Essay:pi contains pi

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Pi contains pi as proved by induction:

1. pi contains pi as one significant digit in a finite representation ("3": 3.141592653....)

2. assume pi contains pi as n significant digits in a finite representation of pi. Pi must also contain pi as n+1 significant digits as the number of digits of pi is stretched to infinity.

Stated another way, no one disputes that pi includes every number that has a finite representation. Thus pi includes itself in all of its increasingly precise representations, without limit, and therefore pi contains pi itself within a vanishingly small margin of error.

Pi has no termination in its digital representation, no patterns in the digits, and no repetitive periodicity. Pi containing pi can be viewed as two strings: the original pi without termination in its digital representation, and another pi that starts within pi but also without any termination point.

Pi containing pi means that there will be a string of numbers that repeat themselves: from the beginning of pi to when it first begins to repeat itself, which then would repeat itself again and again. However, it is generally assumed that there are no repeating patterns in pi.

The flaw in this argument against pi containing pi is that the periodicity may not occur until after an infinite string of digits, and thus this creates no contradiction with pi being an irrational number.

Stumbling into the "pi within pi" amid another otherwise random string of numbers can seem eerily similar to happening upon eternity in heaven or hell. While the math question has a math answer, the implications for the existence of eternity are unmistakable.

• Infinity denial
• Stack Exchange commentary, arguing for "yes"
• Reddit commentary, with a consensus of "no"