Using the Von Neumann representation we can represent the non-negative whole numbers using the empty set, e.g. $1$ as {$\emptyset$}.
How do we represent with this notation numbers like $\sqrt2, -1, \pi$ or $i$?
Asked
Active
Viewed 31 times
0
-
Start from natural numbers, then define integers as pairs, then define rationals as equivalence classes of integers then define reals as Dedekind cuts (a set or rational). – Mauro ALLEGRANZA Jan 28 '22 at 09:57
-
I believe that numbers such as $\sqrt{2}$, $-1$, $\pi$ and $i$ (or integer, rational, real and complex numbers for that matter) are constructed from the natural numbers. But these constructions requires the definition of a certain equivalente relation on the set of natural numbers and the classes will be what we call integer numbers. And process continue to the rational, real and complex numbers. See any book on set theory (the constructions are pretty standard) – ADi Jan 28 '22 at 10:01
-
See e.g. Ethan Bloch, The real numbers and real analysis (Springer, 2011) – Mauro ALLEGRANZA Jan 28 '22 at 10:11
-
I went over the details somewhat in this post here as to how to construct the other systems from the naturals. To then get to the complex numbers, there are many ways to go, e.g. using matrices as seen in this series of YouTube videos. – PrincessEev Jan 28 '22 at 10:20