# On Dedekind Cuts and Decimal Expansions

In grade school and high school, I was taught a real number is a number with a decimal expansion--that is, a finite sequence of digits followed by a decimal point followed by an infinite sequence of digits.

When I moved on to studying analysis, I was introduced to the Dedekind cut construction of the real numbers, and then proved every real number could be expressed as a decimal expansion.

Now presumably, the real numbers could also be rigorously constructed as decimal expansions, very similarly to the Cauchy sequence construction.

Is there a particular reason why, when Dedekind performed his original construction of the reals, he chose to use cuts rather than decimal expansions (or base 2 expansions for that matter)? Is there some unforeseen difficulty in rigorously constructing the reals by means of decimal expansions?

• Dedekind cuts are more natural, in a sense. If we construct the reals using a specific base (base $10$, for example), it's a bit arbitrary. Dec 2, 2016 at 21:59