# What simple rigorous ways are there to incorporate infinitesimals into the number system and are they then useful for basic Calculus?

##### 2 Answers

Here's about the simplest way, but the explanation gets a little long...

#### Explanation:

The Real numbers (*field*.

They are a set containing distinct elements called *abelian group* under addition, the non-zero elements form an *abelian group* under multiplication and multiplication is distributive over addition.

The simplest way to add infinitesimals to any field

Given a field

First note that if

#x < H# for all#x in F#

In particular

If

Now notice something a little subtle: If

#(P(H))/(Q(H))*(Q(H))/(P(H)) = (P(H)Q(H))/(Q(H)P(H)) = 1# .

This requires that we be able to 'cancel out' common factors between the numerator and denominator.

As a result, we consider:

#(P(H))/(Q(H)) = (R(H))/(S(H)) <=> P(H)S(H) = Q(H)R(H)#

That's our field

More formally, it's the set of rational functions with coefficients in

#(P(t))/(Q(t)) -= (R(t))/(S(t)) <=> P(t)S(t) = Q(t)R(t)#

Here's a little discussion of application...

#### Explanation:

Let

#(P(t))/(Q(t)) -= (R(t))/(S(t)) <=> P(t)S(t) = Q(t)R(t)#

Let

Is this field useful for simple Calculus?

Consider

If

#f(a) = a^3#

#f(a+epsilon) = (a+epsilon)^3 = a^3+3a^2epsilon+3a epsilon^2+epsilon^3#

So the average slope of

#(f(a+epsilon)-f(a))/((a+epsilon)-a) = ((a^3+3a^2epsilon+3aepsilon^2+epsilon^3)-a^3)/epsilon = 3a^2+3aepsilon+epsilon^2#

We can introduce another concept to our field

For example, the standard part of

Has this helped any?

It kind of replaces the limit process with infinitesimal arithmetic and taking "standard part".

Note that this discussion has been a little informal. It is fairly easy but lengthy to make it more rigorous, but I do not really like this "standard part" mechanism.

In addition, note that the number system defined in this way does not include

Note that even our simple extension

If:

#f(x) = { (0, x < 0), (1/2, x = 0), (1, x > 0) :}#

Then we can 'approximate' the derivative of

#f'(x) = { (1/epsilon, x in [-epsilon/2, epsilon/2]), (0, x !in [-epsilon/2, epsilon/2]) :}#