Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Imaginary |
ImaginaryAdjective1. Not based on fact; dubious; "the falsehood about some fanciful secret treaties"- F.D.Roosevelt; "a small child's imaginary friends"; "her imagined fame"; "to create a notional world for oneself". Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "imaginary" was first used in popular English literature: sometime before 1509. (references) |
(From Wikipedia, the free Encyclopedia)
The complex numbers are an extension of the real numbers, in which all polynomials have roots. The complex numbers contain a number i, the imaginary unit, with i2= -1. Every complex number can be represented in the form x+iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.
The sum and product of two complex numbers are:
Complex numbers were first introduced in connection with explicit formulas for the roots of cubic polynomials. In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered. For example complex matrix, complex polynomial and complex Lie algebra.
- (a+ib) + (c+id) = (a+c) + i(b+d)
- (a+ib) · (c+id) = ac-bd + i (bc+ad)
History
The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Tartaglia, Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in the 17th century and was meant to be derogatory. The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss. The formally correct definition using pairs of real numbers was given in the 19th century.
Definition
Formally we may define complex numbers as ordered pairs of real numbers (a, b) together with the operations:
So defined, the complex numbers form a field, the complex number field, denoted by C (or in blackboard bold).
- (a, b) + (c, d) = (a + c, b + d)
- (a, b) · (c, d) = (ac - bd, bc + ad)
We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0,1).
C could also be defined as the topological closure of algebraic numbers and the algebraic closure of R.
Geometry
A complex number can also be viewed as a point or a position vector on the two dimensional Cartesian coordinate system. This representation is sometimes called an Argand diagram. In the figure, we have
The latter expression is sometimes shorthanded as r cis φ, where r is called the absolute value of z and φ is called the complex argument of z. By simple trigonometric identities, we see that
- z = x + iy = r (cos φ + i sin φ).
Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching.
- r1 cis φ1 · r2 cis φ2 = r1r2 cis (φ1+φ2);
- r1 cis φ1 / r2 cis φ2 = r1 / r2 cis (φ1-φ2);
Multiplication with i corresponds to a counter clockwise rotation by 90 degrees. The geometric content of the equation i2 = -1 is that a sequence of two 90 degree rotation results in a 180 degree rotation. Even the fact (-1) · (-1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.
Euler's formula states that ei φ = cisφ. The exponential form gives us a better insight then the shorthand rcisφ, which is almost never used in serious mathematical articles.
Absolute value, conjugation and distance
Recall that the absolute value (or modulus or magnitude) of a complex number z = r eiφ is defined as |z| = r. Algebraically, if z = a + ib, then |z| = &radic(a2 + b2 ).
One can check readily that the absolute value has three important properties:
for all complex numbers z and w. By defining the distance function d(z, w) = |z - w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers.
- |z + w| ≤ |z| + |w|
- |z w| = |z| |w|
- |z / w| = |z| / |w|
The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as or z*. As seen in the figure, is the "reflection" of z about the real axis. The following can be checked:
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
- if and only if z is real
- if z is non-zero
The complex argument of z=reiφ is φ. Note that the complex argument is unique up to modulo 2π.
Matrix representation of complex numbers
While usually not useful, alternative representations of complex field can give some insight into their nature. One particularly elegant representation interprets every complex number as 2x2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form
with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as
which suggests that we should identify the real number 1 with the matrix
and the imaginary unit i with
a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to -1.
The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
Some properties
Real vector space
C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field.
Solutions of polynomial equations
A root of the polynomial p is a complex number z such that p(z) = 0. A most striking result is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the Fundamental Theorem of Algebra, and shows that the complex numbers are an algebraically closed field.
Indeed, the complex number field is the algebraic closure of the real number field. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X2 + 1:
This is indeed a field because X2 + 1 is irreducible. The image of X in this quotient ring becomes the imaginary unit i.
- C = R[X] / (X2 + 1).
Complex analysis
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Applications
Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.
If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as the real part of complex valued functions of the form
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
- f(t) = z eiωt
In electrical engineering, this is done for varying voltages and currentss. The treatment of resistors, capacitors and inductors can then be unified by introducing imaginary frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents.)
The residue theorem of complex analysis is often used in applied fields to compute certain improper integrals.
The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.
In Special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.
In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = ert.
See also
quaternions, complex geometry, local fields, phasors, Leonhard Euler, the most remarkable formula in the world, Hypercomplex number, Complex numbers at Wikibooks
Further Reading
- An Imaginary Tale, by Paul J. Nahin; Princeton University Press; ISBN 0691027951 (hardcover, 1998). A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Complex number."
(From Wikipedia, the free Encyclopedia)
In mathematics, an imaginary number is a number whose square is negative. The term was coined by René Descartes in the seventeenth century and was meant to be derogatory: obviously such numbers don't exist. Nowadays we find the imaginary numbers on the vertical axis of the complex number plane. Every imaginary number can be written as where is a real number and the imaginary unit with the property that
(In electrical engineering and related fields, the imaginary unit is often written as to avoid confusion with a changing current, traditionally denoted by .) Every complex number can be written uniquely as a sum of a real number and an imaginary number.
Despite their name, imaginary numbers are just as real as real numbers; see Complex number#Definition on how they can be constructed.
The powers of repeat in a cycle:
- ...
See also
- Quaternion
- Octonion
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Imaginary number."
(From Wikipedia, the free Encyclopedia)
Imagination is in general, the power or process of producing mental images and ideas. The term is technically used in psychology for the process of reviving in the mind percepts of objects formerly given in sense perception. Since this use of the term conflicts with that of ordinary language, some psychologists have preferred to describe this process as "imaging" or "imagery" or to speak of it as "reproductive" as opposed to "productive" or "constructive" imagination. Imagined images are seen with the "mind's eye". One hypothesis for the evolution of human imagination is that it allowed conscious beings to solve problems (and hence increase an individual's fitness) by use of mental simulation.The common use of the term is for the process of forming in the mind new images which have not been previously experienced, or at least only partially or in different combinations. Thus the image of a centaur is the result of combining the common percepts of man and horse; fairy tales and fiction generally are the result of this process of combination. Imagination in this sense, not being limited to the acquisition of exact knowledge by the requirements of practical necessity, is up to a certain point free from objective restraints. In various spheres, however, even imagination is in practice limited: thus a man whose imaginations do violence to the elementary laws of thought, or to the necessary principles of practical possibility, or to the reasonable probabilities of a given case is regarded as insane.
The same limitations beset imagination in the field of scientific hypothesis. Progress in scientific research is due largely to provisional explanations which are constructed by imagination, but such hypotheses must be framed in relation to previously ascertained facts and in accordance with the principles of the particular science.
In spite, however, of these broad practical considerations, imagination differs fundamentally from belief in that the latter involves "objective" control of subjective activity. The play of imagination, apart from the obvious limitations (e.g. of avoiding explicit self-contradiction), is conditioned only by the general trend of the mind at a given moment. Belief, on the other hand, is immediately related to practical activity: it is perfectly possible to imagine myself a millionaire, but unless I believe it I do not, therefore, act as such. Belief always endeavours to conform to objective conditions; though it is from one point of view subjective it is also objectively conditioned, whereas imagination as such is specifically free. The dividing line between imagination and belief varies widely in different stages of mental development. Thus someone from a technologically primitive culture who is ill frames an ideal reconstruction of the causes of his illness, and attributes it to the hostile magic of an enemy. In ignorance of pathology he is satisfied with this explanation, and actually believes in it, whereas such a hypothesis in the mind of someone who understood germ theory it would be treated as a pure effort of imagination, or even as a hallucination. It follows that the distinction between imagination and belief depends in practice on knowledge, social environment, training and the like.
Although, however, the absence of objective restraint, i.e. a certain unreality, is characteristic of imagination, none the less it has great practical importance as a purely ideational activity. Its very freedom from objective limitation makes it a source of pleasure and pain. A person of vivid imagination suffers acutely from the imagination of perils besetting a friend. In fact in some cases the ideal construction is so "real" that specific physical manifestations occur, as though imagination had passed into belief or the events imagined were actually in progress. See, for example, psychosomatic illness.
based on an article from a 1911 encyclopedia
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Imagination."
Synonyms: ImaginarySynonyms: fanciful (adj), imagined (adj), notional (adj). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Imagination | Fabulous, legendary; mythical, mythic, mythological; chimerical; imaginary, visionary; notional; fancy, fanciful, fantastic, fantastical; whimsical; fairy, fairy-like; gestic. |
Number | Positive, negative; rational, irrational; surd, radical, real; complex, imaginary; finite; infinite; impossible. |
Unsubstantiality | Visionary; (imaginary); immaterial; spectral; dreamy; shadowy; ethereal, airy; cloud built, cloud formed; gossamery, illusory, insubstantial, unreal. |
| Source: adapted from Roget's Thesaurus. | |
| Domain | Usage | |
Screenplays | I don't get hurt or bleed, hair doesn't muss; it's one of the advantages of being imaginary. (The Purple Rose of Cairo; writing credit: Woody Allen.) Uh well, I had an imaginary girlfriend (Cartoon Planet; writing credit: Andy Merrill; Pete Smith) Well she really didn't say much but your imaginary friend, Adam, he spilled hit guts (Caroline in the City; writing credit: Angela Carneiro) Maybe you're confusing us with your imaginary friends (The Weekenders; writing credit: Douglas Langdale) I'm not good or real I'm evil, and imaginary. (Will & Grace; writing credit: Evan Weinstein) | |
Lyrics | I'm your imaginary friend (I'm Already There; performing artist: Lonestar) | |
Clever | I was the next door kid's imaginary friend. (references; author: unknown) | |
Tongue Twisters | Can you imagine an imaginary menagerie manager managing an imaginary menagerie? (references; author: unknown) | |
Movie/TV Titles | Her Imaginary Lover (1933) His Imaginary Family (1913) An Imaginary Elopement (1911) Pat Metheny Group: Imaginary Day Live (2001) Imaginary Light (1994) | |
Source: compiled by the editor from various references; see credits. | ||
| Domain | Title |
Books | |
Theater & Movies | |
Music |
|
Source: compiled by the editor from various references; see credits. | |
| Thumbnail | Description & Credit |  | Painting by Creative Arts Studio, prepared for use in an official film on Naval history, circa the early 1960s. It depicts an imaginary meeting of some of the Confederacy's naval leaders, including (seated, left to right): Captain Franklin Buchanan, Captain Josiah Tattnall, and Commander Matthew F. Maury. Shown standing (from left to right) are Captain George N. Hollins, Rear Admiral Raphael Semmes, and Secretary of the Navy Stephen Mallory. Credit: NAVY. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Author | Quotation |
Breville | When real nobleness accompanies the imaginary one of birth, the imaginary mixes with the real and becomes real too. |
John Ruskin | Imaginary evils soon become real one by indulging our reflections on them. |
Oliver Goldsmith | Don't let us make imaginary evils, when you know we have so many real ones to encounter. |
Source: compiled by the editor from various references. | |
| Author | Date | Quotation |
John Locke | 1690 | And the success of the combat will be unavoidably the same he there describes it: ----- Libertas pauperis haec est: Pulsatus rogat, & pugnis concisus, adorat, Ut liceat paucis cum dentibus inde reverti.* This will always be the event of such an imaginary resistance, where men may not strike again. (Second Treatise of Government) |
Source: compiled by the editor from various references. | ||
| Title | Author | Quote |
Emma | Austen, Jane | The event acquitted her of all the fancifulness, and all the selfishness of imaginary complaints |
Trainspotting | Irvine Welsh | Begbie always constructed imaginary qualities in his friends, then shamelessly claimed them for himself |
Source: compiled by the editor from various references. | ||
| Subject | Topic | Quote |
Health | Let's create an imaginary scenario and go through the process step by step. Say you have a storage room in your home that you hardly ever enter. (references) | |
Civil Liberties | Equatorial Guinea | These practices range from the police demanding bribes for imaginary offenses to city, provincial, and federal officials extorting money for "licenses" for which there is no statutory basis. (references) |
Lexicography | Devil's Dictionary | ELYSIUM, n. An imaginary delightful country which the ancients foolishly believed to be inhabited by the spirits of the good. This ridiculous and mischievous fable was swept off the face of the earth by the early Christians -- may their souls be happy in Heaven! |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Imaginary" is generally used as an adjective (general or positive) -- approximately 100.00% of the time. "Imaginary" is used about 744 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) | 100% | 744 | 9,138 |
Source: compiled by the editor from several corpora; see credits.
Expressions using "imaginary": imaginary being ♦ Imaginary calculus ♦ imaginary creature ♦ Imaginary expression ♦ imaginary number ♦ imaginary numbers ♦ imaginary part ♦ imaginary part of a complex number ♦ imaginary place ♦ Imaginary points ♦ imaginary pregnancy ♦ Imaginary quantity ♦ pure imaginary number. Additional references. | |
| Hypenated Usage | |
Ending with "imaginary": semi-imaginary. | |
| 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. |
| Language | Translations for "imaginary"; alternative meanings/domain in parentheses. | |
Albanian | imagjinar (fabulous, fancied, fanciful, fantastic, fantastical, fictional, fictitious, fictive, imaginative, unreal, visional, visionary), i përfytyruar (imaginative), joreal (aerial, aeriform, insubstantial, notional, off balance, unreal, unsubstantial), fiktiv (fictitious, unreal). (various references) | |
Arabic | متخيل (conceived), تصوري (fictitious, ideological), تخيلي (fictive), خيالي (aerial, chimerical, conceptual, fancied, fanciful, fancy, fantast, fantastic, fictional, fictitious, fictive, figment, ideal, ideational, imaginative, impracticable, mythical, notional, quixotic, romantic, unreal, utopian, visionary). (various references) | |
Bulgarian | въображаем (abstract, fancy, fictional, ideal, mythical, mythological, notional, phantom, unreal, visional, visionary), имагинерен. (various references) | |
Chinese | 虚构 (Chimeric, Chimerical, fictional, fictive), 虛構 . (various references) | |
Czech | imaginární (visionary), pomyslný (notional), domnìlý (alleged, assumed, fancied, presumptive, supposed). (various references) | |
Danish | indbildt, imaginær. (various references) | |
Dutch | imaginair. (various references) | |
Farsi | پنداری , وهمی (Bizarre, Illusive, Illusory, Uncanny, Unreal, Unrealistic), تصوری (Brainchild, Conceptual, Imaginable, Romantic, Unreal, Unrealistic), خیالی (Abstract, Bizarre, Brainchild, Dreamy, Image, Phantom, Poetic, Romantic, Unreal, Unrealistic, Visionary), خیال (Design, Dump, Fancy, Fiction, Ghost, Humor, Idea, Ideology, Impression, Intention, Mind, Notion, Plan, Spectrum, Thought, Vision, Whim, Wraith), انگاشتی (Supposition). (various references) | |
Finnish | mielikuvituksellinen (fanciful, fantastic), luuloteltu (fancied), kuvitteellinen, kuviteltu (fancied, fictitious). (various references) | |
French | imaginaire. (various references) | |
German | imaginär (make believe). (various references) | |
Greek | φανταστικός (brilliant, fancy). (various references) | |
Hebrew | מדומה (dummy, false, feigned, mock), דמיוני (airy fairy, fabulous, fairy, fancied, fanciful, fantasy, fictitious, illusory, imaginative, phantom, romantic, unreal, utopian, visionary). (various references) | |
Hungarian | kitalált (fabricated, fabulous, feigned, fictional, fictitious, fictive, invented, made), képzeletbeli (chimerical, fancy, ideal, insubstantial, notional, phantom, visionary), kiagyalt, képzelt (fancied, fantastic, fictional, fictitious, mythical), képzeletben élő, képzelet szülte, képzelet alkotta, elképzelt (imaginative). (various references) | |
Indonesian | khayal (hallucination, image), berantah (fabled). (various references) | |
Italian | immaginario (fanciful, fictitious, fictive, notional, pretentious, visionary). (various references) | |
Japanese Kanji | 空想的 . (various references) | |
Japanese Katakana | くうそうてき. (various references) | |
Korean | 가상 (Aggressor, fictitious, Simulating). (various references) | |
Manx | sheiltynagh (conjectural, fanciful, imaginable, imaginative, notional, presumable, putative, speculative, speculator, theoretic, theoretical, theorist, thinkable), neurieugh, far-smooinaghtagh (chimerical). (various references) | |
Norwegian | innbilt. (various references) | |
Pig Latin | imaginaryay.(various references) | |
Portuguese | imaginrio (ideal), imaginário (aerial, airy, etherial, fancy, fantastic, fictional, notional, of fantasy, overground, unreal). (various references) | |
Romanian | ireal (aerial, aeriform, cardboard, fabled, fabulous, fanciful, phantasmal, unreal, vaporous), imaginar (aerial, fairy, fancied, fanciful, notional, subjective, unreal, visional, visionary, would be), închipuit (conceited, false, fancied, fictitious, notional, overweening, unreal, visional). (various references) | |
Russian | воображаемый (fancied, fictitious, fictive, imaginable, make believe, notional, visional). (various references) | |
Serbo-Croatian | imaginaran (fancied), zamišljen (abstracted, notional, pensive, reflective, ruminant, ruminative, thoughtful, wistful), vazdušast (aeriform, airy, light, puffy, unreal), uobražen (assuming, bighead, conceited, overweening, presuming, pretentious, supercilious, uppish). (various references) | |
Spanish | imaginario (fancied, fanciful, notional, unreal, untrue). (various references) | |
Swedish | inbillad (chimerical, fancied, fanciful, ideal, insubstantial, phantom, would be). (various references) | |
Thai | ซึ่งสมมุติขึ้น. (various references) | |
Turkish | imgesel (fictional, fictitious, imaginative), sanal (conjectural, virtual), hayali (aerial, airy, cardboard box, chimerical, delusive, Faerie, faery, fanciful, fantastic, fantastical, fictional, fictitious, fictive, illusive, illusory, imaginative, insubstantial, phantasmal, spectral, unreal, visionary), gerçek dışı (delusive, fanciful, insubstantial, phantasmal, unreal, unsubstantial), farazi (hypothetical, suppositional, supposititious), düşsel (fictional, unreal, utopian, visionary). (various references) | |
Turkmen | hyяally (supposed). (various references) | |
Ukrainian | уявний (fairy, fancied, fictitious, notional, putative, virtual, visionary), комплексний (complex, synthetic, synthetical), нереальний (aerial, aeriform, airy, delusive, delusory, unreal, unsubstantial, vaporous). (various references) | |
Vietnamese | tưởng tượng (aerial, fictional, fictitious, fictive, imaginative, mythical, mythologic, mythological, visionary), không có thực. (various references) | |
Welsh | dychmygol. (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Late Latin | 300-700 | phantasticus. (various references) |
| Source: compiled by the editor from various references. | ||
Misspellings | |
"Imaginary" is suggested in spellcheckers for the following: Imagina, imaginari, imaginery, immaginary, immaginery. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "imaginary" (pronounced i'ma"june'rē) |
| 5 | -u n e' r ē | bicentenary, cardiopulmonary, cautionary, centenary, concessionary, confectionary, confectionery, coronary, counterrevolutionary, culinary, deflationary, dictionary, disciplinary, discretionary, disinflationary, diversionary, evolutionary, exclusionary, expansionary, expeditionary, extraordinary, functionary, illusionary, inflationary, interdisciplinary, luminary, mercenary, missionary, noninflationary, ordinary, preliminary, probationary, pulmonary, reactionary, recessionary, revolutionary, seminary, stationary, stationery, urinary, veterinary, visionary. |
| 4 | -n e' r ē | quaternary. |
| 3 | -e' r ē | actuary, adversary, ancillary, apothecary, arbitrary, aviary, beneficiary, Blackberry, blueberry, budgetary, capillary, Cassowary, cemetery, cometary, commentary, commissary, Constabulary, contemporary, corollary, cranberry, customary, depositary, Dewberry, dietary, dignitary, itinerary, judiciary, lapidary, Dogberry, dromedary, dysentery, emissary, epistolary, estuary, fiduciary, formulary, fragmentary, funerary, gooseberry, hackberry, hereditary, honorary, Huckleberry, interplanetary, involuntary, legendary, library, literary, military, momentary, monastery, monetary, mortuary, mulberry, necessary, nonmilitary, obituary, paramilitary, pecuniary, pituitary, planetary, primary, proprietary, raspberry, Rosemary, salutary, sanctuary, sanitary, savagery, secondary, secretary, sedentary, semilegendary, solitary, statuary, strawberry, subsidiary, temporary, Tilbury, topiary, tributary, undersecretary, unitary, unnecessary, unsanitary, vocabulary. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-a-g-i-i-m-n-r-y" | |
-3 letters: aiming, airing, airman, angary, arming, grainy, magian, margay, margin, marina, maying, miring, ragman, raying, riming. | |
-4 letters: again, agria, amain, amiga, amnia, angry, anima, animi, gamay, gamin, garni, grain, grama, grana, grimy, inarm, iring, mangy, mania, maria, mayan, mingy, naira, rainy, rangy. | |
-5 letters: agar, agin, agma, airn, airy, amia, amin, amir, anga, aria, army. | |
| Words containing the letters "a-a-g-i-i-m-n-r-y" | |
+2 letters: imaginarily, marginality. | |
+5 letters: organismically. | |
| 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. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Modern | 5. Usage: Commercial 6. Images: Slideshow 7. Images: Photo Album 8. Quotations: Familiar | 9. Quotations: Historic 10. Quotations: Fiction 11. Quotations: Non-fiction 12. Usage Frequency | 13. Expressions 14. Expressions: Internet 15. Translations: Modern 16. Translations: Ancient | 17. Derivations 18. Rhymes 19. Anagrams 20. Bibliography |
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