An immeasurably small amount or quantity.
An infinitesimal is a quantity so small that it cannot be measured by ordinary means, yet it is not zero. It describes something that shrinks without limit, always getting smaller but never quite disappearing.
The idea is easier to picture than it might seem. Imagine cutting a number in half, then cutting that half in half, and continuing forever: 1, 0.5, 0.25, 0.125, … The values keep getting smaller and smaller, approaching zero but never reaching it. Each step in that process is getting closer to an infinitesimal quantity.
Infinitesimals are not a number students will calculate with directly in most math classes. Rather, the concept is a way of thinking about quantities that are vanishingly small, and it forms part of the foundation of calculus.
When Do Students Learn About Infinitesimals?
Students build intuition for infinitely small quantities well before the term itself is formally introduced.
Grades 3–5 – Fractions That Keep Getting Smaller
Students explore fractions and decimals, noticing how numbers can be divided into smaller and smaller parts. This lays the groundwork for understanding infinitesimally small quantities.
Grades 6–8 – Sequences and Shrinking Patterns
Students work with number patterns and sequences that decrease toward zero, developing intuition for quantities that approach but never reach a limit.
Grades 9+ – Infinitesimals in Calculus
Students encounter the concept formally in calculus, where infinitesimals underlie the ideas of limits, derivatives, and integrals.
