Exercise 98Number theory › IrrationalityHigh schoolAdded 3 Nov 2026

Liouville numbers

#Liouville#transcendental#irrationality

Statement

Show that $L=\sum_{n=1}^{+\infty}10^{-n!}$ is irrational.

Hint

Rational numbers have eventually periodic decimals; $L$ has isolated $1$ s at positions $n!$ .

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Structure. $L=0.110001000000000000000001\ldots$ — $1$ s at positions $1,2,6,24,120,\ldots$ and $0$ s elsewhere.

Not eventually periodic. The gaps between $1$ s increase without bound (the positions are factorials), so the decimal is not eventually periodic.

Conclusion. $L$ is irrational.