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

Introduction to the HyperReals zDescriptive introduction z“Pictures” of the HyperReals zAxioms for the HyperReals zSome Properties (Theorems) of HyperReals

Descriptive introduction zA complete ordered field extension of the Reals (in a similar way that the Reals is a complete ordered field extension of the Rationals) zContains infinitely small numbers zContains infinitely large numbers zHas the same logical properties as the Reals

The HyperReals Near the Reals

The HyperReals Far from the Reals

Axioms for the HyperReals zAxioms common to the Reals yAlgebraic Axioms yOrder Axioms yCompleteness Axiom zAxioms unique to the HyperReals yExtension Axiom yTransfer Axiom

Algebraic Axioms zClosure laws: 0 and 1 are numbers. If a and b are numbers then so are a+b and ab. zCommutative laws: a + b = b + a and ab = ba zAssociative laws: a + (b + c) = (a + b) +c and a(bc)=(ab)c zIdentity laws: 0 + a = a and 1a = a zInverse laws: yFor all a, there exists number –a such that a + (-a) = 0 and yif a  0, then there exist number a -1 such that a(a -1 ) = 1 zDistributive law: a( b+c ) = ab + ac

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

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

Properties of < z0 < 1 zTransitive law: If a<b and b<c then a<c. zTrichotomy law: Exactly one of the relations a<b, a = b, b<a, holds. zSum law: If a<b, then a+c<b+c. zProduct law: If a<b and 0<c, then ac<bc. zRoot law: For a>0 and positive integer n, there is a number b>0 such that b n = a.

Completeness Axiom zA number b is said to be an upper bound of a set of numbers A if b  x for all x in A. zA 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. zCompleteness Axiom: zEvery non-empty set of numbers which is bounded above has a least upper bound.

Axioms unique to the HyperReals zExtension Axiom zTransfer Axiom zNote: Actually these axioms are all that are needed as for the HyperReals as the previous axioms can be derived from these two axioms.

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

Transfer Axiom z(Function version) Every real statement that holds for one or more particular real functions holds for the hyperreal extensions of these functions z(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.

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

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

Definitions: Finite and Infinite A HyperReal number b is said to be: yfinite if b is between two real numbers. ypositive infinite if b is greater than every real number. ynegative infinite if b is smaller than every real number. yinfinite if b is positive infinite or negative infinite.

Theorem: The only real infinitesimal number is 0. zProof: 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.

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

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.

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.

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 r. Let H = 1/ . Thus H > r for all positive real number r. Therefore H is an infinite number.

General Approach to Using the HyperReals zStart with standard (Real) problem zExtend to non-standard (HyperReal) - adding * zFind solution of non-standard problem zTake standard part of solution to yield standard solution - removing * zNote: In practice we normally switch between Real and HyperReal without comment.

Theorem 1: Rules for Infinitesimal, Finite, and Infinite Numbers Th. Assume that  and  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 yNegatives:  -  is infinitesimal. x-b is finite but not infinitesimal. x-H is infinite.

(Theorem cont) yReciprocals:  1/  is infinite. x1/b is finite but not infinitesimal. x1/H is infinitesimal. ySums:   +  is infinitesimal.  b+  is finite but not infinitesimal. xb+c is finite (possibly infinitesimal).  H+  and H+b are infinite.

(Theorem cont) yProducts:   and  b are infinitesimal xb*c is finite but not infinitesimal. xH *b and H*K are infinite. yRoots:  If  >0, is infinitesimal. xIf b>0, is finite but not infinitesimal. xIf H>0, is infinite.

(Theorem cont) yQuotients:   b,  H, and  b  H are infinitesimal xb/c is finite but not infinitesimal.  b/ , H/  and  H/ b are infinite provided

Indeterminate Forms Indetermina te Form infinitesimalfinite (equal to 1) infinite     H/KH/H 2 H/HH 2 /H H*  H*(1/H 2 )H*(1/H)H 2 *(1/H) H+KH+(-H) (H+1)+(-H) H+H Examples

Theorem 2 1.Every hyperreal number which is between two infinitesimals is infinitesimal. 2.Every hyperreal number which is between two finite hyperreal numbers is finite. 3.Every hyperreal number which is greater than some positive infinite number is positive infinite. 4.Every hyperreal number which is less than some negative infinite number is negative infinite.

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

Theorem 3. Let a, b, and c be hyperreal numbers. Then 1.a  a 2.If a  b, then b  a 3.If a  b and b  c then a  b. (i.e.,  is an equivalence relation.)

Theorem 4. Assume a  b, Then 1.If a is infinitesimal, so is b. 2.If a is finite, so is b. 3.If a is infinite, so is b.

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.

Note this means: 1.st(b) is a real number 2.b = st(b) +  for some infinitesimal . 3.If b is real then st(b) =b.

Theorem 5. Let a and b be finite hyperreal numbers. Then 1.st(-a) = -st(a). 2.st(a+b) = st(a) + st(b). 3.st(a-b) = st(a) - st(b). 4.st(ab) = st(a) * st(b). 5.If st(b), then st(a/b) = st(a)/st(b).

(theorem 5 cont.) 6.st(a) n = st(a n )

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

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

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