Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Infinitesimal |
InfinitesimalAdjective1. Infinitely or immeasurably small; "two minute whiplike threads of protoplasm"; "reduced to a microscopic scale". Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "infinitesimal" was first used: 1710. (references) |
Synonyms: InfinitesimalSynonyms: microscopic (adj), minute (adj). (additional references) |
(From Wikipedia, the free Encyclopedia)
An infinitesimal is only a notional quantity - there exists no infinitesimal real number. This can be shown using the least upper bound axiom of the real numbers: consider whether the least upper bound c of the set of all infinitesimals is or is not an infinitesimal. If it is, then so is 2c, contradicting the fact that c is an upper bound. It it is not, then neither is c/2, contradicting the fact that among all upper bounds, c is the least.
The first mathematician to make use of infinitesimals was Archimedes. See how Archimedes used infinitesimals.
When Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go:
It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit, which obviates the need to use infinitesimals.
Nevertheless, the use of infinitesimals continues to be convenient for simplifying notation and calculation.
Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation of the derivative of f(x) = x² can be justified with a minor modification: we have to talk about the standard part of the difference quotient, and the standard part of x + dx is x.
Alternatively, we can have synthetic differential geometry.
See also:
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Infinitesimal."
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Fraction | Adjective: fractional, fragmentary, inconsiderable, negligible, infinitesimal. |
Littleness | Impalpable, intangible, evanescent, imperceptible, invisible, inappreciable, insignificant, inconsiderable, trivial; infinitesimal, homoeopathic; atomic, subatomic, corpuscular, molecular; rudimentary, rudimental; embryonic, vestigial. |
Numeration | Arithmetic, analysis, algebra, geometry, analytical geometry, fluxions; differential calculus, integral calculus, infinitesimal calculus; calculus of differences. |
Smallness | Inappreciable, evanescent, infinitesimal, homeopathic, very small; atomic, corpuscular, microscopic, molecular, subatomic. |
| Source: adapted from Roget's Thesaurus. | |
Crosswords: Infinitesimal |
| English words defined with "infinitesimal": infinitesimal calculus, Infinitesimal increment, Infinitesimally. (references) |
| Specialty definitions using "infinitesimal": bit bang ♦ emittance ♦ Finite Larmor Radius ♦ infinitesimal dipole, Instantaneous Unit Hydrograph ♦ mu factor ♦ radiant emittance ♦ sound energy density ♦ true strain, truncation error ♦ voltage factor. (references) |
| Non-English Usage: "Infinitesimal" is also a word in the following languages with English translations in parentheses. German (infinitesimal), Portuguese (evangelic), Spanish (infinitesimal), Swedish (infinitesimal). |
| Domain | Title |
Books | |
Music |
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Source: compiled by the editor from various references; see credits. | |
| Subject | Topic | Quote |
Economic History | Jamaica | This pattern of food consumption has show only infinitesimal variations since 1997. Meat, poultry and fish account collectively for 22 percent of total food consumption followed by cereals and cereal preparations. (references) |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Infinitesimal" is generally used as an adjective (general or positive) -- approximately 98.72% of the time. "Infinitesimal" is used about 78 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Adjective (general or positive) | 98.72% | 77 | 37,929 |
| Noun (proper) | 1.28% | 1 | 339,140 |
| Total | 100.00% | 78 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "infinitesimal": infinitesimal calculus ♦ infinitesimal dipole ♦ Infinitesimal increment. Additional references. | |
| Hypenated Usage | |
Ending with "infinitesimal": non-infinitesimal. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Expression | Frequency per Day |
infinitesimal | 12 |
| Source: compiled by the editor from various references; see credits. | |
| Language | Translations for "infinitesimal"; alternative meanings/domain in parentheses. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Albanian | pambarimisht i vogël, jashtëzakonisht i vogël. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Arabic | كمية متناهية الصغر, متناهى الصغر, صغير (baby, diminutive, little, lower case, minuscule, minute, niggling, petty, remote, short, small, teeny, tiny, tiny bit, wee, young), دقيق (accurate, careful, close, delicate, elaborate, exact, express, farina, fine, flour, inappreciable, intangible, just, keen, mathematical, meal, nearness, nice, painstaking, particular, pernickety, precise, punctilious, punctual, rigorous, scrupulous, searching, sharp, sound, specific, straightforward, strict, ticklish, tiny, touchy, tricky, true, veracious). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Bulgarian | безкрайно малък (evanescent, indivisible), безкрайно малко количество. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Czech | infinitezimální. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Danish | infinitesimalregning (infinitesimal calculus), infinitesimalpotens (infinitesimal power), elementardipol (doublet, infinitesimal dipole, radiating doublet). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dutch | infinitesimaal-rekening (infinitesimal calculus), infinitesimaalrekening (calculus, infinitesimal calculus), hertz-dipool (doublet, infinitesimal dipole, radiating doublet), elementaire elektrische dipool (doublet, infinitesimal dipole, radiating doublet), elementaire dipool (doublet, infinitesimal dipole, radiating doublet), differentiaal- en integraalrekening (calculus, infinitesimal calculus). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Farsi | بینهایت کوچک , بی اندازه خرد. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Finnish | infinitesimaalilaskenta (infinitesimal calculus), reaalianalyysi (infinitesimal calculus), differentiaali-ja integraalilaskenta (infinitesimal calculus), alkiodipoli (doublet, infinitesimal dipole, radiating doublet). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
French | infinitésimail. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
German | winzig (diminutive, diminutively, imperceptible, lilliputian, midget, minikin, minuscule, minute, pigmy, teenily, teeny, tinily, tiny, undersized, wee, weely, weeny), verschwindend (disappearing, insignificant, vanishing). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Greek | πολλοστημόριο, απειροστόσ, απειροελάχιστοσ. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Hebrew | קטן בתכלית "קט ות, זעיר מא". (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Hungarian | parányi (atomic, diminutive, invisible, microscopic, microscopical, minute, teeny, teeny-weeny, tiny, to miniaturize, wee), elenyészõ (evanescent, minimal, vanishing). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Italian | infinitesimale. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Japanese Kanji | 無限小 , 極微 (microscopic), 極小 , 微少 . (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Japanese Katakana | きょくしょう, きょくび (microscopic), びしょう (eulogistic name, euphemism, microscopic, slight wound, smile), む'"しょう. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Manx | ree-veg. (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Pig Latin | infinitesimalay infinito (boundless, endless, illimitable, immediacy, immense, incomprehension, infinitive, infinity, unbounded, unmeasured). (various references) infinitezimal (evanescent), infim (inappreciable, insignificant, little, tiny), cantitate infinitezimalã. (various references) бесконечно малый (evanescent). (various references) infinitezimalan, beskrajno mala količina. (various references) infinitesimal. (various references) infinitesimal, oändligt liten. (various references) sonsuz küçük değer, sonsuz küçük, son derece küçük, bölünemeyecek kadar küçük değer. (various references) найдрібніший. (various references) số lượng rất nhỏ, rất nhỏ (minuscule, teeny, teeny-weeny, wee). (various references) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | infinitus. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "infinitesimal": infinitesimally, infinitesimals. (additional references) | |
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"Infinitesimal" is suggested in spellcheckers for the following: infinitesimel, infinitessimal, infinitessimaly, infinitestimal, infinitezimal, inifinitesimal. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "infinitesimal" (pronounced i'nfi'nute"sumul) |
| 6 | -e" s u m u l | decimal. |
| 5 | -s u m u l | maximal, proximal. |
| 4 | -u m u l | animal, caramel, minimal, optimal. |
| 3 | -m u l | abnormal, abysmal, baptismal, camel, dermal, dismal, enamel, endodermal, epidermal, formal, geothermal, Hamal, hydrothermal, informal, isothermal, mammal, mesodermal, normal, paranormal, pommel, primal, pummel, thermal, tramel, trammel. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-e-f-i-i-i-i-l-m-n-n-s-t" | |
-2 letters: infantilism. | |
-3 letters: finalities, infinities, initialism. | |
-4 letters: aminities, anilities, filaments, inanities, infantile, infinites, laminitis, liniments, mainlines, semifinal. | |
-5 letters: ailments, alienism, alienist, aliments, anilines, antilife, families, feminist, fetialis, filament, filiates, filmiest, finalise, finalism, finalist, finniest, flamiest, flamines, infamies, infantes, infinite, inflames, inflates, initials, linesman, liniment, litanies, mainline, manifest, manliest, mannites, melanins, melanist, militias, minifies, smaltine. | |
| Words containing the letters "a-e-f-i-i-i-i-l-m-n-n-s-t" | |
+1 letter: infinitesimals. | |
+2 letters: infinitesimally. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
Hexadecimal (or equivalents, 770AD-1900s) (references)49 6E 66 69 6E 69 74 65 73 69 6D 61 6C |
| Leonardo da Vinci (1452-1519; backwards) (references)
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| American Sign Language (origins from 1620-1817 in Italy and, especially, France) (references)
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| Semaphore (1791, in France) (references)
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| Braille (1829, in France) (references)
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Morse Code (1836) (references).. -. ..-. .. -. .. - . ... .. -- .- .-.. |
| Dancing Men (Sir Arthur Conan Doyle, 1903) (references)
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Binary Code (1918-1938, probably earlier) (references)01001001 01101110 01100110 01101001 01101110 01101001 01110100 01100101 01110011 01101001 01101101 01100001 01101100 |
HTML Code (1990) (references)I n f i n i t e s i m a l |
ISO 10646 (1991-1993) (references)0049 006E 0066 0069 006E 0069 0074 0065 0073 0069 006D 0061 006C |
| British Sign Language (Fingerspelling, BSL; 1992, British Deaf Association Dictionary of British Sign Language) (references)
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Encryption (beginner's substitution cypher): (references)43807275807586718575796778 |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Commercial | 5. Quotations: Non-fiction 6. Usage Frequency 7. Expressions 8. Expressions: Internet | 9. Translations: Modern 10. Translations: Ancient 11. Derivations 12. Rhymes | 13. Anagrams 14. Orthography 15. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
