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Topic: Imaginary unit

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Imaginary number

Complex number

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Derivation (abstract algebra)

	Imaginary number - Wikipedia, the free encyclopedia
		One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left.
		In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with a changing current, traditionally denoted by i.
		Imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, and cartography.
	en.wikipedia.org /wiki/Imaginary_number (726 words)

	Imaginary unit - Wikipedia, the free encyclopedia
		If all mathematical textbooks and published literature referring to imaginary or complex numbers were rewritten with −i replacing every occurrence of +i (and therefore every occurrence of −i replaced by −(−i) = +i), all facts and theorems would continue to be equivalently valid.
		In this case, the ambiguity results from the geometric choice of which "direction" around the unit circle is "positive".
		In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with electrical current as a function of time, traditionally denoted by i(t) or just i.
	en.wikipedia.org /wiki/Imaginary_unit (1003 words)

	Tutorial, Electric Geometry - Historical Prelude
		The concept of the imaginary number was introduced in the 1545 works of Girolamo Cardano, the actual words "real" and "imaginary" being added later in 1637 by Decartes.
		Imaginary numbers could be applied to geometry, just as Girard and Decartes had done for the negatives.
		Multiplication by the negative unit, -1, clearly represented a reflection in the origin, and now a multiplication by the imaginary unit, i, took the +x-axis direction into the +y-axis direction.
	www.hypercomplex.com /education/intro_tutorial/hi02.html (406 words)

	IMAGINARY AND COMPLEX NUMBERS
		In using imaginary numbers in electricity, the imaginary unit is often represented by j, instead of i, since i is the common notation for electrical current.
		Imaginary numbers are added or subtracted by writing them using the imaginary unit i and then adding or subtracting the real number coefficients of i.
		Imaginary numbers are multiplied or divided by writing them using the imaginary unit i, and then multiplying or dividing them like algebraic terms.
	www.tpub.com /doemathematics/mathematics82.htm (469 words)

	Imaginary number (Site not responding. Last check: 2007-11-05)
		In mathematics an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero.
		An imaginary number is a number which can be written as $ib b is a real number and i the$ imaginary unit with the property that
		One way of viewing imaginary numbers is consider a standard number line positively increasing in magnitude to the and negatively increasing in magnitude to the At zero on this x axis draw a y axis; "positive" imaginary numbers then increase magnitude upwards and "negative" imaginary numbers increase magnitude downwards.
	www.freeglossary.com /Imaginary_number (637 words)

	ImaginaryNumber at CodePedia
		The imaginary unit is defined as the square root of negative one.
		Imaginary numbers are just as real as any other numbers (the name "imaginary" does not imply anything substandard about their status of existence).
		They are called imaginary numbers simply to contrast them with the set of real numbers, which had been discovered much earlier and therefore were already named.
	www.codepedia.com /1/ImaginaryNumber (180 words)

	Units: I
		In the history of English units the inch seems to come before the foot: after the Norman conquest of 1066 the foot was defined to equal 12 inches, rather than the inch being defined as 1/12 foot.
		In the United States, atmospheric pressure is customarily expressed as the height of a column of mercury exerting the same pressure as the atmosphere.
		The colpa, a traditional Irish unit of pasturage, is approximately equal to the Irish acre.
	www.unc.edu /~rowlett/units/dictI.html (1513 words)

	PlanetMath: imaginary
		A complex number is called imaginary if its real part is 0.
		The term imaginary has had its definition expanded by analogy to several areas of mathematics.
		This is version 5 of imaginary, born on 2002-02-16, modified 2005-01-21.
	planetmath.org /encyclopedia/Imaginary.html (114 words)

	Imaginary unit: Encyclopedia topic (Site not responding. Last check: 2007-11-05)
		In this case, the ambiguity results from the geometric choice of which "direction" around the unit circle (unit circle: in mathematics, a unit circle is a circle with unit radius, i.e., a circle whose...
		The imaginary unit is sometimes written in advanced mathematics contexts (as well as in less advanced popular texts), but great care needs to be taken when manipulating formulas involving radicals.
		The notation is reserved either for the principal square root (square root: A number that when multiplied by itself equals a given number) function, which is only defined for real x ≥ 0, or for the principal branch of the complex square root function.
	www.absoluteastronomy.com /reference/imaginary_unit (925 words)

	Complex number - Wikipedia, the free encyclopedia
		The imaginary unit i is the complex number (0, 1).
		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.
		The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers).
	en.wikipedia.org /wiki/Complex_number (3424 words)

	Lesson Page - Imaginary Unit
		The reason for the name "imaginary" numbers is that when these numbers were first proposed several hundred years ago, people could not "imagine" such a number.
		It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist.
		Electrical engineers use the imaginary unit (which they represent as j) in the study of electricity.
	regentsprep.org /Regents/mathb/2c2/imagineLes.htm (111 words)

	Society Hill at Kilmer Woods II - Rules & Regulations
		Sides: The sides of each Unit are imaginary vertical planes along and coincident with the innermost surface of the studding of the perimeter walls (originally installed by the Sponsor) or where there is not studding, the innermost surface of concrete block perimeter walls or equivalent.
		A unit owner is obligated to advise the Association when a unit is rented and must provide the Association with a copy of the rental agreement or lease.
		Unit Owners with a fireplace shall be required to have the chimney cleaned or inspected from the interior of the unit, every three years.
	www.kilmerwoods2.com /Rules-reg.htm (6236 words)

	unit05-sect01-les02-lessona (Site not responding. Last check: 2007-11-05)
		Before beginning this Focus, it is important that you understand the concept of imaginary and complex numbers.
		A pure imaginary number, you will recall, is in the form bi, where b is a non-zero real number and i is the imaginary unit, where i =
		It is in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
	www.cdli.ca /courses/math3207/unit05/section01/lesson02/3-lesson-a.htm (210 words)

	Define imaginary number - a definition from Whatis.com (Site not responding. Last check: 2007-11-05)
		An imaginary number is a quantity of the form ix, where x is a real number and i is the positive square root of -1.
		The set I of imaginary numbers consists of the set of all possible products iw, where w is an element of the set R of real numbers.
		Imaginary and complex numbers are used in engineering, particularly in electronics.
	searchsmb.techtarget.com /sDefinition/0,,sid44_gci283974,00.html (321 words)

	The Imaginary Unit, i - Encyclopedia FunTrivia
		The definition itself of the imaginary unit ("a solution of the equation x^2 = -1") is ambiguous, because the equation has two solutions.
		On the contrary, it is "imaginary", which means it can't be represented anywhere on the real number line.
		These numbers are called "imaginary numbers", and are, by definition, numbers whose square is less than or equal to zero.
	www.funtrivia.com /en/subtopics/The-Imaginary-Unit-i-200670.html (2138 words)

	Definition of the Unit
		Each Unit is intended to contain all space within the area bounded by the interior surface of the exterior perimeter or party walls of each Unit and the floor and the ceiling of each as follows:
		TOP: The top of each Unit is an imaginary plane along and coincident with the unfinished and unexposed surface of the gypsum board which forms the ceiling of the uppermost story in the Unit where it closes at every side of such Unit.
		No Unit may be partitioned or subdivided without prior written approval of the holder of any first mortgage loan on such Unit and the Board of Trustees of the Association.
	home.comcast.net /~dpvw/master_deed_paragraph_5.htm (508 words)

	program
		The Imaginary Number is a collection of nine single projects and new groupings of works, including drawings, installations, sculptures and video and film installations.
		In mathematics, the imaginary number makes it possible to solve equations that have no solution in the realm of "real numbers" - the square root of -1 is defined as the imaginary unit i.
		Curated by Anselm Franke and co-curated by Hila Peleg, The Imaginary Number is the first of a series of exhibitions and projects in the future KW programme that engages with the social aspects of imagination and the realities of systems of belief.
	www.kw-berlin.de /english/program_theimaginarynumber.htm (596 words)

	Toward a General Theory of Process (Site not responding. Last check: 2007-11-05)
		An imaginary unit vector in time requires a mathematical foundation that is an extension of Hamilton’s quaterions.
		This applies also to the imaginary unit vectors associated with the second- and third-order spacetime del operators given later in the text.
		The fundamental concepts involved here are far more elaborate than the passive notion of imaginary time Stephen Hawking presented a decade later in his book A Brief History of Time (and in the associated technical papers).
	www.geocities.com /moonhoabinh/ithapapers/Gplinks/gplink2.html (444 words)

	Geometric Calculator - Introduction
		The unit basis bivectors are xy, yz, and zx each representing a unit area in the plane named by the basis.
		If a is a unit vector and B is any Clifford number, then -aBa is the Clifford number with all its parts reflected in the plane orthogonal to a.
		If a and b are unit vectors and C is any Clifford number, then abCba is the Clifford number with all its parts rotated in the ab plane by twice the angle between a and b.
	elf.org /calculator/intro.html (690 words)

	Defect report #207
		After some thought, the author of this DR feels that imaginary types are experimental enough that the first approach is better and has worded the Suggested Technical Corrigendum on that basis.
		Though imaginary types are not compatible with the corresponding real type, values of one may be used where the other is expected in the following cases.
		In each case the value is converted to the value of the other type that has the same representation (that is, by multiplying by the imaginary unit when converting to an imaginary type, and by dividing by the imaginary unit when converting to a real type).
	www.open-std.org /jtc1/sc22/wg14/www/docs/dr_207.htm (619 words)

	Unit - Voyager, the free encyclopedia (Site not responding. Last check: 2007-11-05)
		Unit of action, a discrete piece of action (or beat) in a theatrical presentation
		Unit (ring theory), an element that is invertible with respect to ring multiplication
		In category theory, there is a natural transformation called the unit from the identity functor to the composition of two adjoint functors, q.v.
	www.voyager.in /Unit (310 words)

	Math Forum - Ask Dr. Math
		As you probably know, imaginary numbers are just a particular case of complex numbers.
		A complex number is written as a + ib, where i is the imaginary unit (the square root of - 1), and a and b are real numbers.
		An imaginary number is one where the real part (the "a") is zero, so an imaginary number is written as just ib, where b is a real number.
	mathforum.org /library/drmath/view/53810.html (560 words)

	I
		i denotes the imaginary unit, a complex number whose square is -1.
		The imaginary unit is represented by j instead.
		In radiocommunication, I is the ITU prefix allocated to Italy.
	www.brainyencyclopedia.com /encyclopedia/i/ii/i.html (353 words)

	Complex number
		(See imaginary number for a discussion of the "reality" of complex numbers.) The 18th century saw the labors of Abraham de Moivre and Leonhard Euler.
		This generalization is largely due to Kummer, to whom is also due the theory of ideal numbers, which has recently been simplified by Felix Klein (1893) from the point of view of geometry.
		A further complex theory is due to Evariste Galois, the basis being the imaginary roots of an irreducible congruence,
	www.knowledgefun.com /book/c/co/complex_number.html (2267 words)

	HSE Complex number - Wikibooks, collection of open-content textbooks
		It is interesting delve a little further into the reason why this imaginary number was introduced in the first place - it turns out that there was a valid reason why mathematicians realized that such a construct was useful, and could provide deeper insight.
		This was in fact the motivation for considering imaginary numbers, and opened up a fascinating area of mathematics.
		This is true of every complex number; for example, the number 5 has real part 5 and imaginary part 0, while the number 7i has real part 0 and imaginary part 7.
	en.wikibooks.org /wiki/HSE_Complex_number (1930 words)

	<complex.h>
		The choice of I instead of i for the imaginary unit concedes to the widespread use of the identifier i for other purposes.
		The corresponding real type for the imaginary unit is float, so that use of I for algorithmic or notational convenience will not result in widening types.
		On systems with imaginary types, the application has the ability to control whether use of the macro I introduces an imaginary type, by explicitly defining I to be _Imaginary_I or _Complex_I.
	www.opengroup.org /onlinepubs/009695399/basedefs/complex.h.html (409 words)

	Imaginary planet - Uncyclopedia
		These have the form (a * b + c / d ^ e) * i, where i is the imaginary unit.
		For instance, potato chips actually turn out to be imaginary planets.
		Oprah has been said to be an imaginary planet by some, but they were just under the powerful influence of Pop Rocks.
	uncyclopedia.org /wiki/Imaginary_planet (128 words)

	Alternatura - IT, Computing and Sciences Creative Technology
		It is a transformation of the imaginary (or irrational numbers) into the same numeric field of the imaginary numbers.
		Each imaginary number is represented by a pair of values that corresponds to the Cartesian coordinates of the corresponding point in the imaginary plane.
		Since this the point (x, y) in the plane corresponds to the imaginary number x+iy with i the imaginary unit.
	www.alternatura.com /main_equation.htm (342 words)

	All Elementary Mathematics - Study Guide - Algebra - Imaginary and complex numbers...
		So, a number is imaginary, if its second power is a negative number.
		In contrast to imaginary numbers all the rest numbers (positive and negative, integers and fractional, rational and irrational ones) are called real numbers.
		A sum of a real and an imaginary number is called a complex number, and marked as:
	www.bymath.com /studyguide/alg/sec/alg20.html (255 words)

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