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THE HYPERREAL NUMBERS

THE HYPERREAL NUMBERS. We want to introduce two new types of numbers, namely, Infinite and infinitesimal numbers. What we should do here is an extension from R, the *R, the hyper-real numbers.

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THE HYPERREAL NUMBERS

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  1. THE HYPERREAL NUMBERS We want to introduce two new types of numbers, namely, Infinite and infinitesimal numbers. What we should do here is an extension from R, the *R, the hyper-real numbers. Specifically, R should be a subset of *R and still preserve all the properties we know and next point is to make sure that our new field *R is ordered. That is, for every a, b in *R, one and only one of the followings hold: a=b, a<b, or a>b.

  2. Construction of Hyperreal numbers The first point in our agenda is to establish some measure of equality and order between two hyper-real numbers. When comparing two hyper-real numbers, a and b, we can form three disjoint sets: the agreement set (set of indices of corresponding equal terms of the sequences), and a-greater set (set of indices of corresponding terms greater in a than b), and the b-greater set (whatever is left-over.). We want to define a definition of largeness which will allow us to choose exactly one of these three terms, so that we can choose one and only one “large set”.

  3. Let us first consider when we want to say that two infinite sequence are in fact the same hyper-real number. Since we want a measure of equality, we want an equivalence relation. (Reflexivity, Symmetry, transitivity must hold.) The agreement set of a sequence with itself will be the set of natural numbers. This is our first requirement of “largeness”: 1- )The set of natural numbers N={1,2,3,…..} is large. This ensures the reflexivity. Symmetry will be satisfied trivially, but transitivity requires more thought.

  4. Now suppose that we know that a=b, b=c. This means that the agreement set between a and b, Eab, is large, and so is the agreement set Ebc. The intersection of Eab and Ebc is a subset of Eac. We want Eac to be large, so that we may say that a=b and b=c implies a=c. This is our next requirement of largeness. 2- ) If two subsets of N are large, then all supersets of their intersection are also large. In particular, this condition entails that if A and B are large, then so is their intersection. Also, any superset of a large set is large. 3- )The empty set is not large.

  5. 4- ) Set B is large if and only if its complement is not large. It is now apparent that we will need some sort of rigorous criterion to be able to filter out which subsets of N are large and which are not. A filter F is a set that only includes all large sets, where large sets are only as those described in condition 1 and 2 above. If a filter includes all large sets as defined by conditions 1-4 above, It is called an ultra filter.

  6. 5- ) A finite set is not large. An ultra-filter that also satisfies condition 5 is enough to define an order and equivalence relation between any two hyper real numbers. A non-principal ultra-filter F is a set of subsets of the natural numbers that satisfies the following: 1. F contains the set of natural numbers N. 2. If F contains the subsets B and C, then it also contains their intersection. 3. If F contains the subset B, then it contains all of the supersets of B as well. 4. F does not contain a subset B if and only if it contains its complement.

  7. 5. F does not contain any finite subsets of N. With these remarks, the most important task of our construction is complete. /==========================\ HYPER-REAL ARITHMETIC \==========================/ Let F be a fixed non-principal ultra-filter on N. Define the following three relations module F, for any infinite sequences of real numbers a and b. a=b if the agreement set between a and b is in F,

  8. a>b if the a-greater set is in F, a<b if the b-greater set is in F. From now on, saying that a subset of N is “large” means that it is included in our ultra-filter F. Define the following equivalence class on an infinite real sequence r: C [r]:={all real infinite sequences s such that s=r},

  9. and a hyper-real number r is the equivalence class C [r]. The set of hyper-real numbers is denoted by *R. Hyper-real arithmetic is all done term-by-term. To add two hyper-real numbers, we add all of the corresponding entries, to multiply them, we multiply corresponding terms. The set of real numbers R is a subset of *R, and a member r of R is the equivalence class identified by the constant sequence r. (I. e, r:=(r, r, r, r, r, r, ……..)).

  10. Let us give a few examples, where the choice of F is irrelevant, since I will only use the fact that it contains all cofinite sets. The hyper-reals we will use in these examples: 0=(0, 0, 0,…….) 1= (1, 1, 1, ……..) 2=(2, 2, 2, …….) a= (1, ½, 1/3, …..)

  11. b= (1, 2, 3,….…,n,….) c= (2, 4, 6, ….,2n,…..) d= (0, 0, 0, .....fifty more zero, 54, 55, 56……) f= (3, 6, 9, …..,3n,….) Now, we can see the following order and equivalence relations among them. 0<1, d=b, f>c>b>a.

  12. Examples of addition, subtraction, multiplication, division: a+0=(1+0, ½+0, …….1/n+0,…….) =a fx1=(3x1, 6x1, ……,3nx1,……..) =f b + c =(1+2, 2+4, 3+6,……,n+2n,…..) =(3, 6, 9, ………………………) =f

  13. References • 1-Lectures on the Hyperreals, by robert Goldblatt • 2-Nonstandard Analysis: Theory and Application by Leif O. Arkeryd. • 3-Non-standard Analysis by Abraham Robinson. • 4-The math forum- Ask Dr. Math: A Mathematical Essay: Non-standard Analysis and the Hyper-real numbers.

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