational. One can also see that adding one digit to the decimal representation of a rational number, without loss of generality, does not make an irrational number.
Therefore, by induction on the number of decimal places, pi is rational. Q.E.D.
Note that this proof can be used to prove the rationality of other mathematical constants, such as e, √2, etc.
[And finally, an author's call for papers, of sorts. If you send something to the author, then please send it to me as well so I can put it here. — dse]
As every true distance in the physical universe is a finite number with a finite number of digits, and as the circumference and the diameter of a circle are, supposed to be, in reallity, distances in the physical universe, then if
pi = c/d, as the product of any finite number multiplied by any other finite number CANNOT equal infintity, and as true infinities are impossible in the physical universe, pi's digits cannot go on forever.
I can, however, disprove my theory, but only with abstract circles. In the physical universe, however, it holds.
Here it is: in a perfect circle, there would be an infinite precision, as there would have to be an infinite amount of points. So, you could get an irrational number. However, there is no such thing as a perfect circle in the physical universe, so I think I'm safe.
P.S. — Oh, I'm looking for some geometry-based (NOT INDUCTION based. Sorry, but I hate inductions) data on my theory. I wonder if there is something wrong with it. After all, in thousands of years, has no one has ever thought of it before?
Ben now uses this Theorem to prove that pi is rational:
The Fundamental Postulate of Alternative Algebra. One may divide by zero.
The Fundamental Theorem of Alternative Algebra. Any two numbers are equal.
Proof: Consider two numbers, a and b, which do not equal zero. The steps follow:
a = b
a2 = ab
a2 − b2 = ab - b2
(a − b)(a + b) = b(a - b)
0(a + b) = 0b
a + b = b
2b = b
2 = 1
Considering this Theorem, that 2 = 1, one may prove that any two numbers are equal. Then, take a circumference c and a diameter d:
2 = 1
1 = 0
1(c − d) = 0(c − d)
c − d = 0
c = d
Since pi is the ratio of the circumference to the diameter, pi now equals one. Thus pi is rational. And an integer for that matter. In fact, pi equals e. You get the picture.
Jacob himself once argued that this is wrong:
Another belief I hold is that either irrational numbers can not be expressed as the sum of an infinite number of rationals, or that irrational numbers are truly rational. The proof for this is easy. A rational number plus a rational number is rational. Thus no matter how many times you add a rational to a rational, even an infinite number of times you will still have a rational. But if this were true what would be the infinite sum of 3.14... or in other words 3/1 + 1/10 + 4/100 + ... if every digit corresponded with the digits of Pi? I don't know...
The problem with my argument, is that it is true that a rational plus a rational is a rational for any finite number of iterations. I can prove that. But when one goes to an infinite number of iterations, things may get skewed, like the fact that when you add an infinite number of 0's you maybe won't get 0. (see his second paragraph)
I just want to say that his insights are really quite interesting. What you just read was a mere teaser. Go and visit that site. Because even if you disagree it will at the very least provide you with food for thought.
Since this page was published on the Internet sometime in November 1994 (it was available as my .plan file for a while before then), a few visitors have explained to me a supposed hole in my theorem concerning division by zero, or something else. If you are one of those people, then you can safely rest assured that that this page is a joke, and that the only reason I didn't respond to some of you was simply that I couldn't figure out what to say.