Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Complex |
ComplexAdjective1. Complicated in structure; consisting of interconnected parts; "a complex set of variations based on a simple folk melody"; "a complex mass of diverse laws and customs". Noun1. A conceptual whole made up of complicated and related parts. 2. A compound described in terms of the central atom to which other atoms are bound or coordinated. 3. (psychoanalysis) a combination of emotions and impulses that have been rejected from awareness but still influence a person's behavior. 4. A whole structure (as a building) made up of interconnected or related structures. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "complex" was first used in popular English literature: sometime before 1550. (references) |
| Domain | Definition |
Aerospace | 1. Short for launch complex, as in Complex 25B at Cape Kennedy.2. Pertaining to a magnitude composed of a real number and an imaginary number. (references) |
Chemistry | Any compound in which the bonding is by interaction of the electrons of the donor with empty orbitals of the acceptor. Source: European Union. (references) |
Medicine | May develop as a result of long term conditioning, and/or early traumatic experience. Source: European Union. (references) |
Mining | A. A large-scale field association or assemblage of different rocks of any age or origin, having structural relations so intricately involved or otherwise complicated that the rocks cannot be readily differentiated in mapping, e.g., a volcanic complex. See also:igneous complex; basement complex b. A unit that consists of a mixture of rocks of two or more genetic classes, i.e., igneous, sedimentary, or metamorphic, with or without highly complicated structure; example: Franciscan Complex c. Said of an ore that carries several metals difficult to extract d. An assemblage of rocks of any age or origin that has been folded together, intricately mixed, involved, or otherwise complicate e.g., a volcanic complex. See also:igneous complex; basement complex i.e., igneous, sedimentary, or metamorphic, with or without highly complicated structure; example: Franciscan Complex. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
A complex is a collection of buildings closely packed together normally with a perimeter wall or fence surrounding them. This article is about complexes in chemistry; see complex number for the mathematical and engineering usage.
In chemistry, a complex is a structure composed of a central metal atom or ion, generally a cation, surrounded by a number of negatively charged ions or neutral molecules possessing lone pairs. A complex may also be called a coordination compound or metal complex.
The ions/molecules surrounding the metal are called ligands. A ligand that is bound to a metal ion is said to be coordinated with the ion. The process of binding to the metal ion with more than one coordination site per ligand is called chelation. Compounds that bind avidly to form complexes are thus called chelating agents (for example, EDTA).
Simple ligands like water or chlorine form only one link with the central atom and are said to be monodentate. Some ligands are capable of forming multiple links to the same metal atom, and are described as bidentate, tridentate etc. EDTA is hexadentate, which accounts for the great stability of many of its complexes.
Typically, the chemistry of complexes is dominated by interactions between s and p orbitals of the ligands and the d (or f) orbitals of the metal ions. Because of this, simple octet bonding theory fails in the case of complexes and to understand the chemistry of these systems, a deeper understanding of chemical bonding rules is necessary.
One such rule is called electron counting, or the rule of 18. Crystal field theory, introduced by Hans Bethe in 1929, is a more quantum mechanically based attempt at understanding complexes. But crystal field theory treats all interactions in a complex as ionic. Ligand field theory, introduced in 1935 and built from molecular orbital theory, can handle a broader range of complexes and can explain complexes in which the interactions are covalent. The chemical applications of group theory can aid in the understanding of crystal or ligand field theory, by allowing simple, symmetry based solutions to the formal equations.
Naming complexes
The basic procedure for naming a complex:eg. [NiCl4]2- = tetrachloronickelate (II) ion
- Write the number of ligands surrounding the central metal ion/atom using: mono-, di-, tetra-, penta-, and hexa.
- Write the name of the ligands (in alphabetical order) appending an o for anions: fluoro, chloro, cyano. Neutral ligands keep their original name.
- Name the central atom/ion appending -ate if the atom is negative. Neutral or positive atoms/ions keep their original name.
Transition metals make good central ions for complexes.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Complex."
(From Wikipedia, the free Encyclopedia)
Complex analysis is the branch of mathematics investigating holomorphic functions, i.e. functions which are defined in some region of the complex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, every holomorphic function is representable as power series in every open disc in its domain of definition, and is therefore analytic. In particular, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, such as all polynomials, the exponential function, and the trigonometric functions, are holomorphic. See also : holomorphic sheaves and vector bundles.One central tool in complex analysis is the path integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary (Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues is useful. If a function has a pole or singularity at some point, meaning that its values "explode" and it does not have a finite value there, then one can define the function's residue at that pole, and these residues can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem. Functions which have only poles but no essential singularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities.
A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed.
An important property of holomorphic functions is that if a function is holomorphic throughout a simply connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions such as the Riemann zeta function which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
There is also a very rich theory of complex analysis in more than one complex dimension where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) are no longer true. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, maybe the most important result in the one-dimensional theory, fails dramatically in higher dimensions.
Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and some even before. Important names are Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Traditionally, complex analysis, in particular the theory of conformal mappings, has many applications in engineering, but it is also used throughout analytical number theory. In modern times, it became very popular through a new boost of Complex dynamics and the pictures of fractals produced by iterating holomorphic functions, the most popular being the Mandelbrot set. Another important application of complex analysis today is in string theory which is a conformally invariant quantum field theory.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Complex analysis."
(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)
There are different senses of complexity:
- In computer science, the study of how much time and memory a computer algorithm may take is the field of computational complexity theory.
- Complexity is often used as a shorthand for the field that developed in the late 1980s around the use of mathematical and computational modeling of biological, economic and technological systems known as "complex systems" (sometimes complex adaptive systems).
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Complexity."
| The following table is compiled from various sources, across various languages. When English abbreviations or acronyms come from a non-English source, this is noted. | |||
| Entry | Source | Expression | Field |
| CONUS | English | Continental United States Complex | N/A |
Source: compiled by the editor, based on several corpora (additional references). | |||
Synonyms: ComplexSynonyms: building complex (n), composite (n), coordination compound (n). (additional references) |
| Antonym: simple (adj). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Complexity | Adjective: gnarled, knarled. complex, complexed; intricate, complicated, perplexed, involved, raveled, entangled, knotted, tangled, inextricable; irreducible. |
Disorder | Complex; a. |
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. |
Positive, negative; rational, irrational; surd, radical, real; complex, imaginary; finite; infinite; impossible. | |
Whole | Alpha and Omega, " be all and end all "; complex, complexus; lock stock and barrel. |
| Source: adapted from Roget's Thesaurus. | |
| Domain | Usage | |
Screenplays | If I didn't know better, I'd say he developed an edifice complex. (Tomorrow Never Dies; writing credit: Bruce Feirstein) This could be a lot more uh, uh, uh, complex, I mean it just might, it might not be such a simple, uh, you know (The Big Lebowski; writing credit: Ethan Coen; Joel Coen) Ah, you've got an inferiority complex, you have (A Hard Day's Night; writing credit: Alun Owen) I'm not complex, you're complex (Bait; writing credit: Andrew Scheinman; Adam Scheinman) Libido, inferiority complex, stinko, blotto, free love, bathtub gin, monkey glands, Karl Marx is he one of the Marx Brothers (Auntie Mame; writing credit: Betty Comden; Patrick Dennis) | |
Lyrics | As funny as it be by you, it not that complex (It Wasn't Me; performing artist: Shaggy) And in the day, everything's complex (When You're Gone; performing artist: The Cranberries) Or you may find yourself with a quite complex complex, (Oedipus Rex; performing artist: Tom Lehrer) You may have heard about his odd complex. (Oedipus Rex; performing artist: Tom Lehrer) | |
Clever | To every complex problem there is an easy answer, and it is wrong! (references; author: unknown) | |
Movie/TV Titles | Naked Complex (1963) Kokaku kidotai: Stand Alone Complex (2002) Love Complex (2000) The Lucifer Complex (1978) Haldia Dock Complex (1978) | |
Source: compiled by the editor from various references; see credits. | ||
| Domain | Title | ||
References | |||
Books |
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Periodicals |
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Theater & Movies | |||
Music |
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Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | Proteins in the cytoplasm target and activate biochemical reactions in specific cellular components, such as the mitochondria and the nucleus. Within the nucleus, gene expression is activated and leads to new protein synthesis, which are formed on the endoplasmic reticulum. These proteins are transported via the Golgi complex for packaging and distribution to cellular membrane and for extracellular release. See artwork: (on disk in file). Credit: Jeannie Kelly (artist). |  | Members of the first National Advisory Cancer Council at the groundbreaking ceremonies at the NCI's building 6 in June, 1938. (Left to right) Francis Wood, C.C Little, James Ewing, Arthur Compton, James Conant, Thomas Parran, and Ludwig Hektoen. This new building, erected on land donated by Mrs. Luke J. Wilson was the fourth to be constructed in the complex that is now the National Institutes of Health. The structure was unique in that year of 1939, with its physical equipment and facilities designed solely for scientific research in a specialized field of science. Building 6 was to house the National Cancer Institute, the first of the nine specialized institutes that would comprise NIH. See also ar003810. Credit: Unknown photographer/artist. |
 | Histopathology of lung showing ferruginous body. A fiber of asbestos or related substance is coated by an iron-protein complex and surrounded by macrophages. Prussian blue iron stain. Credit: CDC. |  | Engine test in Ames Full-Scale Aerodynamic Complex. Credit: NASA. |
 | Canberra Deep Dish Communications Complex. Credit: NASA. |  | This Hubble telescope image shows one of the most complex planetary nebulae ever seen, NGC ... Credit: NASA. |
 | The western region of Australia's Great Sandy Desert is an area almost devoid of sand, but characterized by complex geology. Credit: NASA. |  | Guinea-Bissau is a small country in West Africa. Complex patterns can be seen in the shallow waters along its coastline, where silt carried by the Geba and other rivers washes out into the Atlantic Ocean. Credit: NASA. |
 | The Kennedy Center (to right) and the Watergate Complex. Credit: America's Coastlines. |  | A bird's eye view of a passing hydrofoil and most of the netting complex. The mouth of the trap and the cod end extending landward can be clearly seen. Credit: Fisheries. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
 |  |
| "Temple Complex" by Sam Segar Commentary: "Abandoned Temple complex at Orcha, Rajastan, India." | "Skyscraper and tree" by Jen Dixon Commentary: "A shot looking upward at the PPG One building in Pittsburgh, Pennsylvania, USA. This building, as well as the several others in the complex, is made almost entirely of glass." |
Source: photographs selected by the editor, with permission from the photographers. | |
| Play | Caption | Play | Caption |
 | Lots of complex synthesized work similar to an early Peter Gabriel style. | | Complex piano excerpt very typical of the Russian school. |
| Source: compiled by the editor from various references; see credits. | |||
| Author | Quotation |
Albert Einstein | No matter how we may single out a complex from nature...its theoretical treatment will never prove to be ultimately conclusive... I believe that this process of deepening of theory has no limits. |
Bradford | For every complex problem there is a simple solution -- and it's wrong. |
Laurence J. Peter | Some problems are so complex that you have to be highly intelligent and well informed just to be undecided about them. |
Oscar Wilde | I adore simple pleasures. They are the last refuge of the complex. |
Remy De Gourmont | Simple ideas lie within the reach only of complex minds. |
Source: compiled by the editor from various references. | |
| Title | Author | Quote |
Sylvie and Bruno Concluded | Carroll, Lewis | We often see one mind a most complex correction of limbs and organs, when joined together |
So Long, and Thanks For All the Fish | Douglas Adams | He opened the bottle and wiped the top of it with the towel, which had the opposite effect to the one intended, in that the Ol' Janx Spirit instantly killed off millions of the germs which had been slowly building up quite a complex and enlightened civilization on the smellier patches of the towel |
Les Miserables | Hugo, Victor | As to convents, they present a complex question |
Portrait of the Artist as a Young Man | Joyce, James | Dublin was a new and complex sensation |
Source: compiled by the editor from various references. | ||
| Subject | Topic | Quote |
Health | The route is complex. (references) | |
A large immune complex. (references) | ||
Swallowing is a complex process. (references) | ||
Business | The media environment is complex. (references) | |
Complex IT solutions include software and hardware supplies. (references) | ||
The relationship of the United States with Mexico is broad and complex. (references) | ||
Children | Brazil | An NGO in Niteroi began construction of a sports complex for the disabled. (references) |
Russia | A complex and cumbersome system was developed to manage the life-long institutionalization of some children; three different ministries (Education, Health, and Labor and Social Development) assumed responsibility for different age groups and categories of orphans. (references) | |
Civil Liberties | Latvia | Noncitizens may own land only under complex procedures but may not purchase land in the border zones. (references) |
Discrimination | Belgium | With Dutch, French, and German as official languages, the country has a complex linguistic regime, including language requirements for various elective and appointive positions. (references) |
Israel and the occupied territories | Under the complex mixture of laws and regulations that apply to the occupied territories, Palestinians are disadvantaged under Israeli law and practices compared with the treatment received by Israeli settlers. (references) | |
Economic History | Djibouti | It is complex and far from transparent. (references) |
Human Rights | Netherlands | When the entire complex is renovated, it is expected to hold 700 prisoners. (references) |
India | Extensive, complex patterns of violence continued in many of the seven northeastern states. (references) | |
Solomon Islands | The only prison complex in use during the year was completed with the assistance of a foreign government. (references) | |
Political Economy | OMAN | Oman's customs procedures are complex. (references) |
Nicaragua | Party politics are complex and fragmented in this young democracy. (references) | |
Georgia | The Russian-Georgian relationship is extremely tense and complex at this time. (references) | |
Political Rights | Hungary | Members of Parliament are elected every 4 years through a complex, multistate process, in which voters cast ballots for individual candidates and party lists. (references) |
Mauritania | A countrywide census, taken at the end of 1998 and designed to register all citizens and standardize the current complex system of names, also was aimed in part at providing the basis for free and fair elections. (references) | |
Trade | Syria | Customs procedures are cumbersome, tedious, and time-consuming because of complex regulations. (references) |
Travel | Kenya | However, Kenya is a developing country with a complex market. (references) |
Japan | Japanese society is complex, structured, hierarchical and group-oriented with strong emphasis on maintaining harmony and avoiding surface confrontation. (references) | |
Vietnam | One bedroom at a serviced apartment complex in the center of town goes for about US $900 a month, while individual houses range from US $300 to US $4,000 depending on location, size, and standard. (references) | |
Women | Micronesia | Such actions were deemed offenses against the family, not just the individuals within them, and were addressed by a complex system of familial sanctions. (references) |
Worker Rights | Australia | For them a complex body of government regulations, as well as decisions of applicable federal or state industrial relations commissions, prescribe a 40-hour or shorter workweek, paid vacations, sick leave, and other benefits. (references) |
Egypt | The minimum wage does not provide for a decent standard of living for a worker and family; however, base pay commonly is supplemented by a complex system of fringe benefits and bonuses that may double or triple a worker's take-home pay. (references) | |
Lexicography | Devil's Dictionary | ROPE, n. An obsolescent appliance for reminding assassins that they too are mortal. It is put about the neck and remains in place one's whole life long. It has been largely superseded by a more complex electrical device worn upon another part of the person; and this is rapidly giving place to an apparatus known as the preachment. |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| Speaker | Phrase(s) |
Ann Richards | What you have to do, if you're going to send any medicine to Cuba, you have to qualify and register with the Department of Commerce in what is a rather complex registration procedure. |
Donald Rumsfeld | This job is complex. It is multidimensional. It is important, because people's lives are at risk. It is challenging. You'd never use the word fun. |
Laura Schlessinger | When you live in a condo complex with people next door, I don't know how you can be dead for four months without anybody noticing you not coming and going. |
Peter Jennings | Precisely. The trouble with polling, as we both know, is they polls don't always ask the complex questions. And when they do, people answer in different forms. |
Phil McGraw | Well, first off, you'd be amazed how much I agree with what they say sometimes, because sometimes there are problems that are terribly, terribly complex. So I would embrace their ideology in that regard many times. |
Rush Limbaugh | Whatever the answer, it is clear that the relationship between Chirac and Hussein is long and complex, and not altogether easy to understand. |
Source: compiled by the editor from various references; see credits. | |
| Speaker | Term | Phrase(s) |
Theodore Roosevelt | 1901-1909 | Modern life is both complex and intense, and the tremendous changes wrought by the extraordinary industrial development of the last half century are felt in every fiber of our social and political being. |
Herbert C. Hoover | 1929-1933 | We can not hope to succeed in directing this increasingly complex civilization unless we can draw all the talent of leadership from the whole people. |
Harry S. Truman | 1945-1953 | The bigger and more complex the Federal Program, the more necessary it is for the Chief Executive to submit a single budget for action by the Congress. |
Dwight Eisenhower | 1953-1961 | In the councils of government, we must guard against the acquisition of unwarranted influence, whether sought or unsought, by the military-industrial complex. |
Lyndon B. Johnson | 1963-1969 | Today, too, the work of government is far more complex than in our early years, requiring more time to learn and more time to master the technical tasks of legislating. |
Gerald Ford | 1974-1977 | Complex welfare programs cannot be reformed overnight. |
Jimmy Carter | 1977-1981 | Our problems are too complex for simple slogans or for quick solutions. |
Ronald Reagan | 1981-1989 | You know, we only have a military-industrial complex until a time of danger, and then it becomes the arsenal of democracy. |
Source: compiled by the editor from various references. | ||
| "Complex" is generally used as an adjective (general or positive) -- approximately 94.30% of the time. "Complex" is used about 7,642 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) | 94.3% | 7,206 | 1,345 |
| Noun (singular) | 5.7% | 436 | 13,209 |
| Total | 100.00% | 7,642 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "complex": adsorbing complex ♦ adsorption complex ♦ AIDS dementia complex ♦ AIDS related complex ♦ AIDS-Related Complex ♦ amiga Persecution Complex ♦ Antigen-Antibody Complex ♦ apartment complex ♦ B complex ♦ basal complex ♦ basement complex ♦ bovine respiratory disease complex ♦ building complex ♦ clay complex ♦ Coat Protein Complex I ♦ cold store complex ♦ Complement Membrane Attack Complex ♦ complex absence ♦ complex arrays ♦ complex body part ♦ Complex ether ♦ complex fraction ♦ Complex homeopathy ♦ complex impedance ♦ complex instruction set computer ♦ complex instruction set computing ♦ Complex integer ♦ complex low ♦ complex low pressure ♦ complex number ♦ Complex of lines ♦ complex ore ♦ complex programmable logic device ♦ complex quantity ♦ Complex Regional Pain Syndromes ♦ complex sentence ♦ complex situations ♦ complex sound ♦ complex tone ♦ Complex unit ♦ complex word ♦ Eisenmenger Complex ♦ electra complex ♦ fundamental complex ♦ Golgi complex ♦ guilt complex ♦ hospital complex ♦ imaginary part of a complex number ♦ Immune Complex Diseases ♦ industrial complex ♦ inferiority complex ♦ Iron-Dextran Complex ♦ lateral occipital complex ♦ LCM-Lassa Complex Viruses ♦ Major Histocompatibility Complex ♦ Mycobacterium avium Complex ♦ oedipal complex ♦ Oedipus complex ♦ persecution complex ♦ persecutory complex ♦ Platelet Glycoprotein GPIb-IX Complex ♦ Platelet Glycoprotein GPIIb-IIIa Complex ♦ pneumoenteritis complex ♦ port complex ♦ Pyruvate Dehydrogenase Complex ♦ Pyruvate Dehydrogenase Complex Deficiency Disease ♦ QRS complex ♦ ranalian complex ♦ school complex ♦ soil complex ♦ Sucrase-Isomaltase Complex ♦ superiority complex ♦ Synaptonemal Complex ♦ Tacaribe Complex Viruses ♦ target complex ♦ time Complex Simulator ♦ vitamin B complex. Additional references. | |
| Hyphenated Usage | |
Beginning with "complex": complex-feature-based, complex-seeming, complex-shaped, complex-stabilization. | |
Ending with "complex": non-complex, over-complex. | |
Containing "complex": B-complex vitamin. | |
| 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 "complex"; alternative meanings/domain in parentheses. | |
Albanian | tërësi (altogether, ensemble, entirety, integer, integral, integrality, integrity, massiveness, plenitude), kompleks (campus, compound), fiksim (fastening, fixation, fixing, hitch), bllok (bloc, block, book, lump, pad, tablet, unit). (various references) | |
Arabic | مجموعة (aggregate, aggregation, assemblage, association, band, battery, bloc, block, body, case, collection, combination, community, company, compilation, gathering, group, list, pack, party, series, set, squad, suit, system, team, troop), مركب (assembled, built up, combination, combined, component, composite, composition, compound, installed, put together, synthesis), عقدة (clove, ganglion, hitch, kink, knob, knurl, loop, nub, tangle, umbo), المجمع (gleaner). (various references) | |
Bulgarian | съставен (built up, component, composite, compositive, constituent, constitutive, elemental, integrate), сложен (aggregate, complicated, compound, convoluted, daedalian, elaborate, integrate, intricate, knotty, multiplex, perplexed, perplexing, sinuous, subtle, tangly, tricky), цялост (continuity, entirety, fullness, fulness, integrality, oneness, plenitude, thoroughness, totality, whole, wholeness), труден (arduous, awkward, difficile, difficult, dodgy, formidable, heavy, knotty, laborious, lively, painful, perplexed, perplexing, pick-and-shovel, rocky, sore, spiny, stroppy, taxing, thorny, ticklish, tight, tough, tricky, troublesome, uphill, warm), комплексен, комплекс, забъркан (complicated, embroiled, inextricable, intricate, involved), заплетен (anfractuous, convoluted, inextricable, intricate, involute, involved, knotty, labyrinthine, mazy, sinuous, tangly, tortuous). (various references) | |
Chinese | 複雜 (complicated), 複合 (compound), 褔 (double), 复杂 (Complexities, Complexity, Complicated, Intricacies, intricacy, intricate, sophisticate, Sophisticating). (various references) | |
Czech | celek (all, altogether, gross, total, totality, unit, whole), soubor (body, company, ensemble, file, package, set, suite, syllabus), složitý (complicated, intricate, involved), podřadný (crummy, inferior, subordinate), komplikovaný (complicated, convoluted, elaborate, intricate, involved, many-sided, sophisticated), komplexní (global), komplex (hang up, pack). (various references) | |
Danish | komplexudløst, kompleks (comprehensive, ensemble, entity, extensive, group), blanding (admixture, mixture). (various references) | |
Dutch | complex, samengesteld (complicated, compound). (various references) | |
Esperanto | malsimpla (complicated), komplekso, kompleksa. (various references) | |
Farsi | پیچیده(.adj), مجموعه (Bundle, Caboodle, Collection, Repertory, Scrapbook, Set, Yardage), مرکب ازچندجزء , هم تافت , گروهه (Team), اچار (Spanner, Wrench), بغرنج (Intricate, Obscurant). (various references) | |
Finnish | monimutkainen (complicated), monimuotoinen (complicated), kompleksi, homeopaattinen kompleksi. (various references) | |
French | complexe (college, complicated). (various references) | |
German | komplex (complexly), kompliziert (complicated, complicates, compound, difficult, elaborate, intricate, involved, sophisticated, tricky). (various references) | |
Greek | σύμπλεγμα (cluster, group, ligature), συγκρότημα (cluster, group), πολύπλοκος (complicated), πολυσύνθετος. (various references) | |
Hebrew | מורכב (combined, complicated, composed, composite, compound, intricate), מכלל (assembly, perfection), מכלול (compound, ensemble, entirety, generality, integrity, perfection, splendor, totality, wholeness), מסובך (complicated, intricate, involved, knotty, sophisticated, tangled, twisted), תצמיד, תסביך (complication). (various references) | |
Hungarian | bonyolult (complicated, difficult, elaborate, inextricable, intricate, involute, involved, knotty, mazy, problematic, thorny, tricky), összetett (combined, composite, compound, mixed, multiple). (various references) | |
Indonesian | ruwet (intricate, knotty), kompleks, himpunan (compoilation). (various references) | |
Italian | complesso (complicated, ensemble, entirety, group, intricate, manifold, system, totality, unit, whole). (various refe |
