Introduction to Non-Standard Analysis

Here is the long-awaited article for all my "math" buddies that I promised to publish months ago. It took that long to research, boil-and-reduce Non-Standard Analysis to its present form ... enjoy!

I'll begin by providing a brief bio of Abraham Robinson, the daddy of this branch of mathematics. Abraham Robinson was born in Waldenburg, Germany (now Walbrzych, Poland), the second child of a Jewish family. When Abraham was 14, Hitler came to power and the family left for Palestine and settled in Jerusalem. There Abraham graduated from the Hebrew University of Jerusalem with a major in mathematics. In 1939 he began graduate studies in Paris, France, but had to flee when the Nazis invaded. He enlisted in the Free French Air Force and began teaching himself aerodynamics, becoming an expert on airfoils. He was finally awarded a graduate degree from the Hebrew University in 1946 and became a pioneer in model theory. Later, he became well known for his approach of using methods of mathematical logic to attack problems in analysis and abstract algebra. He died April 1974 at the age of 55.

 Non-Standard Analysis, or NSA, sometimes referred to as infinitesimal analysis, is the "alternate lifestyle" of mathematics. It allows one to solve calculus problems without the use of limits, and instead by making use of infinitesimals and infinites. This set of "infinites" are called hyperreal numbers, inferring that the numbers exist in more dimensions than real numbers, much like the word "hyperspace" means existing in more than three dimensions. Hyperreals have two components, the infinite number, and the infinitesimal number. They are defined as follows:

 Infinite Number

Numbers whose absolute value is greater than any positive real number

Infinitesimal Number

Numbers whose absolute value is less than any positive real number, and also greater than zero

 The Hyperreal Line

 Because these numbers are hyperreal, they exist beyond the real number line. Therefore, a hyperreal line, denoted as *R, can be constructed from the real number line, denoted as R. Drawing the hyperreal line might look something like this:

                              -2 -1   0  1   2

Negative  _ _ _ _ | _ | _ | _ | _ | _ _ _ _ Positive

Infinite                         finite                      Infinite

 Hyperreal numbers that are not infinite or infinitesimal on this line are called finite numbers. *R is different from R because it contains an infinite number, usually called H, and an infinitesimal number, 1/H, or more commonly denoted as ^x (delta-x). This, however, is a simplified drawing of *R. If we were to train a microscope on just one number, say, the number 1, *R will look like this:

           1-2^x   1- ^x     1     1+ ^x   1+2^x  

_ _ _ _ | _ _  _ | _ _ _ | _ _ _ | _ _ _ | _ _ _ _

 This means that there is a "cloud" of hyperreal numbers floating infinitesimally close to each number on the line. If we now look at the finite portion of *R through an imaginary

telescope, *R appears thus:

         H-2    H-1  H  H+1  H+2

_ _ _ _ | _ _ | _ _ | _ _ | _ _ | _ _ _ _

 Basic Definitions

Any hyperreal number x is said to be:

positive infinitesimal - x is positive but less than every positive real number

negative infinitesimal - x is negative but greater than every real negative number

infinitesimal - x is either positive infinitesimal, negative infinitesimal, or zero

finite - x is between two real numbers

positive infinite - x is greater than every real number

negative infinite - x is less than every real number

 If two hyperreal numbers, say a and b, whose difference a - b is infinitesimal, we can then say that a is infinitely close to b.

 The Extension Principle

This principle is defined as follows:

For every real function f of one or more variables there exists a corresponding hyperreal function *f of equal variables

This definition is simply saying that we can apply hyperreal numbers to real functions

... 'nuff said.

 The Transfer Principle

Every real statement that holds for one or more particular real functions holds for the hyperreal natural extentions of these functions

A "real statement" can be just about any truth about math, but the best examples would be some math rules and laws, such as the Commutative Law for Addition, or the rule that states that division by zero is never allowed. The Transfer Principle states that each example also holds whenever the variables are hyperreal numbers, and allows us to compute in exactly the same way as real numbers. For a quick example, we know that an infinitesimal number, say d ², is a positive infinitesimal that is smaller than d because the real function 0 < d² < d is true for all real d between zero and 1.

 Standard Part Principle

The real numbers are sometimes referred to as "standard" numbers, which gave rise to naming this new form of math "Non-Standard Analysis", because the hyperreals that are not real numbers are referred to as "nonstandard". For this reason, the real number that is infinitely close to ^x is said to be the "standard part" of ^x. Hence, the Standard Part Principle:

Let b be a finite infinitesimal number. The standard part of b, denoted as st(b), is the real number that is infinitely close to b. Every finite hyperreal number is infinitely close to exactly one real number. Infinite hyperreal numbers do not have standard parts.

 We can infer a couple of things from this definition. Obviously, st(b) is a real number. This means that b must be a compound number made up of one standard part and one (or more) parts infinitesimal. Hence, b = st(b) +  ^x.

 Computing Standard Parts

Finding the standard part in a calculus problem is a basic skill that must be introduced before we can "solve for x", so-to-speak. With that in mind, we begin.

The next set of theorems are presented without proofs. They enumerate some basic truths about infinitesimals.

 Let x and y be finite hyperreal infinitesimals. Let the standard parts be st(x) and st(y)

1. if st(-x) then -st(x)

2. if st(x + y) then st(x) + st(y)

3. if st(x - y) then st(x) - st(y)

4. if st(xy) then st(x) times st(y)

5. if st(y) doesn't equal 0, then st(x/y) = st(x)/st(y)

6. if st(xª) then [st(x)]ª

 With this set of rules before us, let's try to find the standard part of an expression.

When ^x is an infinitesimal and x is real, find the standard part in: 3x² + 3x(^x)  + (^x)².

Using the rules established above we get:

st[(3x² + 3x(^x)  + (^x)²] =

st(3x²) + st[3x(^x)]  + st[(^x)²] =

3x² + st(3x) · st(^x) + st(^x) ² =

3x² + 3x · 0 + 0 ² = 3x²

 Arriving at the standard part is usually tackled in 3 steps. The first step involves computations with hyperreal numbers, the second step requires computations involving standard parts, and the third step involves computations with ordinary real numbers.

 If H is a positive infinite hyperreal number, find the standard part of c.

        2H³ + 5H² - 3H

c =   7H³ - 2H² + 4H

First Stage

Remembering that infinite numbers do not have standard parts, we must manipulate the expression into a form that has standard parts. Multiplying both portions of the expression by H¯³ yields

      H¯³ · (2H³ + 5H² - 3H)       2 + 5H¯¹ - 3H¯²

c = H¯³ · (7H³ - 2H² + 4H)  =   7 - 2H¯¹ + 4H¯²

Second Stage

The hyperreals with negative exponents are now infinitesimal, and we can find there standard parts.

            st(2 + 5H¯¹ - 3H¯²)     st(2) + st(5H¯¹) ? st(3H¯²)       2 + 0 - 0

st(c) =  st(7 - 2H¯¹ + 4H¯²)  = st(7) ? st(2H¯¹) - st(4H¯²)   =  7 - 0 + 0

Third Stage                2

Simple math yields   7

 As a final example, let's consider the following to be an infinite hyperreal number and that ^x is a nonzero infinitesimal.

    3 + ^x

4^x + ^x²

The numerator and denominator have standard parts, that being 3 and 0. However, the quotient is an infinite number, and therefore has no standard part. In other words -

                                        3 + ^x

The standard part of 4^x + ^x² is undefined.

 Now for some fun. Let's find the derivative of x³ using the delta-epsilon method of taking limits, and also by using nonstandard analysis.

 Using Limits

lim           (x + h)³ - x³
h >0                 h
Expanding the (x + h)³ portion gives us

x³ + 3x²h + 3xh² + h³ - x³

               h

and canceling out the x³ in the numerator and the h in the denominator leaves us with a limit that looks like this:

lim           3x² + 3xh + h²
h >0

Taking the limit (or essentially substituting zero for h) leaves 3x².

 Using NSA

The derivative in NSA terms is as follows:

 Slope = st[f(x + ^x)  - f(x) ]  therefore:
                      st[ ^x]
st[ (x + ^x)³ - x³ ]

          st[^x]

Expanding the numerator gives [x³ + 3x²(^x) + 3x(^x) ² + (^x)³] - x³ and canceling out the x³ in the numerator and the ^x in the denominator leaves

st[3x² + 3x (^x) + (^x) ²] = st(3x²) + st[3x (^x)] + st[(^x) ²] = 3x² + 0 + 0 = 3x²

 Although the steps look similar using both methods, using NSA gives a much more concrete and believable result than does the limit method. The limit method uses h (or "as h approaches zero") as a real number during the portion in which we were simplifying the expression (especially when h is in the denominator), but then when the limit is taken, the h is dismissed as trivial, or being taken as zero. This strains at common sense, which mathematics is fanatical for. However, in the NSA method, the ^x is an infinitesimal, and therefore IS a hyperreal number in the expression and can be manipulated as usual. There is no "limit" taken (or assuming that it is zero) because the standard part of a single infinitesimal IS zero. This makes much more logical sense, and calculus now regains its elegance. If this method was taught to me in school, I think I would have enjoyed it much more.

 Source: Elementary Calculus: An Infinitesimal Approach, H. Jerome Keisler