Surreal numbers as hyperseries

Abstract: Surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all ordinal numbers. Hyperseries can be regarded as the ultimate formal device for representing regular growth rates at infinity. In this paper, we show that any surreal number can naturally be regarded as the value of a hyperseries at the first infinite ordinal $\omega$. This yields a remarkable correspondence between two types of infinities: numbers and growth rates.