Introduction to the HyperReals

1. Introduction to the HyperReals An extension of the Reals with infinitely small and infinitely large numbers.

2. Introduction to the HyperReals • Descriptive introduction • “Pictures” of the HyperReals • Axioms for the HyperReals • Some Properties (Theorems) of HyperReals

3. Descriptive introduction • A complete ordered field extension of the Reals (in a similar way that the Reals is a complete ordered field extension of the Rationals) • Contains infinitely small numbers • Contains infinitely large numbers • Has the same logical properties as the Reals

4. The HyperReals Near the Reals

5. The HyperReals Far from the Reals

6. Axioms for the HyperReals • Axioms common to the Reals • Algebraic Axioms • Order Axioms • Completeness Axiom • Axioms unique to the HyperReals • Extension Axiom • Transfer Axiom

7. Algebraic Axioms • Closure laws:0 and 1 are numbers. If a and b are numbers then so are a+b and ab. • Commutative laws: a + b = b + a and ab = ba • Associative laws: a + (b + c) = (a + b) +c and a(bc)=(ab)c • Identity laws: 0 + a = a and 1a = a • Inverse laws: • For all a, there exists number –a such that a + (-a) = 0 and • if a  0, then there exist number a-1 such that a(a-1) = 1 • Distributive law: a( b+c ) = ab + ac

8. Order Axioms The is a set P of positive numbers which satisfies: • If x, y are elements of P then x + y is an element of P. • If x, y are elements of P then xy is an element of P. • If x is a number then exactly one of the following must hold: • x = 0, • x is an element of P or • -x is an element of P.

9. Definition of < and > • a < b if and only if (b - a) is an element of P i.e. (b - a) is positive • a > b if and only if b < a

10. Properties of < • 0 < 1 • Transitive law: If a<b and b<c then a<c. • Trichotomy law: Exactly one of the relations a<b, a = b, b<a, holds. • Sum law: If a<b, then a+c<b+c. • Product law: If a<b and 0<c, then ac<bc. • Root law: For a>0 and positive integer n, there is a number b>0 such that bn = a.

11. Completeness Axiom • A number b is said to be an upper bound of a set of numbers A if b  x for all x in A. • A number c is said to be an least upper bound of the set of numbers A if c is an upper bound of A and b  c for all upper bounds b of A. • Completeness Axiom: • Every non-empty set of numbers which is bounded above has a least upper bound.

12. Axioms unique to the HyperReals • Extension Axiom • Transfer Axiom • Note: Actually these axioms are all that are needed as for the HyperReals as the previous axioms can be derived from these two axioms.

13. Extension Axiom • The set R of real numbers is a subset of the set R* of hyperreal numbers. • The order relation <* on R* is an extension of the order < on R. • There is a hyperreal number  such that 0 <* and <*r for each positive real number r. • For each real function f there is a given hyperreal function f* which has the following properties • domain(f) = R domain(f*) • f(x) = f*(x) for all x in domain(f) • range(f) = R range(f*) (Extension actually applies to any standard set built from the Reals.)

14. Transfer Axiom • (Function version) Every real statement that holds for one or more particular real functions holds for the hyperreal extensions of these functions • (Full version) Every standard statement (about sets built from the Reals) is true if and only if the corresponding non-standard statement (about sets built from HyperReals - formed by adding the * operator) is true.

15. Example: Deriving the Commutative Laws from the Extension and Transfer Axioms • Commutative laws for the Reals: S: aR bR a+b=b+a and a•b=b•a. The Extension axiom gives us R*, +*, •*, and * and the Transfer axiom tells us that S*, the commutative laws for the HyperReals is true. • Commutative laws for the HyperReals: S*: a*R* b*R* a+*b=b+*a and a•*b=b•*a.

16. Definition: Infinitesimal A HyperReal number b is said to be: • positive infinitesimal if b is positive but less than every positive real number. • negative infinitesimal if b is negative but greater than every negative real number. • infinitesimal if b is either positive infinitesimal, negative infinitesimal, or 0.

17. Definitions:Finite and Infinite A HyperReal number b is said to be: • finite if b is between two real numbers. • positive infinite if b is greater than every real number. • negative infinite if b is smaller than every real number. • infinite if b is positive infinite or negative infinite.

18. Theorem: The only real infinitesimal number is 0. • Proof: Suppose s is real and infinitesimal. Then exactly one of the following is true: s is negative, s = 0, or s is positive. If s is negative then it is a negative infinitesimal and hence r < s for all negative real numbers r. Since s is negative real then s < s which is nonsense. Thus s is not negative. Likewise if s is positive we get s < s. So s is not positive. Hence s = 0.

19. The Standard Part Principle • Theorem: For every finite HyperReal number b, there is exactly one real number r infinitely close to b. • Definition: If b is finite then the real number r, with r b, is called the standard part of b. We write r = std( b ).

20. Proof of the Standard Part Principle Uniqueness: Suppose r, s  R and r  b and s  b. Hence r  s. We have r-s is infinitesimal and real. The only real infinitesimal number is 0. Thus r-s = 0 which implies r = s.

21. Existence: Since b is finite there are real numbers s and t with s < b < t. Let A = { x | x is real and x < b }. A is non-empty since it contains s and is bounded above by t. Thus there is a real number r which is the least upper bound of A. We claim r  b. Suppose not. Thus r  b and Hence r-b is positive or negative. Case r-b is positive. Since r-b is not a positive infinitesimal there is a positive real s, s < r-b which implies b < r-s so that r-s is an upper bound of A. Thus r-s  r but r-s < r. Thus r-b is not positive. Case r-b is negative. Since r-b is not a negative infinitesimal there is a negative real s, r-b<s which implies r-s < b so r-s is in A and hence r  r-s but r < r-s, since s<0. Thus r-b is infinitesimal. So r  b.

22. Infinite Numbers Exist Let be a positive infinitesimal. Thus 0 <  < r for all positive real number r. Let r be a positive real number. Then so is 1/r. Therefore 0 <  < 1/r and so 1/ > r. Let H = 1/ . Thus H > r for all positive real number r. Therefore H is an infinite number.

23. General Approach to Using the HyperReals • Start with standard (Real) problem • Extend to non-standard (HyperReal) - adding * • Find solution of non-standard problem • Take standard part of solution to yield standard solution - removing * • Note: In practice we normally switch between Real and HyperReal without comment.

24. Theorem 1: Rules for Infinitesimal, Finite, and Infinite Numbers Th. Assume that e and d are infinitesimals; b,c are hyperreal numbers which are finite but not infinitesimal; and H, K are infinite hyperreal numbers; and n an integer . Then • Negatives: • -e is infinitesimal. • -b is finite but not infinitesimal. • -H is infinite.

25. (Theorem cont) • Reciprocals: • 1/e is infinite. • 1/b is finite but not infinitesimal. • 1/H is infinitesimal. • Sums: • e +d is infinitesimal. • b+e is finite but not infinitesimal. • b+c is finite (possibly infinitesimal). • H+e and H+b are infinite.

26. (Theorem cont) • Products: • e*d and e*bareinfinitesimal • b*c is finite but not infinitesimal. • H *b and H*K are infinite. • Roots: • If e >0, is infinitesimal. • If b>0, is finite but not infinitesimal. • If H>0, is infinite.

27. (Theorem cont) • Quotients: • e/ b, e/ H, andb/ Hareinfinitesimal • b/c is finite but not infinitesimal. • b/e , H/e, andH/b are infinite provided

28. Indeterminate Forms Examples

29. Theorem 2 • Every hyperreal number which is between two infinitesimals is infinitesimal. • Every hyperreal number which is between two finite hyperreal numbers is finite. • Every hyperreal number which is greater than some positive infinite number is positive infinite. • Every hyperreal number which is less than some negative infinite number is negative infinite.

30. Definitions:Infinitely Close • Two numbers x and y are said to be infinitely close ( written x  y) if and only (x-y) is infinitesimal.

31. Theorem 3. Let a, b, and c be hyperreal numbers. Then • a  a • If a  b, then b  a • If a  b and b  c then a  b. (i.e.,  is an equivalence relation.)

32. Theorem 4. Assume a  b, Then • If a is infinitesimal, so is b. • If a is finite, so is b. • If a is infinite, so is b.

33. Definition: Standard Part Let b be a finite hyperreal number. The standard part of b, denoted by st(b), is the real number which is infinitely close to b.

34. Note this means: • st(b) is a real number • b = st(b) + e for some infinitesimal e. • If b is real then st(b) =b.

35. Theorem 5. Let a and b be finite hyperreal numbers. Then • st(-a) = -st(a). • st(a+b) = st(a) + st(b). • st(a-b) = st(a) - st(b). • st(ab) = st(a) * st(b). • If st(b) , then st(a/b) = st(a)/st(b).

36. (theorem 5 cont.) • st(a)n = st(an). • . • .

37. Example 1: st(a) Assume c  4 and c 4.

38. Example 2: st(a) Assume H is a positive infinite hyperreal number.

39. Example 3: st(a) Assume e is a nonzero infinitesimal.