Definition

1. Adjective:

   • Immeasurably or incalculably minute; extremely small: Describes a quantity or degree so small that it is negligible or approaches zero. It is often used in a technical or mathematical context to denote values approaching a limit of zero.
   • Relating to or involving infinitesimals: Pertaining to the mathematical concept of an infinitely small quantity, which is foundational to calculus.

2. Noun:

   • (Mathematics) A variable or quantity that approaches zero as a limit: A fundamental concept in calculus, representing a value that can be made arbitrarily close to zero. It is used in the definitions of derivatives and integrals.

Usage Examples

• Adjective:

  • The probability of such an event is infinitesimal. (The chance is so small it is practically zero.)
  • They detected only an infinitesimal amount of the chemical in the sample. (The quantity found was immeasurably tiny.)
  • The difference between the two measurements was infinitesimal. (The discrepancy was extremely minute.)

• Noun:

  • In calculus, we treat the area under a curve as the sum of an infinite number of infinitesimals. (It is the sum of infinitely small quantities.)
  • The concept of the infinitesimal was crucial to the development of analysis. (The idea of a quantity approaching zero was key.)

Advanced Usage

• "Infinitesimal calculus": The branch of mathematics, namely differential and integral calculus, that deals with limits, derivatives, and integrals using the concept of infinitesimals.

  • Newton and Leibniz independently developed the foundations of infinitesimal calculus.

• "To an infinitesimal degree": To an immeasurably small extent.

  • The two theories differ only to an infinitesimal degree.

Variants and Related Words

• Infinitesimally (adverb): In an immeasurably small manner or degree.
  • The results were infinitesimally close to the predicted values.

Synonyms

• Minute: Extremely small in size or amount.
• Microscopic: So small as to be visible only with a microscope; figuratively, very small.
• Negligible: So small or unimportant as to be not worth considering.
• Inappreciable: Too small to be significant or perceived.

Antonyms

• Enormous: Very large in size, quantity, or extent.
• Immense: Extremely large or great.
• Significant: Sufficiently great or important to be worthy of attention.

Related Phrases and Concepts

• "Orders of magnitude": A phrase used in scientific contexts to compare sizes; an infinitesimal difference is many orders of magnitude smaller.

  • The error was infinitesimal, several orders of magnitude below the threshold.

• "Vanishingly small": A descriptive phrase similar in meaning to infinitesimal, often used in physics and mathematics.

  • As the input approaches zero, the output becomes vanishingly small.

Adjective

1. infinitely or immeasurably small
   • two minute whiplike threads of protoplasm
   • reduced to a microscopic scale

Noun

1. (mathematics) a variable that has zero as its limit