Pi contains pi as showing by proof 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, pi includes every number that has a finite representation. Thus pi includes itself in all of its increasingly precise representations, without limit, and therefore contains pi within a vanishingly small margin of error.
Analogy to Life and Eternity
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.