Is this number rational?

elementary-number-theory

A number given in decimal (or any $n$-ary representation other than $n=1$) is rational if, and only if, its representation contains a block that from some point onwards repeats indefinitely. The number you describe clearly can't contain any such repeating block, so it is irrational.


Perhaps it could be mentioned that if you space out more drastically your ones $$ \sum_{i=1}^{\infty} 10^{- i!} $$ you get Liouville's number which is transcendental over the rationals.

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