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

Question

Emilia Aguilar · Accepted Answer

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

The Real numbers (#RR#) are an example of a field. They are a set containing distinct elements called #0# and #1#, equipped with addition #+# and multiplication #*# that behave in the ways that you are used to. In technical language, they form an 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 #F# is to add one, then add anything else you need to make the resulting set closed under addition, multiplication, additive inverse, and multiplicative inverse of non-zero elements. If #F# is an ordered field (as the Real numbers are), then you can extend #F# in a way consistent with the ordering. Given a field #F#, we will construct a field #bar(F)# containing infinitesimal elements. We will also specify that #F# is an ordered field, having a #<# binary relation. First note that if #bar(F)# is a field containing infinitesimals, then it will contain the reciprocals of those elements and thus infinite elements too. Rather than adding an infinitesimal, I will start by adding a designed infinite element, let's call it #H# (short for Huge). #H# satisfies: #x < H# for all #x in F# In particular #overbrace(1+1+...+1)^"N times" < H# for any positive integer #N#. So #bar(F)# is a non-Archimedean field. If #P(x)# is any polynomial with coefficients in #F#, then #P(H)# must also be in #bar(F)#. If #Q(x)# is another non-zero polynomial with coefficients in #F#, then #(P(H))/(Q(H))# must also be in #bar(F)#. Now notice something a little subtle: If #P(H) != 0# then the multiplicative inverse of #(P(H))/(Q(H))# must be #(Q(H))/(P(H))#, so we must have: #(P(H))/(Q(H))*(Q(H))/(P(H)) = (P(H)Q(H))/(Q(H)P(H)) = 1#. To achieve this, we must be able to "cancel out" common factors that exist between the denominator and the numerator. Consequently, we think about: #(P(H))/(Q(H)) = (R(H))/(S(H)) <=> P(H)S(H) = Q(H)R(H)# That's our field #bar(F)# or in particular #bar(RR)# More formally, it's the set of rational functions with coefficients in #F# modulo the equivalence relation: #(P(t))/(Q(t)) -= (R(t))/(S(t)) <=> P(t)S(t) = Q(t)R(t)#

Isaiah Allen · Answer

Here's a little discussion of application...

Let #bar(RR)# be the field of rational expressions #(P(t))/(Q(t))# with Real coefficients modulo the equivalence relation: #(P(t))/(Q(t)) -= (R(t))/(S(t)) <=> P(t)S(t) = Q(t)R(t)# Let #epsilon = 1/t# be our designated infinitesimal. Is basic calculus useful in this field? Consider #f(x) = x^3# If #a in RR# then: #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(x)# over the interval #[a, a+epsilon]# is: #(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 #bar(RR)#, which is that of "standard part". The standard part of a finite number in #bar(RR)# is the part formed by throwing away any infinitesimal terms. For example, the standard part of #3a^2+3aepsilon+epsilon^2# is #3a^2#, which is the derivative of #f(x) = x^3# at #x=a#. Has anyone benefited from this? In a way, it takes the "standard part" and uses infinitesimal arithmetic to replace the limit process. I apologize for the somewhat informal nature of this discussion, but I do not particularly like this "standard part" mechanism, and it would be fairly simple, albeit time-consuming, to make it more rigorous. In addition, note that the number system defined in this way does not include #sqrt(epsilon)#. For such objects, you need more sophisticated fields incorporating infinitesimals, such as the Levi-Civita field. Note that even our simple extension #bar(RR)# supports some interesting ideas: If: #f(x) = { (0, x < 0), (1/2, x = 0), (1, x > 0) :}# Then we can 'approximate' the derivative of #f(x)# with: #f'(x) = { (1/epsilon, x in [-epsilon/2, epsilon/2]), (0, x !in [-epsilon/2, epsilon/2]) :}#

Charles Anctil · Answer

undefined One simple rigorous way to incorporate infinitesimals into the number system is through non-standard analysis, which was developed by Abraham Robinson. In this approach, infinitesimals are treated as actual numbers, called hyperreal numbers, which are larger than zero but smaller than any positive real number. These hyperreal numbers can be used to extend the real number system and provide a framework for calculus.

Non-standard analysis allows for a more intuitive treatment of calculus concepts, such as limits, derivatives, and integrals. It provides a formal framework to reason with infinitesimals and allows for more concise and elegant proofs in calculus. However, it is important to note that non-standard analysis is an alternative approach to traditional calculus based on limits, and both approaches are mathematically equivalent.

In basic calculus, the use of infinitesimals can simplify calculations and provide a more intuitive understanding of concepts. For example, infinitesimals can be used to define derivatives as ratios of infinitesimal changes, leading to a more direct interpretation of the derivative as the instantaneous rate of change. Infinitesimals can also be used to define integrals as sums of infinitesimal contributions, making it easier to understand the concept of area under a curve.

Overall, incorporating infinitesimals through non-standard analysis can be a useful tool in basic calculus, providing a different perspective and facilitating calculations and conceptual understanding.

Elijah Abram · Answer

undefined One simple and rigorous way to incorporate infinitesimals into the number system is through non-standard analysis, specifically using the hyperreal numbers. In this approach, infinitesimals are treated as numbers that are smaller in magnitude than any standard real number but are not zero.

Hyperreal numbers extend the real numbers to include infinitesimal and infinite quantities while maintaining the usual rules of arithmetic. This allows for a more intuitive and rigorous treatment of concepts like limits, derivatives, and integrals in calculus.

Hyperreal numbers can be useful in basic calculus for providing a more intuitive framework for understanding concepts such as instantaneous rates of change (derivatives) and areas under curves (integrals). They can help bridge the gap between algebraic manipulations and geometric interpretations, making calculus concepts more accessible and intuitive for students.