Infinitesimal: Understanding Non-Standard Analysis

Overview

An infinitesimal is a quantity that is arbitrarily close to zero but not actually zero. It’s a fundamental concept in non-standard analysis, which provides a rigorous foundation for calculus using infinitesimals.

Key Concepts

In non-standard analysis, infinitesimals are treated as actual numbers, unlike in traditional calculus where they are often treated as limits. The set of hyperreal numbers includes standard real numbers, infinitesimals, and infinite numbers.

Deep Dive

Non-standard analysis, developed by Abraham Robinson, formalizes the intuitive idea of infinitesimals. A number $\epsilon$ is infinitesimal if $|\epsilon| < r$ for every positive real number $r$. This contrasts with standard analysis where a quantity approaching zero is defined by a limit process.

Applications

Infinitesimals are used to provide a more intuitive and direct approach to solving problems in differential and integral calculus. They offer an alternative framework for understanding concepts like derivatives and integrals.

Challenges & Misconceptions

A common misconception is that infinitesimals are simply zero. However, in non-standard analysis, they are distinct from zero. The challenge lies in understanding their formal definition and properties within the hyperreal number system.

FAQs

• What is the difference between an infinitesimal and zero? An infinitesimal is not zero, but is smaller in absolute value than any positive real number.
• Who developed non-standard analysis? Abraham Robinson.
• Are infinitesimals used in modern calculus? While not typically taught in introductory courses, they form the basis of a rigorous alternative to standard calculus.