Infinitesimals

Definition

An Infinitesimal is a quantity that is closer to zero than any standard real number, yet is not zero itself. It is the fundamental conceptual unit of Calculus, representing a “change so small it is beyond measurement.”

Why It Matters

The infinitesimal is the “microscope” of mathematics. It allows us to zoom in so far that a jagged curve looks like a straight line, enabling us to measure “instantaneous” change—the speed of a car at a single moment or the slope of a hill at a single point. Without this conceptual leap, we would be stuck in a “static” world, unable to model or master anything that moves or flows.

Core Concepts

• The Non-Archimedean View: Historically, infinitesimals were treated as numbers $dx$ such that $0 < dx < 1/n$ for every natural number $n$.
• Leibniz’s Notation: The $dx$ and $dy$ used in differential notation represent infinitesimal changes in $x$ and $y$.
  • How to read: “The differentials d x and d y.”
  • Meaning: Infinitesimal changes in $x$ and $y$ — quantities smaller than any positive real, used in ratios like $\frac{dy}{dx}$ for instantaneous rate of change.
• The Ghost of Vanished Quantities: Bishop Berkeley famously critiqued infinitesimals as “the ghosts of departed quantities”—quantities that are treated as non-zero to allow division, then treated as zero to finalize a result.
• Formalization via Limits: In modern standard analysis, infinitesimals are replaced by the Limit Process. In non-standard analysis (Robinson), they are rigorously defined within the hyperreal number system.

Connected Concepts

• Limit Process
• Calculus
• Conceptual Foundation of Infinite Series
• Tangent Problem and Velocity Problem