Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Conjugate |
ConjugateAdjective1. Joined together especially in a pair or pairs. 2. Of a pinnate leaflet; having only one pair of leaflets. 3. (chemistry) formed by the union of two compounds; "a conjugated protein". 4. (chemistry) of an organic compound; containing two or more double bonds each separated from the other by a single bond. Noun1. A mixture of two partially miscible liquids A and B produces two conjugate solutions: one of A in B and another of B in A. Verb1. Unite chemically so that the product is easily broken down into the original compounds. 2. Of verbs. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "conjugate" was first used in popular English literature: sometime before 1613. (references) |
(From Wikipedia, the free Encyclopedia)
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number z = a + ib is defined to be z* = a - ib. It is also often denoted by a bar over the number, rather than a star.For example, (3-2i)* = 3 + 2i, i* = -i and 7* = 7.
One usually thinks of complex numbers as points in a plane with a cartesian coordinate system. The x-axis contains the real numbers and the y-axis contains the multiples of i. In this view, complex conjugation corresponds to reflection at the x-axis.
Properties
The following are valid for all complex numbers z and w, unless stated otherwise.
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
- (z + w)* = z* + w*
- (zw)* = z* w*
- (z/w)* = z* / w* if w is non-zero
- z* = z if and only if z is real
- |z*| = |z|
- |z|2 = z z*
- z-1 = z* / |z|2 if z is non-zero
If p is a polynomial with real coefficients, and p(z) = 0, then p(z*) = 0 as well. Thus the roots of real polynomials outside of the real line occur in complex conjugate pairs.
The function φ(z) = z* from C to C is continuous. Even though it appears to be a "tame" well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R. This Galois group has only two elements: φ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation.
Generalizations
Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C-star algebras.
One may also define a conjugation for quaternions: the conjugate of a + bi + cj + dk is a - bi - cj - dk.
Note that all these generalizations are multiplicative only if the factors are reversed:
Since the multiplication of complex numbers is commutative, this reversal is "invisible" there.
- (zw)* = w* z*
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Complex conjugate."
(From Wikipedia, the free Encyclopedia)
In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a group's structure.
Definition
Suppose G is a group. Two elements a and b of G are called conjugate iff there exists an element g in G with g-1ag = b. It can be readily shown that conjugacy is an equivalence relation and therefore partition G into equivalence classes. The equivalence class that contains the element a in G is
and is called the conjugacy class of a. Every element of the group belongs to precisely one conjugacy class. The classes Cl(a) and Cl(b) are equal if and only if a and b are conjugate, and disjoint otherwise.
- Cl(a) = {x in G : there exists g in G such that x = g-1ag}
Properties
If G is abelian, then g-1ag = a for all a and g in G; so Cl(a) = {a} for all a in G in the abelian case.
If two elements a and b of G belong to the same conjugacy class (i.e. if they are conjugate), then they have the same order. More generally, every statement about a can be translated into a statement about b=g-1ag, because the map φ(x) = g-1xg is an automorphism of G. Similarly, if H and K are subgroups of G and H and K are conjugate, then H is isomorphic to K (note that the converse is not true; consider any isomorphic subgroups of an abelian group).
An element a of G lies in the center Z(G) of G if and only if its conjugacy class has only one element, a itself. More generally, if CG(a) denotes the centralizer of a in G, i.e. the subgroup consisting of all elements g such that ga = ag, then the index [G : CG(a)] is equal to the number of elements in the conjugacy class of a (this is a special case of a result for conjugacy of subsets given later, as can be seen by letting S = {a} in the equation below).
Conjugacy class equation
If G is a finite group, then the previous paragraphs, together with the Theorem of Lagrange, implies that the number of elements in every conjugacy class divides the order of G.
Furthermore, for any group G, we can define a representative set S = {xi} by picking one element from each equivalence class of G which has more than one element. S then has the property that G is the disjoint union of Z(G) and the conjugacy classes Cl(xi) of the elements of S. One can then formulate the following important class equation:
where the sum extends over Hi = CG(xi) for each xi in S. [G : Hi] is the number of elements in class i, a proper divisor of |G| bigger than one. If the divisors of |G| are known, then this equation can often be used to gain information about the size of the center or of the conjugacy classes.
- |G| = |Z(G)| + ∑i [G : Hi]
As an example of the usefulness of the class equation, consider a group G with order pn, where p is a prime number and n > 0. Since the order of any subgroup of G must divide the order of G, it follows that each Hi also has order some power of p( ki ). But then the class equation requires that |G| = pn = |Z(G)| + ∑i (p( ki )). From this we see that p must divide |Z(G)|, so |Z(G)| > 1, and therefore we have the result: every finite p-group has a non-trivial center.
Conjugacy of subgroups
One can also look at conjugation as a group action on the set of all subgroups of a group G. They fall into orbitss, called again conjugacy classes of subgroups. The stabilizer of a given subgroup H for this action is its normalizer.
Conjugacy of general subsets
More generally, given any subset S of G (S now not necessarily a subgroup), we define a subset T of G to be conjugate to S if and only if there exists some g in G such that T = g-1Sg (the notation S x = x-1Sx is often used). We can define Cl(S) as the set of all subsets T of G such that T is conjugate to S.
A frequently used theorem is that, given any subset S of G, the index of N(S) (the normalizer of S) in G equals the order of Cl(S):
This follows since, if x and y are in G, then S x = S y if and only if xy -1 is in N(S), in other words, if and only if x and y are in the same coset of N(S).
- |Cl(S)| = [G : N(S)]
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Conjugacy class."
(From Wikipedia, the free Encyclopedia)
Conjugate can be:
- in terms of complex numbers, the complex conjugate
- in group theory, Conjugacy class or Conjugate closure
- in chemistry, Conjugate acid or conjugate base
- in linguistics, Grammatical conjugation
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Conjugate."
Synonyms: ConjugateSynonyms: conjugated (adj), coupled (adj), conjugate solution (n). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Accompaniment | Coupled; Verb: conjugate. |
Number | Sum, difference, complement, subtrahend; product; multiplicand, multiplier, multiplicator; coefficient, multiple; dividend, divisor, factor, quotient, submultiple; fraction, rational number; surd, irrational number; transcendental number; mixed number, complex number, complex conjugate; numerator, denominator; decimal, circulating decimal, repetend; common measure, aliquot part; prime number, prime, relative prime, prime factor, prime pair; reciprocal; totient. |
Repetition | Rehearse; do over again, say over again; ring the changes on; harp on the same string; din in the ear, drum in the ear; conjugate in all its moods tenses and inflexions, begin again, go over the same ground, go the same round, never hear the last of; resume, return to, recapitulate, reword. |
Word | Adjective: verbal, literal; nominal. conjugate, paronymous; derivative. |
| Source: adapted from Roget's Thesaurus. | |
Crosswords: Conjugate |
| English words defined with "conjugate": Axis of a curve ♦ conjugate solution, Conjugating ♦ Double point ♦ Glycocholic acid ♦ Hyperboloid of revolution ♦ Paronymous ♦ Self-conjugate ♦ Taurocholic. (references) |
| Specialty definitions using "conjugate": 4-Chloro-7-nitrobenzofurazan ♦ anterior internuclear palsy, available power ♦ base length ♦ Cascade Blue, concomitant strabismus, conduction-band minima ♦ Dodge About ♦ gear generation, generating method ♦ heterotropia, heterotropy ♦ ideal transducer, internuclear anterior ophthalmoplegia 2.anterior internuclear ophthalmoplegia, internuclear posterior ophthalmoplegia 2.posterior internuclear ophthalmoplegia ♦ Lhermitte's ophthalmoplegia, Lymphocyte Function-Associated Antigen-1 ♦ photo base, plait point, posterior intern internuclear palsy, Pursuit, Smooth ♦ Vaccines, Conjugate. (references) |
| Etymologies containing "conjugate": Biconjugate. (references) |
| Domain | Title |
Books |
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Source: compiled by the editor from various references; see credits. | |
| Thumbnail | Description & Credit |  | Chorioallantoic membrane infected with variola, stained with rabbit anti-vaccinia conjugate, then viewed using immunofluorescent microscopy technique. Credit: CDC. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Subject | Topic | Quote |
Health | NIAID supports studies to develop improved pneumococcal conjugate vaccines for children worldwide. (references) | |
In children <5 years old, over 80% invasive isolates are included in the 7-valent conjugate vaccine. (references) | ||
Incorporation of serogroup C meningococcal conjugate vaccine in the U.S. routine childhood vaccination programs. (references) | ||
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Conjugate" is generally used as a noun (singular) -- approximately 97.96% of the time. "Conjugate" is used about 49 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 |
| Noun (singular) | 97.96% | 48 | 49,194 |
| Lexical Verb (base form) | 2.04% | 1 | 339,140 |
| Total | 100.00% | 49 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "conjugate": conjugate axis ♦ Conjugate axis of a hyperbola ♦ Conjugate diameters ♦ Conjugate focus ♦ conjugate image points ♦ Conjugate mirrors ♦ Conjugate point ♦ conjugate points ♦ conjugate roots ♦ conjugate solution ♦ Wheat Germ Agglutinin-Horseradish Peroxidase Conjugate. Additional references. | |
| Hypenated Usage | |
Ending with "conjugate": Self-conjugate. | |
Containing "conjugate": Self-conjugate triangle. | |
| 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 "conjugate"; alternative meanings/domain in parentheses. | |
Albanian | zgjedhoj (inflect), të rrjedhur nga e njëjta rrënjë, i bashkuar (associated, cohesive, combined, confederate, conjoint, conjunct, connected, consolidated, incorporate, joining, linked, United), bashkoj (add together, ally, amalgamate, band together, close up, combine, compound, conglomerate, conjoin, connect, consolidate, couple, fuse, fuze, glue, hook up, inosculate, integrate, interlock, join, joint, knit, link, merge, piece, piece together, pool, solder, splice, unify, unite, weld), çift (couple, couplet, double, doublet, duet, match, pair, span, twin, two, twosome). (various references) | |
Arabic | قرن (connection, connexion, couple, coupling, horn, joint, pairing), وحد (combine, confederate, consolidate, graft, integrate, knit, marry, pool, unify, unite), صرف الافعالة. (various references) | |
Bulgarian | съединен (conjoint, coupled, incorporate, linked), свързвам се полово, свързвам (ally, articulate, associate, bandage, bond, brace, bracket, catenate, colligate, concatenate, conjoin, construe, couple, gather, interconnect, join, knot, lash, link, marry, mate, pack, piece out, piece together, put through, run on, truss, weld), сливам се (coalesce, interflow), спрягам (inflect), спрегнат, копулирам (copulate). (various references) | |
Chinese | 共轭 (Conjugated, Conjugating). (various references) | |
Czech | èasovat (inflect). (various references) | |
Danish | conjugatus, conjugata (conjugate diameter), konjugat, forbundet (connected, connexion). (various references) | |
Dutch | vervoegen, conjugeren. (various references) | |
Esperanto | konjugacii. (various references) | |
Farsi | توام (Duad, Geminate, Gemini, Joint), صرف کردن (Expend, Have, Inflect, Spend), درهم امیختن (Interlace). (various references) | |
Finnish | yhteenkytketty, taivuttaa (bend, bow, decline, fold, induce, inflect, persuade). (various references) | |
French | conjuguer. (various references) | |
German | konjugieren. (various references) | |
Greek | κλίνω (cant, careen, decline, hang up, heel, incline, inflect, lean, list, shelve, slant, slope, tend, tilt, tip, trend, verge). (various references) | |
Hebrew | לצמ" (couple, fasten, link, pair off, twin). (various references) | |
Hungarian | párosított (coupled, double), egyesült (combined, conjoint, incorporate, incorporated, United). (various references) | |
Indonesian | menafsirkan (construe, interprete). (various references) | |
Italian | coniugare (combine). (various references) | |
Manx | co-whingaghey (conjunction). (various references) | |
Papiamen | konhugá. (various references) | |
Pig Latin | onjugatecay.(various references) | |
Portuguese | conjugado (bed-sitter, conjugated, jugate, united). (various references) | |
Romanian | cuvânt derivat din aceeaşi rãdãcinã, conjuga, unit (associate, conjoint, conjunct, conjunctively, connected, joint, uniat, uniate, United), se uni (ally, associate, band together, coalesce, combine, commingle, conjoin, connect, inosculate, join, join hands, unite, weld), se împerechea (copulate, couple, rut), obiect care formeazã o pereche cu un alt obiect, derivat din aceeaşi rãdãcinã, împerecheat (geminate, twin). (various references) | |
Russian | соединенный (amalgamated, conjoint, conjunct, connected, coupled, incorporate, jnt joint, linked, tandem), спрягать соединенный, спрягать, парный (double, jugate, mating, pair, pair-horse, twin), диаметр тазового дна. (various references) | |
Serbo-Croatian | srodan (affined, agnate, akin, close, collateral, related, similar), spajati se (inosculate, merge), sjedinjavati se, konjugiran, izmenjati (exchange, inflect). (various references) | |
Spanish | conjugado (conjugated). (various references) | |
Swedish | konjugeras, konjugat, förenad i par, böjas, av samma stam. (various references) | |
Turkish | karşılıklı (inter-, mutual, opposed, opposing, opposite, reciprocal, reciprocating, tete a tete), eşlenik, birleşmek (affiliate, agree, ally, ally oneself, amalgamate, associate oneself, band together, be made one, coalesce, combine, confederate, congregate, conjoin, converge, couple, fasten, federate, fuze, gang, incorporate, inosculate, join, knit, knit up, link, make one, meld, merge, reunite, unite, weld), birleşik (adjunctive, combined, confederate, conjoint, conjunct, connected, federate, integrated, joint, jointed, unified, United), aynı kökten türemiş sözcük, aynı kökten türemiş, çift (ambi-, amphi-, bi-, binary, brace, couple, dipl-, diplo, double, dual, duet, duple, duplicate, dyad, even, geminate, pair, twin, yoke), çekmek (abide, absorb, arrest, attract, be a sufferer by, be a sufferer from, be cursed with smth., bear, bear with, captivate, carry, catch, charm, claw, decline, drag, draw, draw off, draw on, draw out, dwindle, engage, engross, extract, fetch, glamor, glamour, go through, go to scale at, grind, grip, haul, heave, hitch, hoist, hold, inflect, inhale, invite, last out, lug, lump, magnetize, pass through, pique, pluck, prepossess, pull, pull away, pull over, pull up, receive, record, run up, scale in, scale out, schlep, schlepp, shoot, shrink, sip, siphon, siphon off, soak, soak in, sop up, stand the racket, stretch, suck, suffer, sustain, syphon, syphon off, take, take after, take one's medicine, throw back, toss, tow, tow away, tug, turn the scale at, undergo, up with, whisk, withdraw). (various references) | |
Ukrainian | сполучений (amalgamated, conjunct, connected, coupled, incorporate), сполучатися (blend, communicate, conjoin, connect, couple), відмінювати, з'"днуватися (agglutinate, articulate, coalesce, come together, conjoin, connect, couple, interlock, join, knit, mix, tie in, unite), з'"днаний (aggregated, amalgamate, amalgamated, associated, conglomerate, congregate, coupled, joined, joint, linked). (various references) | |
Vietnamese | số liên hợp, kết hợp (annectent, combinative, conjoint, conjunctional, unitedly), đường kính liên hợp. (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | declina, declinabant, declinabat, declinabimus, declinabis, declinabit, declinabitis, declinabo, declinabunt, declinans, declinant, declinante, declinantes, declinantum, declinare, declinarent, declinaret, declinassent, declinasset, declinasti, declinastis, declinat, declinate, declinatur, declinaverat, declinaveris, declinaverit, declinaverunt, declinavi, declinavit, declinemus, declinent, declines, declinet, declinetis. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "conjugate": conjugated, conjugately, conjugateness, conjugatenesses, conjugates. (additional references) | |
Words containing "conjugate": nonconjugated, unconjugated. (additional references) | |
| |
"Conjugate" is suggested in spellcheckers for the following: congugate, conjucate, Conjugale, Conjugo, conjungate, conjutate. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "conjugate" (pronounced kÄ'njugā"t or kÄ"njugā't) |
| 3 | -g ā" t | gait, gate, negate. |
| 5 | -j u g ā' t | subjugate. |
| 4 | -u g ā' t | abrogate, castigate, congregate, corrugate, delegate, desegregate, fumigate, instigate, interrogate, investigate, irrigate, mitigate, navigate, obligate, propagate, relegate, segregate. |
| 3 | -g ā' t | arrogate, floodgate, litigate, profligate, tailgate. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-c-e-g-j-n-o-t-u" | |
-2 letters: coagent, cognate, jaconet. | |
-3 letters: cangue, cogent, coteau, jaunce, jounce, jugate, nougat, octane, outage, tongue, toucan, uncage. | |
-4 letters: acute, agent, agone, atone, cajon, canoe, canto, cento, centu, conga, conge, conte, cotan, count, enact, gaunt, genoa, genua, guaco, guano, jaunt, jeton, junco, junta, junto, oaten, ocean, octan, ounce, tango, tegua, togae, togue, tonga, ungot. | |
-5 letters: acne. | |
| Words containing the letters "a-c-e-g-j-n-o-t-u" | |
+1 letter: conjugated, conjugates. | |
+2 letters: conjugately. | |
+3 letters: unconjugated. | |
+4 letters: conjugalities, conjugateness, nonconjugated. | |
| 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: Commercial | 5. Images: Photo Album 6. Quotations: Non-fiction 7. Usage Frequency 8. Expressions | 9. Expressions: Internet 10. Translations: Modern 11. Translations: Ancient 12. Derivations | 13. Rhymes 14. Anagrams 15. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
