Rasheed-Ahmad-Rasheed-Soomro-For-f-x-where-x-and-f-x-both-are-real-the-x-f-x-can-be-plotted-as-a-point-easily-Now-consider-F-X-where-X-and-F-X-are-complex-numbers-How-can-X-F-X-be-plo

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Question Number 1239 by Rasheed Ahmad last updated on 17/Jul/15

$$RasheedAhmad\left(RasheedSoomro\right)$$$$\bullet \mathrm{For}\mathrm{f}\left(\mathrm{x}\right)\mathrm{where}\mathrm{x}\mathrm{and}\mathrm{f}\left(\mathrm{x}\right)\mathrm{both}are$$$$\mathrm{real}\mathrm{the}(\mathrm{x},f\left(x\right))canbeplottedas$$$$apointeasily.\bullet NowconsiderF\left(X\right)$$$$whereXandF\left(X\right)arecomplex$$$$numbers.Howcan(X,F\left(X\right))be$$$$plotted?Foraparticularexample:(3+2i,4-5i)$$$$howcanbeplotted?$$$$$$$$$$

Commented by 123456 last updated on 18/Jul/15

$$\mathrm{its}\mathrm{need}\mathrm{a}{\mathbb{R}}^4\mathrm{graph}\mathrm{to}\mathrm{proot}\mathrm{it},\mathrm{however}$$$$\mathrm{we}\mathrm{can}\mathrm{plot}\mathrm{it}\mathrm{at}{\mathbb{R}}^3\mathrm{by}\mathrm{making}\mathrm{this}$$$$\mathrm{lets}f\left(z\right):\mathbb{C}\to \mathbb{C}\mathrm{then}$$$$\mathrm{the}x\mathrm{axis}\mathrm{is}\mathrm{real}\mathrm{part}\mathrm{of}z$$$$\mathrm{the}y\mathrm{axis}\mathrm{is}\mathrm{imaginary}\mathrm{art}\mathrm{of}z$$$$\mathrm{and}\mathrm{we}\mathrm{can}\mathrm{prot}f\left(z\right)as\mathrm{two}\mathrm{fuction}\mathrm{like}$$$$a\left(z\right)=\mathrm{\Re }\left(f\left(z\right)\right)\mathrm{\Re }\mathrm{is}\mathrm{real}\mathrm{part}$$$$b\left(z\right)=\mathrm{\Im }\left(f\left(z\right)\right)\mathrm{\Im }\mathrm{is}\mathrm{imaginary}\mathrm{part}$$$$x=\mathrm{\Re }\left(z\right)$$$$y=\mathrm{\Im }\left(z\right)$$$$\mathrm{we}\mathrm{are}\mathrm{taking}\mathrm{it}\mathrm{like}\mathrm{ploting}\mathrm{two}\mathrm{function}$$$$f\left(z\right)=u(x,y)+iv(x,y)$$$$a\left(z\right)=u(x,y)$$$$b\left(z\right)=v(x,y)$$

Commented by Rasheed Soomro last updated on 18/Jul/15

$$Soitmeans,(x_1+{iy}_{1,},x_2+{iy}_2)cannotberepresentedasa$$$$singleentityingeometrywhileitisasingleentityinalgebra.$$$$Actuallythisisaspecialcaseoforderedpair.Itcanbe$$$$writtenas((x_1,y_1),(x_2,y_2))i-eanorderedpairofordered$$$$pairs.\mathbf{An}\mathbf{ordered}\mathbf{pair},\mathbf{whose}\mathbf{coordinates}\mathbf{are}\mathbf{themselves}$$$$\mathbf{ordered}\mathbf{pairs}.\mathit{A}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{l}\mathit{e}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{t}\mathit{y}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathit{g}\mathit{e}\mathit{b}\mathit{r}\mathit{a}.Butthereis$$$$nowaytorepresentitas\mathit{A}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{l}\mathit{e}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{t}\mathit{y}\mathit{i}\mathit{n}\mathit{g}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{y}!!!$$

Commented by 123456 last updated on 18/Jul/15

$$\mathrm{you}\mathrm{can}\mathrm{use}\mathrm{the}\mathrm{idea}\mathrm{of}\mathrm{vector}\mathrm{fields}$$$$\mathrm{since}\mathrm{complex}\mathrm{numbers}\mathrm{are}\mathrm{similiar}$$$$\mathrm{to}\mathrm{it}$$$$\mathrm{then}\mathrm{you}\mathrm{can}\mathrm{plot}\mathrm{it}\mathrm{in}{\mathbb{R}}^2$$$$x=\mathrm{\Re }\left(z\right)\mathrm{real}\mathrm{part}$$$$y=\mathrm{\Im }\left(z\right)\mathrm{im}\mathrm{part}$$$$\mathrm{then}\mathrm{every}\mathrm{point}\mathrm{of}\mathrm{it}\mathrm{has}\mathrm{a}\mathrm{vector}$$$$f\left(z\right)=u\left(z\right)+iv\left(z\right)$$$$u(x,y)+iv(x,y)$$$$\overrightarrow{u}\left(z\right)=u\left(z\right){\overrightarrow{e}}_x+v\left(z\right){\overrightarrow{e}}_y$$$$\overrightarrow{u}(x,y)u(x,y){\overrightarrow{e}}_x+v(x,y){\overrightarrow{e}}_y$$

Commented by Rasheed Soomro last updated on 18/Jul/15

$$ThanksSir\left(123456\right)$$

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