# Definition of rational numbers from real numbers

Usually the set of numbers are introduced starting from integers, from wich the rational numbers are defined using equivalence classes of couples of integer numbers. Than, from these rational numbers, we can construct the real numbers via Dedekind cuts or Cauchy sequences.

But real numbers can be defined in purely abstract way as a set that is a field with a total order that is Dedekind- complete (we can refer to Tarski's axiomatization).

Is it possible, starting from this abstract definition of reals, define the subfield of rationals (and the subset of integers)?

In other words can we define the usual sets of numbers starting from the real numbers instead of from integers?

• The integers are the least subring of $\mathbb{R}$; the rationals are the least subfield of $\mathbb{R}$. – egreg Jan 11 at 14:01
• You should clarify what kind of definitions you allow. egreg's comment gives you an algebraic definition, but, for example, we can show there is no definition of $\mathbb Q$ in $\mathbb R$ in the first-order language of rings. – Wojowu Jan 11 at 14:04
• The last paragraph in the page you refer to seems to imply that the integers (and the rationals) are used for defining the multiplication. The integers are the subgroup generated by $1$. – egreg Jan 11 at 14:06

Yes, of course. Because $$\mathbb R$$ is a field, it has a unit, $$1$$. Then you can immediately construct the integers by considering the additive subgroup generated by $$1$$: you have $$0$$, $$1+2$$, $$2+1=3$$, etc., and $$-1$$, $$-1+(-1)=-2$$, etc.
And now you can consider the subfield of $$\mathbb R$$ generated by $$\mathbb Z$$: that requires you to have the multiplicative inverses of the nonzero integers: $$1/n$$, for all $$n\in\mathbb Z$$, and $$m/n=1/n+\cdots+1/n$$ for all $$m\in\mathbb N$$.
• @EmilioNovati If you agree that this construction of $\mathbb Z$ works (and is satisfactory to you), then you can take quotients of elements of $\mathbb Z$ and this will give you $\mathbb Q$. – Wojowu Jan 11 at 14:53