Difference between surreal and hyperreal numbers

Surreal numbers and hyperreal numbers both go beyond the classical real number system. Although they may seem similar at first glance, they originate from different mathematical traditions and are designed for different purposes.

• Surreal numbers
  Surreal numbers were introduced by John Conway in the 1970s. They form an extraordinarily rich and flexible class of numbers that includes all real numbers, as well as infinite and infinitesimal quantities. Their construction is based on a transfinite, recursive process in which each number is defined as an ordered pair of sets. This approach leads to a numerical system that is strictly larger than the real numbers, allowing the definition of “numbers” that are greater than every real number or smaller than any positive real number.
• Hyperreal numbers
  Hyperreal numbers were developed in the context of nonstandard analysis, mainly by Abraham Robinson in the 1960s. Like surreal numbers, they extend the real numbers by introducing infinite and infinitesimal elements. However, their construction follows a very different path, relying on techniques from mathematical logic and model theory. Hyperreal numbers are specifically intended to provide a rigorous foundation for infinitesimal calculus. They make it possible to work consistently with infinitely small and infinitely large quantities, offering an alternative yet precise formulation of classical differential and integral calculus. Unlike surreal numbers, the hyperreal numbers form an ordered field that satisfies all the usual algebraic properties of the real numbers. The central innovation is the introduction of infinitesimals (ε), that is, numbers that are strictly greater than 0 but smaller than any positive real number. $$ \forall \ n \ \in R^+ \ : \ 0 < ε < n $$

  Example. Consider a sequence whose terms are equal to 1 in the real numbers. In the hyperreal framework, the same sequence can be written as 1±ε, where ε is an infinitesimal (1±ε is read as “one plus or minus an infinitesimal”). This highlights a key idea of nonstandard analysis: in the hyperreal number system, the numbers 1 and 1+ε or 1-ε are infinitely close numbers, yet they are still distinct.

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