2D ویکٹروں اور پیچیدہ تعداد میں بنیادی فرق کیا ہے؟


جواب 1:

اصلی ویکٹر کی جگہوں کے طور پر ، کوئی خاص فرق نہیں ہے۔

لیکن پیچیدہ اعداد ، اصل اعداد کے ذریعہ ضرب لگانے کے علاوہ ، i کے ذریعہ ضرب بھی رکھتے ہیں۔ یہی بنیادی فرق ہے۔ پیچیدہ اعداد میں ایک اصلی الجبرا ڈھانچہ ہوتا ہے جس کی وضاحت دو بنیادی عناصر 1 اور i کے ذریعہ ہوتی ہے ، ضرب 1 1 = 1 ، 1 i = i 1 = i ، اور i i = -1 کے ساتھ۔ عام 2d ویکٹر کی جگہ کسی خاص الجبرا ڈھانچے کے ساتھ نہیں آتی ہے۔


جواب 2:

ThekeydifferenceisthatCisafield,whereas[math]R2[/math]isjustavectorspace.The key difference is that \mathbb{C} is a field, whereas [math]\mathbb{R^2}[/math] is just a vector space.

C\mathbb{C}

Thisisusefulforalotofthings.Forexample,ifyouwantedtodefineafunctionthatrepresentedarotationof90degreescounterclockwiseinthe2Dplane,youcouldmultiplyeach2Dvectorbythematrix(0110),oryoucouldjustmultiplyeachcomplexnumberby[math]i[/math].Beingabletodividecomplexnumbersalsoallowsustodefinecomplexderivatives,whichareoneofthemainpointsofconcernforcomplexanalysis.This is useful for a lot of things. For example, if you wanted to define a function that represented a rotation of 90 degrees counterclockwise in the 2D plane, you could multiply each 2D vector by the matrix \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}, or you could just multiply each complex number by [math]i[/math]. Being able to divide complex numbers also allows us to define complex derivatives, which are one of the main points of concern for complex analysis.

Inallotheraspects,particularlygeometricortopologicproperties,Cand[math]R2[/math]areequivalent.Youcanmapeachandeverycomplexnumbertoauniquevectorin2Dspaceandviceversa.In all other aspects, particularly geometric or topologic properties, \mathbb{C} and [math]\mathbb{R^2}[/math] are equivalent. You can map each and every complex number to a unique vector in 2D space and vice versa.

Infact,alotoftimesincomplexanalysisyouendupwritingcomplexvaluedfunctionsintermsoffunctionsin2Dspace.Givensomefunctionf:CC,youcanalwayswrite[math]f(z)=u(x,y)+iv(x,y)[/math],where[math]x[/math]and[math]y[/math]aretherealandimaginarypartsofzrespectively,andboth[math]u[/math]and[math]v[/math]arefunctionsfrom[math]R2[/math]to[math]R[/math].Knowinghowtodealwiththisbackandforthbetweencomplexnumbersand2Dspaceisveryuseful,sinceitallowsustowritepropertiesofonespaceintermsofpropertiesoftheother(SeeCauchyRiemannequations)In fact, a lot of times in complex analysis you end up writing complex valued functions in terms of functions in 2D space. Given some function f:\mathbb{C}\rightarrow\mathbb{C}, you can always write [math]f(z)=u(x,y)+iv(x,y)[/math], where [math]x[/math] and [math]y[/math] are the real and imaginary parts of z respectively, and both [math]u[/math] and [math]v[/math] are functions from [math]\mathbb{R^2}[/math] to [math]\mathbb{R}[/math]. Knowing how to deal with this back and forth between complex numbers and 2D space is very useful, since it allows us to write properties of one space in terms of properties of the other (See Cauchy-Riemann equations)

ItsalsoimportanttonotethatnothingisstoppingyoufromdefiningamutliplicationoperationinR2thatsequivalenttocomplexnumbermultiplication:It’s also important to note that nothing is stopping you from defining a mutliplication operation in \R^2 that’s equivalent to complex number multiplication:

(ab)(cd)=(acbdad+bc)\begin{pmatrix} a \\ b \end{pmatrix}\cdot\begin{pmatrix} c\\d \end{pmatrix} =\begin{pmatrix} ac-bd\\ ad+bc\end{pmatrix}

Infactthisisaveryrigorouswayofdefiningcomplexnumbersthatyoumayfindinsomealgebratextbooks,thatcompletelycircumventstakingsquarerootsofnegativenumbers.YousimplydefineCas[math]R2[/math]giftedofthisproduct,andsay[math]i=(01)[/math].In fact this is a very rigorous way of defining complex numbers that you may find in some algebra textbooks, that completely circumvents taking square roots of negative numbers. You simply define \mathbb{C} as [math]\R^2[/math] gifted of this product, and say [math]i=\begin{pmatrix} 0 \\ 1 \end{pmatrix}[/math].

C\mathbb{C}

andR2areequivalent.Youcanmapeachandeverycomplexnumbertoauniquevectorin2Dspaceandviceversa. and \mathbb{R^2} are equivalent. You can map each and every complex number to a unique vector in 2D space and vice versa.

درحقیقت ، پیچیدہ تجزیہ میں آپ 2D جگہ میں افعال کی شرائط میں پیچیدہ قیمتی افعال تحریر کرتے ہیں۔ کچھ فنکشن دیا

f:CCf:\mathbb{C}\rightarrow\mathbb{C}

,youcanalwayswritef(z)=u(x,y)+iv(x,y),where[math]x[/math]and[math]y[/math]aretherealandimaginarypartsofzrespectively,andboth[math]u[/math]and[math]v[/math]arefunctionsfrom[math]R2[/math]to[math]R[/math].Knowinghowtodealwiththisbackandforthbetweencomplexnumbersand2Dspaceisveryuseful,sinceitallowsustowritepropertiesofonespaceintermsofpropertiesoftheother(SeeCauchyRiemannequations), you can always write f(z)=u(x,y)+iv(x,y), where [math]x[/math] and [math]y[/math] are the real and imaginary parts of z respectively, and both [math]u[/math] and [math]v[/math] are functions from [math]\mathbb{R^2}[/math] to [math]\mathbb{R}[/math]. Knowing how to deal with this back and forth between complex numbers and 2D space is very useful, since it allows us to write properties of one space in terms of properties of the other (See Cauchy-Riemann equations)

یہ بھی نوٹ کرنا ضروری ہے کہ آپ کو کسی بھی طرح کے عمل کو بیان کرنے سے نہیں روک رہا ہے

R2\R^2

 جو پیچیدہ نمبر ضرب کے مترادف ہے:

(ab)(cd)=(acbdad+bc)\begin{pmatrix} a \\ b \end{pmatrix}\cdot\begin{pmatrix} c\\d \end{pmatrix} =\begin{pmatrix} ac-bd\\ ad+bc\end{pmatrix}

حقیقت میں یہ پیچیدہ اعداد کی وضاحت کرنے کا ایک بہت ہی سخت طریقہ ہے جو آپ کو کچھ الجبرا کی نصابی کتب میں مل سکتا ہے ، جو منفی اعداد کی مربع جڑوں کو مکمل طور پر ختم کرتے ہیں۔ آپ صرف وضاحت کریں

C\mathbb{C}

asR2giftedofthisproduct,andsay[math]i=(01)[/math]. as \R^2 gifted of this product, and say [math]i=\begin{pmatrix} 0 \\ 1 \end{pmatrix}[/math].