# What properties do we lose when moving from the rational numbers to the real numbers?

When we pass from the real numbers to the complex numbers, we lose total ordering. But what do we lose when we move from the rational numbers to the real numbers?

• In this question I claimed that we lose "finite representability", but I'm not sure what that actually means or how to state it rigorously. – Joe Z. Jan 31 '14 at 3:12
• Would this be precision or exactitude? – CAGT Jan 31 '14 at 3:15
• We lose total disconnectedness. – Matemáticos Chibchas Feb 1 '14 at 7:42
• That seems like a relatively weird property to lose, though, compared to the loss of total ordering in complex numbers and commutativity in quaternions. – Joe Z. Feb 1 '14 at 21:53
• Rationals have nice property: every additive function is linear. Reals doesnt. – user135508 Apr 29 '14 at 17:27

What you mean by "finite representability" could be this. Or, you might mean that for any rational number $q$, there are (finite) integers $a$, $b$ where $a/b=q$, whereas if we define the reals as the limit points of Cauchy sequences of rational numbers, then for any real number $r$, we can only be sure that there are (infinite) integer sequences $(a_n)$, $(b_n)$ so that $a_n/b_n\to r$.