Re: Last digit of pi [Was: complete, sure] (Score 1)

Actually, the last digit of pi either 5, 9 or 4.

This can be proven, if somewhat bizarrely, by showing that since pi is the sum of an infinite number of rationals, and there are an infinite number of them that have decimals which repeat forever, that for a hypothetical digit position that is infinitely far away from the decimal point, each digit from 1 through 9 would occur infinitely many times. The sum of 1 through 9 is 45. Since each digit occurs the same number of times, the sum in this column must be 45 times some number which has a last digit of 5 or 0. Since this hypothetical infinityith digit is the last digit of pi, there is no carryover from following digits sums, so the last digit must either be 5 or 0. If it were 0, then you could drop this digit and perform the same task as before, but this time since you are adding an extra 4, the sum must either end in a 4 or a 9 for this digit.

QED

Two big problems (in addition to the fundamental issue that since pi is irrational it can have no 'last' digit):

First, just because each digit occurs infinitely often doesn't imply that each occurs equally often. For a simple example, it's trivial to write an inifinite series of digits that contains each digit an infinite number of times and never sums to a multiple of 45: 1 0123456789 0123456789 0123456789 0123456789 ...

Second, you later state that the final non-zero digit of any multiple of 45 must be a 4, 5, or 9, but this is also false: 180, for example. In fact, the last non-zero can be any digit (except for zero, of course): i.e. 90, 180, 270, ...

In the future, sir, please remember that you are allowed only one completely outrageous statement per post, and comport yourself accordingly.