Now that we have considered rational numbers as repeating decimals, we can also ask ourselves what to do with an infinite sequence of digits that does not contain a repetitive part. We can easily construct such a number: \[0.12\,112\,1112\,11112\,111112\ldots\] The white space in the above number indicates the construction better: first a one and a two, then two ones and a two, then three ones and a two, then four ones and a two, and so on. For this number, say \(x\), we know that \[\begin{array}{c}0.1<x<0.2\\ 0.12<x<0.13\\ 0.121<x<0.122\\0.1211\,<x<0.1212\\ 0.12112<x<0.12113\\ 0.121121<x<0.121122\\ 0.1211211<x<0.1211212\\ 0.12112111<x<0.12112112\\ 0.121121112<x<0.121121112\\ \ldots\ldots\ldots\end{array}\] You see that we have always sandwiched the number \(x\) between two values that get ten times closer together at each step. So we can round this number correctly to any decimal place we want. In other words, on the number line we can indicate with any precision within which interval the number should lie. Other than knowing that it is not a rational number, we do not much about this number. We call an infinite non-repeating decimal an irrational number.

Well-known irrational numbers are \[\begin{aligned}\sqrt{2}&=1.4142135623730950488\ldots \\ \sqrt{3}&=1.7320508075688772935\ldots \\ \pi&=3.1415926535897932385\ldots\end{aligned}\]

The rational and irrational numbers together form the set of real numbers.