Surreal Numbers Presentation Outline 18.304 Term Paper Revision: Professor Kleitman Paul Chou May 15, 2006Abstract Surreal numbers are a field that contain both the reals as well as infinitely large and infinitely small numbers. Similar to Dedekind cuts, surreals are cosntructed using two sets of previously created numbers, and the subsequent properties that emerge create a rich set of properties. While a still largely unexplored area, this pa-per discusses some elementary properties of surreals, construction, comparisons, and operations. In addition, interesting computations involving various forms of infinity will be described, as well as applications in the field of games and combinatorial game theory.Contents 1 Introduction to Surreal Numbers 2 1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Comparison Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Transitive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Reasoning Directly/Indirectly . . . . . . . . . . . . . . . . . . . . . . 5 2 Equivalence and Arithmetic 6 2.1 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Addition and Subtraction with Surreals . . . . . . . . . . . . . . . . . 7 2.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Examples and Applications 9 3.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Surreals and Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Surreal Numbers and Game Theory . . . . . . . . . . . . . . . . . . . 12 11 Introduction to Surreal Numbers All of us are familiar with the very basic number systems; these would include the integers, rationals, reals, complex, etc. However, there is another set of numbers in-troduced recently by John Horton Conway via Donald Knuth’s 1974 Mathematical novelette, “Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.” More formally, these numbers form a field that contain reals as well as infinite and infinitesimal numbers. They have interesting properties as well as useful applications that we will talk about today. 1.1 Construction Every number in the surreals corresponds to two sets of previously created numbers with one restriction: no member of the left set is greater than or equal to any members of the right set: x = (XL, XR) with the restriction that XL � XR This simply means that given an surreal xL in XL, and for any number xR in XR, we must have xL � xR for this to be a valid surreal. Note: for the purposes of notation, I will choose to use lowercase letters for numbers, while uppercase letters will usually mean sets of numbers. The obvious question is what numbers exist in the first place to get these set con-structed numbers started. Conway realized that the natural start would be when both XL and XR are taken to be the empty set ∅. Therefore we can see that zero can be defined as follows: 0 = (∅, ∅) We must be careful that this is indeed a valid surreal number. Based on the defini-tion, we know it must satisfy XL � XR. Since both sets are empty, we see that this is immediately true, as no element of the left set is greater than an element of the right set as there are no elements at all. 2From here, it is not hard to build more and more surreals. We now have 0 avail-able to us to use as part of the left or right set. For example, we can define -1 as follows: −1 = (∅, {0}) and similarly we can define 1 as follows: 1 = ({0}, ∅) We do a quick check with each of these numbers to confirm they are well formed. Since both of them have an empty set, we again see that the condition is satisfied, as there are no elements in either the left or right set for comparison. One interesting note is that if either XL or XR is the empty set, the condition XL � XR is always true, and therefore we can see immediately that infinitely many numbers can be created in this fashion. 1.2 Comparison Definition We now define what it means for surreal numbers to be less than or equal. The usual notation for something like this is x ≤ y. For this to be true in the surreal numbers, we must have: XL � y and x � YR Specifically, every element of XL satisfies the condition xL � y, and that x is not greater than or equal to any element of YR. With this definition, we can show some elementary properties of the numbers we have shown to exist so far. For example, using our definitions we can show that 0 ≤ 0. Since 0 = (∅, ∅), we see that indeed no number of the left set of 0 is greater than or equal to 0, and no number of the right set of 0 is less than or equal to 0. Therefore we have shown that 0 ≤ 0. In addition, we should be able to show other properties of these numbers that make intuitive sense, such as -1 not being greater than or equal to 0. That is, we aim to show 0 � −1 where −1 = (∅, {0}). 3For 0 ≤ −1 to be true, we require that x � YR, where YR in this case is {0}. But we have shown earlier that indeed, 0 ≥ 0, so therefore this is not true and we have that -1 is not greater than or equal to 0. One final example to convince that these rules produce an intuitive ordering for the surreals, we will show that -1 is less than 1. By the inequality rule, we see that −1 ≤ 1. We want to also show that 1 � −1. 1 ≤ −1 implies that 0 � −1, but we have shown that this is false. Therefore it must be that −1 < 1. 1.3 Transitive Law In comparing numbers so far, we have taken the general approach of making sure that, for example, every element of XL …