How can pi have infinite number of digits and never repeat them?

First, I saw this "Infinite Monkey Theorem" that says given infinite number of tries, a monkey could write a play of Shakespeare exactly. If so, why not "given infinite number of digits of pi, a pattern will form?"

Second, "if the digits are finite, then I can write this number as the ratio of two integers." And I found out that pi is the ratio of the circumference and diameter of a circle. If it this is so, then how come pi is proved to be infinite?

• There are several misconceptions involved (I'm hopeful you'll be set straight), but I do think it's a nice question. I like well-phrased curiosity :) – pjs36 Oct 1 '15 at 23:55
• For the last question: It is not possible for the circumference and the radius to both be integers. Consider the $\sqrt{2}$ as another example - it is the ration of the diagonal of a square to the edge of the square, but it is not possible for both quantities to both be integers. – Thomas Andrews Oct 1 '15 at 23:57
• Maybe you want to calculate the probability of selecting a specific string of digits of length k (pattern) from the digits 0-9 N times with replacement, and with respect to order. This will show if the probability is high or not. – NoChance Oct 2 '15 at 0:17

For the first question: I think you are confusing things here. The Infinite Monkey Theorem does state that, but it doesn't state that the monkey should keep writing the same play of Shakespeare's over and over again in some particular pattern. The same thing goes for $\pi$.

For the second question: the thing is that neither the circumference nor the diameter need be integers.

Do you see a repeating pattern ih the digits of the following number, which uses only two digits: $$0.101001000100001000001\dots?$$

Assuming $\pi$ is normal (which seems likely, since most real numbers are normal, but is not proved), then yes, given enough digits of $\pi$, any pattern of finite length will be found. However, in order for a number to be rational—that is, the ratio of two finite integers—such a pattern would have to be unending. That is, either the digits of a rational number must be all zero after a certain point, or they must repeat forever after a certain point. The normality of $\pi$ (if it is in fact normal) cannot guarantee such an eventuality; in fact, it forbids it.

As to your other question, yes, $\pi$ is the ratio of circumference to diameter. If both of those were of integer length, then you would be right: $\pi$ would be rational. However, the fact that $\pi$ is irrational means that either circumference or diameter can be of integer length, or neither, but never both at the same time.

A complete explanation of a proof that $\pi$ is indeed irrational is beyond the scope of this answer. Further details can be found here: https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

For the second question, remember that integers are whole numbers. If the diameter of a circle is an integer, then the circumference will not be a whole number and vice versa.