In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a manner similar to the Latin phrase et cetera to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:
- "The sequence 1, 2, 3, ... continues ad infinitum."
- "The perimeter of a fractal may be iteratively drawn ad infinitum."
- The 17th-century writer Jonathan Swift wrote lightheartedly the idea of self-similarity in natural philosophy with the following lines in his poem "On Poetry: A Rhapsody":
- "So nat'ralists observe, a flea
- Has smaller fleas that on him prey;
- And these have smaller fleas to bite 'em.
- And so proceeds Ad infinitum."
- "Great fleas have little fleas upon their backs to bite 'em,
- And little fleas have lesser fleas, and so ad infinitum.
- And the great fleas themselves, in turn, have greater fleas to go on,
- While these again have greater still, and greater still, and so on."
This text is part of the nursery rhyme "The Siphonaptera".
- Jonathan Swift, "On Poetry: A Rhapsody", pub. 1733
- Augustus de Morgan, A Budget of Paradoxes, pub. 1872, p. 377
- The dictionary definition of ad infinitum at Wiktionary