How do surreal numbers relate to real numbers?

Question

I had the impression that surreal numbers were a subset of reals, being the smallest possible interval away from any other number you could specify. Now, after reading the book, “Surreal Numbers”, it seems that reals are a subset of surreals since 'surreal construction’ in the book includes, the naturals, the rationals, and, eventually, the reals. Just how surreal numbers relate to reals?

Answer 1

The surreal numbers contain a copy of the reals, and many numbers unlike what the reals have, too. The first sentence of the wikipedia page for the surreals confirms.

Specifically, the surreals bounded between integers (like $- 100<x<100$) which are the "simplest" between all (dyadic) rational shifts (like $x=\{x-\frac{1}{2^n}|x+\frac{1}{2^n}\}$) can serve as the real numbers.

"being the smallest possible interval away from any other number you could specify" I'm not sure what you meant by this, but if you can specify some interval, you can generally divide it by 2 to get a smaller one. And in the surreals, you could divide it by something greater than every positive integer to get a much smaller one.

Answer 2

One could say that the surreal numbers are in a sense the ordered field consisting of the largest superset of the real numbers, were it not that there are so many surreal numbers that they don't even fit into a set!

One might characterize the real numbers as the largest subfield of the surreal numbers that does not contain any infinite numbers.