How can irrational numbers be written as decimals

Question

It is commonly stated that irrational numbers can be written as decimals. But the thing is, the decimal would have to be infinite in length.

So why can an irrational number be written as a decimal if one is not able to complete it?

Thanks!

Answer 1

Not only can we not write irrational numbers as decimals, depending on what you consider to be a valid "decimal", we can't even write all rational numbers as a decimal. $$ \frac{1}{9} = 0.111111.... = 0.\overline{1} $$ do you consider the right-most expression a valid decimal? All it really is, is a shorthand notation for telling you how you could start writing down a sequence of decimal numbers that approximate the rational number ${\frac{1}{9}}$.

The same is true for irrational numbers, but the rules for approximating them via decimals is (mostly) more complicated than approximating rationals via decimals. Rational numbers will eventually repeat themselves in decimal notation, and any decimal that eventually keeps repeating will be rational. For example, $$ 0.1122453453274\overline{231} $$ I can tell you will be rational without any further calculation simply because it ends with a repeating sequence of ${231231231231....}$. Irrational numbers will not eventually keep repeating themselves.

Answer 2

Let $x$ be a real number. When we say $x$ can be written as a decimal, it means there exists an integer $N$ and a sequence $(a_{k})_{k=1}^{\infty}$ of digits, elements of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, such that $$ x = N + \sum_{k=1}^{\infty} \frac{a_{k}}{10^{k}} = N + \sup_{n \geq 1} \sum_{k=1}^{n} \frac{a_{k}}{10^{k}}. $$ If $x$ is irrational, this representation is unique. (The situation is slightly more complicated for rationals with terminating decimal expansion: To get a unique representation, we must pick between a representation ending in an infinite string of $9$s or an infinite string of $0$s, as in $1.0\overline{0} = 0.9\overline{9}$.)

What seems to be bothering you (and if so, you are not alone!) is that we cannot in general know all the digits $(a_{k})$.

If $x$ is is the limit of some known sequence of rational numbers, such as roots of rational polynomials, or $\pi$, or $e$, or insert your favorite irrational number here (unless you have especially esoteric favorites), then we can in principle calculate as many of the $a_{k}$ as we want.

To address your question:

Since any two calculating entities will arrive at the same sequence of digits of a particular irrational $x$, the digits of $x$ exist. (Or if you prefer, ...it's convenient to speak of the digits of $x$ as if they exist.)

Since each such sequence $(a_{k})_{k=1}^{\infty}$ defines a unique real number by the "completeness axiom", any infinite decimal represents a real number.