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- ...addition]], subtraction, [[multiplication]], and division can be used with complex numbers. They also follow [[commutative property|commutative]], [[associati Complex numbers were invented to answer special equations that have [[exponent]]s i ...6 KB (917 words) - 00:53, 5 February 2017
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- ...vre's formula''', named after [[Abraham de Moivre]], states that for any [[complex number]] ''z'' and [[integer]] ''n'', ..., with <math>|z|e^{i.\arg z}</math> often being called the polar form of a complex number. ...985 bytes (158 words) - 09:05, 20 June 2017
- ...with [[Multiplicity (mathematics)|multiplicity]] greater than one) in the complex numbers. If the discriminant is smaller than zero, then there are two distinct [[complex number]]s. ...918 bytes (140 words) - 23:26, 12 July 2017
- ...addition]], subtraction, [[multiplication]], and division can be used with complex numbers. They also follow [[commutative property|commutative]], [[associati Complex numbers were invented to answer special equations that have [[exponent]]s i ...6 KB (917 words) - 00:53, 5 February 2017
- ...rong colors are for values close to zero and [[hue]] encodes the value's [[complex number|argument]]. [[File:Complex coloring.jpg|thumb|300px|upright=1.2|The coloring of the complex function-values used above: positive real values are presented in red.]] ...3 KB (534 words) - 10:42, 20 June 2017
- ...f(x)</math> with [[real]] or [[complex number|complex]] coefficients has a complex root. A root is a number ''x'' which gives a solution <math>f(x)=0</math>. * [[Complex analysis]] ...3 KB (574 words) - 23:27, 15 January 2016
- ===A more complex example=== ...2 KB (379 words) - 14:57, 30 April 2017
- ...ric domains supported often include [[real number|real]], [[complex number|complex]], [[interval (mathematics)|interval]], [[rational number|rational]], and a ...asily calculate the total possible error of an inexact arithmetic system. Complex number arithmetic is generally supported by allowing the [[imaginary unit]] ...3 KB (424 words) - 02:38, 27 March 2013
- ...=Thumbnail-of-Riemann Zeta.jpg.png|thumb|The Riemann zeta function, in the complex plane. The real part <math>\operatorname{Re}(s)</math> of the number is dra ...ction]], the [[Riemann zeta function]]. This function inputs and outputs [[complex number]] values. The inputs that give the output zero are called [[root (m ...2 KB (337 words) - 10:32, 20 June 2017
- == Complex Exponentiation == A complex number can also be taken to the power of another complex number. The relationship between imaginary numbers and trigonometry, which ...8 KB (1,393 words) - 01:43, 31 May 2017
- ...oid, the final combined result will be the same. This also works with more complex Monoids e.g. a map of word counts. ...729 bytes (116 words) - 19:01, 14 June 2016
- {{complex|date=January 2012}} ...915 bytes (111 words) - 07:20, 3 November 2013
- ...tput. An example of a multivalued function is the [[square root]]. Every complex number (except for 0), has two square roots. Example: <math>\sqrt{4}</math ...881 bytes (123 words) - 22:40, 23 March 2016
- ...number '''0''' is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all <math>a\in\Bbb{R},</math> ...r '''1''' is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all <math>a\in\Bbb{R},</math> ...3 KB (425 words) - 19:31, 17 March 2013
- {{complex|date=January 2012}} ...846 bytes (137 words) - 09:46, 20 October 2014
- ===The real line and complex plane=== ...infty</math>. This is treated as addition to the [[real numbers]] or the [[complex numbers]]. It is the result of division by zero, or to indicate that a [[se ...6 KB (828 words) - 00:07, 11 November 2015
- ...such as the oxidation of sugar in the human body, through a series of very complex [[electron transfer]] processes.</ref> ...3 KB (389 words) - 08:00, 8 January 2015
- | <nowiki>|x|</nowiki> is the distance along the real line (or across the complex plane) between x and zero | [[Complex numbers]] ...8 KB (1,182 words) - 07:31, 3 November 2013
- ...mpedance.svg|200px|thumb|A graphical representation of the [[Complex plane|complex impedance plane]]]] (see the 2nd figure, "complex impedance plane") ...5 KB (850 words) - 06:03, 26 June 2016
- {{complex|date=May 2012}} ...1,002 bytes (151 words) - 20:37, 2 January 2014
- {{complex|date=May 2012}} ...1 KB (172 words) - 07:05, 3 November 2013
- ...[[complex number]]s. Every real number is a complex number, but not every complex number is a real number. ...6 KB (971 words) - 01:36, 21 August 2017
- In a synthesis reaction, two or more simple substances combine to form a more complex substance. ...e is simple hydrogen gas combined with simple oxygen gas to produce a more complex substance, such as water.<ref name="to react or not to react">[http://utahs ...5 KB (701 words) - 12:12, 4 June 2017
- ...e root is a [[complex number]]. For example, square root of '''-4''' is a complex number '''2''[[imaginary unit|i]]''''' :<math display="block">\sqrt{-4} = 2 ...3 KB (526 words) - 04:59, 30 June 2016
- {{complex|date=March 2012}} {{complex|date=March 2012}} ...3 KB (427 words) - 19:05, 27 March 2017
- {{complex|date=April 2012}} ...c energy, although this may not be helpful to those not familiar with more complex physics. ...3 KB (561 words) - 06:11, 16 June 2016
- {{complex|date=April 2012}} ...1 KB (154 words) - 07:17, 12 January 2015
- A [[real number|real]] or [[complex number]] is called a '''transcendental number''' if it cannot be found as a ...1 KB (199 words) - 20:58, 16 March 2013
- ...e given area, since the normal answer for the wave function is usually a [[complex number]]. The wave function concept was first introduced in the legendary [ ...1 KB (187 words) - 14:02, 12 November 2014
- === Complex numbers === ...rt and an ''imaginary'' part. Every type of number written above is also a complex number. ...14 KB (2,057 words) - 01:36, 21 August 2017
- {{complex|date=March 2012}} The Big O Notation is often used in identifying how complex a problem is (also known as the problem's [[complexity class]]). ...4 KB (700 words) - 14:03, 20 September 2015
- {{complex|date=April 2012}} ...1 KB (189 words) - 07:34, 13 September 2016
- {{complex|date=March 2012}} ...1 KB (200 words) - 15:36, 12 March 2013
- {{Complex|date=June 2012}} ...1 KB (199 words) - 07:36, 12 January 2015
- {{complex|date=September 2011}} ...1 KB (222 words) - 13:32, 21 June 2017
- ...he model. Since individual outcomes might be of little practical use, more complex ''events'' are used to characterize groups of outcomes. The collection of a ...2 KB (250 words) - 19:23, 8 November 2016
- {{complex|date=June 2012}} ...1 KB (195 words) - 08:01, 30 March 2017
- <!-- comment out really complex information ...2 KB (231 words) - 00:24, 15 February 2014
- {{complex|date=June 2014}} ...1 KB (200 words) - 00:02, 2 January 2015
- {{complex|date=July 2012}} ...1 KB (198 words) - 04:44, 5 May 2016
- ...e [[fundamental theorem of algebra]], in the form of a generalisation in [[complex numbers]]. ...1 KB (269 words) - 18:04, 4 May 2016
- * some of the roots may be [[complex number]]s ...2 KB (288 words) - 19:05, 9 March 2015
- Wavelet is (complex) function from the [[Hilbert space]] <math>\psi \in L^{2}(\mathbb{R})</math ...2 KB (280 words) - 16:47, 26 March 2017
- ...[[rational numbers]] by including square roots of positive numbers. The [[complex numbers]] are a closure of the real numbers by including square roots of ne ...2 KB (271 words) - 03:33, 15 January 2015
- ...z'')) is an [[wikt:extend|extension]] of the [[factorial]] function to all complex numbers except negative, integers. The argument of the function is shifted ...lex numbers]]. But it is not defined for negative integers and zero. For a complex number whose real part is not a negative integer, the function is defined b ...8 KB (1,133 words) - 09:45, 26 June 2017
- * [[Complex number]] ...2 KB (303 words) - 17:40, 15 February 2016
- {{complex}} ...2 KB (308 words) - 18:29, 1 November 2016
- {{complex|date=October 2011}} ...^\infty {\alpha \choose i} x^i\quad\mbox{ for all } |x| < 1 \mbox{ and all complex } \alpha\!</math> ...7 KB (1,090 words) - 18:10, 19 October 2014
- {{Article issues|complex=July 2013|wikify=April 2009}} ...2 KB (282 words) - 23:05, 31 May 2017
- {{complex}} ...2 KB (249 words) - 16:54, 23 January 2015
- ...ically numbers that cannot be square rooted and get a real result. Every [[complex number]] except 0 has 2 square roots. For example: −1 has two square roots. ...2 KB (290 words) - 09:33, 2 February 2016
- * <math>*</math> symbol denotes [[complex conjugate]]. ...2 KB (317 words) - 16:57, 23 January 2015
- {{complex|date=November 2012}} ...2 KB (376 words) - 23:39, 8 March 2013
- *[[Dielectric complex reluctance]] *[[Magnetic complex reluctance]] ...5 KB (744 words) - 13:27, 25 September 2014
- {{complex|date=November 2011}} ...2 KB (309 words) - 00:58, 2 January 2015
- {{complex|date=April 2012}} ...2 KB (386 words) - 03:46, 18 October 2014
- {{Complex}} ...2 KB (431 words) - 21:21, 22 August 2017
- ...ave no physical meaning (in fact, they are mathematically [[complex number|complex]]), yet the wave function contains all information that can be known about ...7 KB (1,105 words) - 21:06, 26 June 2017
- ...frequency]], or picking out a signal at a particular frequency from a more complex signal. An ideal LC circuit does not have [[resistance]]. ...2 KB (395 words) - 17:40, 17 December 2015
- ...[[diophantine equation]]. One may also look for solutions in the field of complex numbers. One can also look for solutions in real numbers. ...3 KB (410 words) - 06:25, 6 July 2016
- {{complex|date=February 2012}} ...he discrete group of all <math> p</math>-power [[roots of unity]] in the [[complex number]]s. ...5 KB (835 words) - 22:27, 8 March 2013
- ...The function ''ƒ'' can be either [[real number|real]] or [[complex number|complex]]. The Taylor series is then used to describe what the function looks like ...7 KB (1,081 words) - 10:05, 24 June 2017
- {{complex|date=November 2011}} ...3 KB (414 words) - 06:04, 2 September 2015
- ...or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. ([[Up to]] adding integer multiples of 2π''i'' to t ...7 KB (1,062 words) - 18:16, 9 March 2015
- ...that show why they are the same number, but many of these proofs are very complex.<ref name="Katz">{{cite journal |last1=Katz |first1=K. |last2=Katz |first2= ...3 KB (505 words) - 16:43, 25 May 2017
- {{complex|date=Oct 2016}} ...4 KB (515 words) - 18:40, 2 October 2016
- {{complex|date=April 2012}} ...3 KB (499 words) - 22:30, 24 April 2017
- {{complex|date=April 2012}} ...3 KB (474 words) - 09:45, 20 October 2014
- {{complex|date=June 2012}} ...3 KB (574 words) - 14:25, 26 October 2015
- .... For instance the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in t ...<span style="text-decoration: overline">''b<sub>i</sub>''</span> is the [[complex conjugate]] of ''b<sub>i</sub>''. Then the scalar product of any vector wit ...17 KB (2,689 words) - 00:42, 2 June 2016
- ...as ''3i'' to create ''2+3i''. These kinds of mixed numbers are known as [[complex number]]s. ...3 KB (567 words) - 22:50, 3 February 2016
- hard an algorithm is called ''complexity''. When we ask how complex an algorithm is, often we want to know how long it will take a computer to ...he steps again many times. One would say that sorting by numbers is more ''complex''. ...10 KB (1,837 words) - 21:56, 31 August 2017
- {{complex|date=January 2012}} ...4 KB (495 words) - 19:17, 27 May 2017
- For [[Complex (chemistry)|coordination complexes]] containing [[transition metal]]s, the ...3 KB (511 words) - 00:27, 12 August 2017
- * the [[complex number]]s <math>\mathbb{C}</math> ...4 KB (740 words) - 01:36, 21 August 2017
- ...0 light years distant in the [[Rho Ophiuchi cloud complex|ρ Ophiuchi cloud complex]].<ref>{{cite book| author=Wilking B A.; Gagné M. & Allen L E. 2008. Star f ...6 KB (950 words) - 11:37, 2 November 2016
- {{complex|date=December 2011}} ...3 KB (510 words) - 06:32, 17 October 2014
- {{complex|date=October 2011}} ...4 KB (570 words) - 21:33, 20 October 2014
- ...del]], or [[computer]] [[simulation]] of a relatively complex system. For complex systems, there are many equations and many variables, not just two or three ...10 KB (1,386 words) - 15:51, 19 June 2016
- For more complex signals, an oscilloscope can be used the measured the DC and/or AC voltage, ...4 KB (668 words) - 21:23, 2 March 2016
- {{complex|date=November 2011}} ...3 KB (462 words) - 22:41, 18 October 2014
- {{complex|date=April 2012}} ...4 KB (633 words) - 05:47, 14 February 2016
- {{complex|date=February 2012}} ...4 KB (642 words) - 19:51, 24 November 2016
- {{complex|date=December 2011}} ...4 KB (677 words) - 11:57, 25 July 2016
- {{complex|date=April 2012}} ...4 KB (741 words) - 19:32, 22 July 2016
- {{complex|date=May 2013}} ...5 KB (707 words) - 11:24, 18 May 2013
- {{complex|date=April 2012}} ...5 KB (766 words) - 22:24, 8 February 2015
- {{complex|date=September 2011}} ...5 KB (705 words) - 06:46, 13 June 2017
- {{complex|date=June 2012}} ...4 KB (527 words) - 00:40, 9 March 2013
- {{complex|date=April 2012}} ...5 KB (736 words) - 02:59, 18 October 2014
- ** C - [[Complex number]] |style="text-align:center;"|'''''[[Complex number|C]]''''' ...41 KB (4,624 words) - 01:45, 25 December 2015
- ...n <math>{\rm F}</math> does not have to be ''invertible'', and can be very complex. ...5 KB (757 words) - 08:22, 4 April 2017
- {{complex|date=May 2013}} ...5 KB (784 words) - 20:07, 20 May 2016
- {{complex|date=June 2012}} ...} F^{cd}</math>. This changes the sign of some of ''F'''s components; more complex metric dualities can be seen in [[general relativity]]. ...15 KB (2,355 words) - 23:13, 6 September 2016
- Chess is very complex. It is better to think of easier games. Nim, for example, is much simpler ...6 KB (1,097 words) - 23:35, 19 September 2015
- {{complex|date=April 2012}} ...multiplication. For example, instead of [[real numbers]], scalars may be [[complex numbers]]; instead of arrows, vectors may be [[function (mathematics)|func ...13 KB (1,723 words) - 07:13, 11 November 2016
- {{complex|date=December 2011}} ...7 KB (1,136 words) - 16:27, 4 November 2016
- {{complex|date=December 2011}} ...9 KB (1,483 words) - 20:40, 29 October 2016
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...tp://en.wikipedia.org/wiki/Abstract_simplicial_complex abstract simplicial complex]), <math>\mathcal{F}_k</math> is formed by the shadow of <math>\mathcal{F}_ ...25 KB (4,480 words) - 08:23, 16 August 2011
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...tp://en.wikipedia.org/wiki/Abstract_simplicial_complex abstract simplicial complex]), <math>\mathcal{F}_k</math> is formed by the shadow of <math>\mathcal{F}_ ...25 KB (4,480 words) - 04:58, 17 November 2010
- {{complex|date=December 2011}} ...8 KB (1,007 words) - 05:56, 15 September 2016
- {{complex|date=July 2012}} ...6 KB (950 words) - 03:43, 21 August 2017
- {{complex|2=The article needs more making-it-easy-to-understand or it doesn't need to ...12 KB (1,234 words) - 02:16, 5 September 2016
- ...math besides [[geometry]], which studies shapes. Some of these areas are [[complex analysis]], [[trigonometry]], and [[Series (mathematics)|series]]. ...8 KB (1,232 words) - 17:17, 31 July 2017
- {{complex|date=June 2017}} ...10 KB (1,551 words) - 10:37, 24 June 2017
- ...number]]s || [[Integer]]s || [[Rational number]]s || [[Real number]]s || [[Complex number]]s ...9 KB (1,088 words) - 18:04, 22 August 2017
- * <math>\mathbb{C}</math>, denoting the set of all [[complex number]]s. ...10 KB (1,884 words) - 16:03, 30 June 2015
- ...ochondria of cells. The large number of reactions involved are exceedingly complex and not described easily. (They include [[cellular respiration]], [[anaerob ...8 KB (1,196 words) - 17:01, 24 May 2017
- {{articleissues|redlinks=September 2011|complex=September 2011|refimprove=September 2011}} ...9 KB (1,269 words) - 01:38, 22 March 2015
- ...thermodynamic characteristics that play an important role in understanding complex processes during the formation of substances; for instance, polymer cross-l ...10 KB (1,538 words) - 19:06, 20 August 2017
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...tp://en.wikipedia.org/wiki/Abstract_simplicial_complex abstract simplicial complex]), <math>\mathcal{F}_k</math> is formed by the shadow of <math>\mathcal{F}_ ...50 KB (8,991 words) - 12:23, 21 May 2023
- ...famous equations, like <tt>f = ma</tt> and <tt>e=mc^2</tt> (although more complex math beyond algebra was needed to come up with that last equation). ...13 KB (2,204 words) - 07:13, 30 July 2017
- ...applications in Computer Science require to efficiently check whether two complex objects are identical, while the objects are presented implicitly (e.g., as ...16 KB (2,886 words) - 07:49, 13 November 2011
- :Taylor approximation of non-vanishing complex-valued graph polynomials. ...21 KB (2,460 words) - 17:28, 7 March 2023
- {{complex|date=December 2011}} ...13 KB (2,026 words) - 03:33, 17 October 2014
- {{complex|date=December 2013}} ...purposes only, whereas in quantum mechanics the wave function is generally complex.]] ...42 KB (7,065 words) - 02:42, 24 August 2017
- ...applications in Computer Science require to efficiently check whether two complex objects are identical, while the objects are presented implicitly (e.g., as ...20 KB (3,617 words) - 01:24, 8 June 2010
- ...mallest things in nature like [[proton]]s, [[neutron]]s and [[electron]]s. Complex [[mathematics]] is used to study subatomic particles and electromagnetic wa ...[[calculus]]. It also requires [[Matrix (mathematics)|matrix algebra]], [[complex number]]s, [[probability theory]], and [[partial differential equation]]s. ...36 KB (5,991 words) - 08:00, 24 August 2017
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...32 KB (5,780 words) - 02:49, 24 November 2016
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...32 KB (5,780 words) - 13:32, 2 December 2017
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...32 KB (5,780 words) - 07:54, 28 November 2019
- ...'''hereditary''' (also called an '''ideal''' or an '''abstract simplicial complex'''), if ...32 KB (5,800 words) - 07:57, 21 May 2014
- {{complex|date=January 2012}} ...19 KB (2,913 words) - 21:36, 4 November 2015
- ...infinite precision operation. However, no such guarantee is given for more complex functions and they are typically only accurate to within the last bit at be ...46 KB (7,060 words) - 01:36, 21 August 2017