Hyperreal NumbersBy: Jeffrey Wong

What Are Hyperreal  Numbers? (Simply Speaking)

Numbers that  are infinitely large (reciprocals of infinitesimals): |ω| > n for all n ≥ 1

Numbers that are infinitely small (infinitesimals): |ε| < 1/n for all n ≥ 1 (thus, this includes 0)

The set of real numbers

Hyperreals On The Number Line

Source(s): http://faculty.otterbein.edu/TJames/Into%20to%20HyperReals.ppt and Keisler pg. 25

Hyperreals (in more detail)Hyperreals are based on two main principles where the other properties are derived

The Extension Principle1) The real numbers are a subset of the hyperreals2) The < relation for real numbers is the same for hyperreals3) There is a hyperreal greater than 0 and less than all positve real numbers4) Every real numbers function has an extended hyperreal function

The Transfer PrincipleIf a sentence in a given language is true for real numbers, then it is true for hyperreals-This is where the hyperreals get all the laws and properties of real numbers-Ie. x + y = y + x (commutative), x/0 is invalid, etc.

Arithmetic Rules Of HyperrealsLet x, y, be infinitesimals, A,B, be hyperreals that are finite but not infinitesimal, g,h be hyperreals that are infinite

Summationx + y -> inifinitesimalx + A -> finite but not infinitesimalA + B -> finite (may be infinitesimal)A + x or A + g -> infinite

Multiplicationx * y or x * B -> infinitesimalA* B -> finite but not infinitesimalg * h or g * A -> infinite

Divisionx/A or g/A or x/g -> infinitesimalA / B -> finite but not infinitesimalg / x or A / x or g / A -> infinite (x ≠ 0)

Note: The results of the following casesx / yg / hg + hg * x

Can be infinitesimal, finite but not infinitesimal, or infiniteHence they are indeterminate

Proof of Product Rule in Calculus (standard)

http://en.wikipedia.org/wiki/Product_rule

How To Use The HyperrealsStandard PrincipleEvery finite hyperreal number is infinitely close to a real number- Real numbers regarded as standard numbers, hyperreals are called nonstandard- if A is a hyperreal number, we will use st(A) to denote the real number A is close to

Steps:1) Start with a standard problem2) Extend problem to nonstandard3) Find solution in nonstandard4) Take the standard part of the solution (if required)

Proof of Product Rule in Calculus (nonstandard)

How Euler Calculated e

How Euler Calculated e (cont’d)

Derivative Of e (standard)

Derivative Of e (nonstandard)

Questions?

ReferencesLectures on the Hyperreals: An Introduction To Nonstandard Analysis (Graduate Texts In Mathematics), Robert Goldblatt. Springer-Verlag. 1998. New York.

Intro To Hyperreals, Dr. Thomas R. Jamesfaculty.otterbein.edu/TJames/Into%20to%20HyperReals.ppt

Fun With NonStandard Analysis, David. A. Rosshttp://www.math.hawaii.edu/~ross/fun_with_nsmodels.pdf

Elementary Calculus: An Approach Using Infinitesimals, H. Jerome Keislerhttp://www.math.wisc.edu/~keisler/calc.html

Wikipedia ArticlesProduct Rule: http://en.wikipedia.org/wiki/Product_rule