メイドインアビス

Introduction#

The rational number system is inadequate for analysis concepts (such as convergence, continuity, differentiation, integration etc.). For instance, it is well known that there is no rational number $p$ such that $p^2 = 2$.

Rudin considers this situation further.

Proposition

Let $A$ be the set of all positive rational numbers $p$ such that $p^2 < 2$ and $B$ be the set of all positive rational numbers $p$ such that $p^2 > 2$. There are no maximum in $A$ and no minimum in $B$.

In other words, we need to prove that for any $p \in A$, there is $q$ such that $q>p$ but $q\in A$ and vice versa. We need to construct such $q$ from $p$ while having $q^2-2<0$ (or $q^2-2>0$).

Observation

In Newton’s method, to find the root to the equation $p^2 -2 = 0$, given initial guess $p'$, we can guess a better estimation to the root $q'$ with:

$$q'=p-\frac{p'^2-2}{2p'}$$

Similarly, Rudin suggests in a way similar to a damped Newton’s method:

We can verify that this construction is valid:

which signedness depends on only $p^2 - 2$, i.e. $q^2 - 2$ has the same sign as $p^2 - 2$. This above discussion shows that there are certain gaps in rational numbers.

Ordered Set#