''Ad infinitum'' is a
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
phrase
In syntax and grammar, a phrase is a group of words or singular word acting as a grammatical unit. For instance, the English expression "the very happy squirrel" is a noun phrase which contains the adjective phrase "very happy". Phrases can consi ...
meaning "to
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
" or "forevermore".
Description
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
''Et Cetera'' ( or (proscribed) , ), abbreviated to ''etc.'', ''etc'', ''et cet.'', ''&c.'' or ''&c'' is a Latin expression that is used in English to mean "and other similar things", or "and so forth". Translated literally from Latin, means 'an ...
'' to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:
* "The
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
1, 2, 3, ... continues ''ad infinitum''."
* "The perimeter of a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
may be iteratively drawn ''ad infinitum''."
The 17th-century writer
Jonathan Swift
Jonathan Swift (30 November 1667 – 19 October 1745) was an Anglo-Irish Satire, satirist, author, essayist, political pamphleteer (first for the Whig (British political party), Whigs, then for the Tories (British political party), Tories), poe ...
incorporated the idea of
self-similarity
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In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
in the following lines from his satirical poem ''On Poetry: a Rhapsody'' (1733):
The vermin only teaze and pinch
Their foes superior by an inch.
So, naturalists observe, a flea
Has smaller fleas that on him prey;
And these have smaller still to bite 'em,
And so proceed ''ad infinitum''.
Thus every poet, in his kind,
Is bit by him that comes behind
The mathematician
Augustus De Morgan included similar lines in his rhyme ''
Siphonaptera''.
See also
*
Mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
*
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
*
Self-reference
Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philoso ...
* "
The Song That Never Ends
"The Song That Doesn't End" (also referred to as "The Song That Never Ends") is a self-referential and infinitely iterative children's song. The song appears in an album by puppeteer Shari Lewis titled '' Lamb Chop's Sing-Along, Play-Along'', r ...
"
*
Turtles all the way down
References
External links
*
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{{Infinity
Infinity
Latin logical phrases
Latin words and phrases