^{ Type post Author Robert J. Marks Date January 21, 2021 Categorized Mathematics Tagged __featured, Coin flips, David Hilbert, Real numbers }

## Most Real Numbers Are Not Real, or Not in the Way You Think

_{Typical real numbers contain an encoding of all of the books in the US Library of Congress }

_{ Robert J. Marks January 21, 2021 Mathematics }

Pick a random real number between zero and one. The number you choose, with probability one, will contain an encoding of all of the books in the US Library of Congress. This sounds absurd, but real numbers require infinite precision and every time you deal with the infinite, things get absurd. Infinities, including the infinite number of digits to express almost every real number, don’t exist. Curiously then, real numbers are not real. How do we choose a random number between zero and one? The easiest way to explain is using binary decimals. The binary number 0.1000… with zeros forever denotes the number ½ or, in base 10 notation, 0.5. The binary decimal 0.01000… with zeros forever is the number…