Let-A-denotes-the-Set-of-Algebraic-Numbers-and-T-the-Set-of-Trancedental-Numbers-Discuss-the-following-Are-A-and-T-closed-with-respect-to-addition-and-multiplication-Are-A-0-and-T-clos

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Question Number 4027 by Rasheed Soomro last updated on 27/Dec/15

$$Let\mathbb{A}denotestheSetofAlgebraicNumbers$$$$and\mathbb{T}theSetofTrancedentalNumbers.$$$$Discussthefollowing:$$$$\bullet Are\mathbb{A}and\mathbb{T}\mathit{c}\mathit{l}\mathit{o}\mathit{s}\mathit{e}\mathit{d}withrespectto$$$$\mathit{a}\mathit{d}\mathit{d}\mathit{i}\mathit{t}\mathit{i}\mathit{o}\mathit{n}and\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{p}\mathit{l}\mathit{i}\mathit{c}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}?$$$$\bullet Are\mathbb{A}-\left\{0\right)and\mathbb{T}\mathit{c}\mathit{l}\mathit{o}\mathit{s}\mathit{e}\mathit{d}withrespectto$$$$\mathit{d}\mathit{i}\mathit{v}\mathit{i}\mathit{s}\mathit{i}\mathit{o}\mathit{n}?$$

Commented by Yozzii last updated on 27/Dec/15

$$Algebraicnumbersaresolutions$$$$tofinite,non-zeropolynomialswhose$$$$coefficientsarerationalnumbers$$$$andareinonevariable.$$$$Thus,x\in \mathbb{A}\Rightarrow x\in \mathbb{Q}orx\in \mathbb{C}-\left\{transendentalcomplexnumbers\right\}$$$$x\in \mathbb{T}\Rightarrow x\notin \mathbb{A}.$$$$Notethat\mathrm{\exists }x\in \mathbb{A}whichareirrational$$$$e.gthesolutionto{x}^2-2=0.$$$$Alsox\in \mathbb{A}couldbeatrigonometric$$$$functionappliedtoarationalmultiple$$$$of\pi .$$$$e.gcos3r=4{cos}^{3}r-3cosr.$$$$letp=cosrandcos3r=0\Rightarrow 3r=2n\pi \pm \frac{\pi }{2}(n\in \mathbb{Z})$$$$r=\frac{1}{3}(2n\pi \pm \frac{\pi }{2})$$$$\therefore 4p^3-3p=0hassolutions$$$$p=cos\left(\frac{1}{3}(2n\pi \pm \frac{\pi }{2})\right)(n=0,1)$$$$$$$$Underthebinaryoperation\ast for$$$$addition,closureof\mathbb{A}isprovided$$$$since\mathbb{A}\subset \mathbb{C}exhibitsclosure.$$$$\mathbb{T}isnotclosedunder\ast since,for$$$$example,-\pi ,\pi \in \mathbb{T}\Rightarrow \left(\pi \right)+(-\pi )=0\notin \mathbb{T}.$$$$$$$$Define\oplus asthebinaryoperationof$$$$multiplication.\mathbb{A}doesexhibitclosure$$$$byvirtueof\mathbb{C}exhbitingclosure$$$$under\oplus and\mathbb{A}\subset \mathbb{C}.\mathbb{T}isnotclosed$$$$undermultiplicationsince,for2\pi ,\frac{1}{\pi }\in \mathbb{T},$$$$2\pi \oplus \frac{1}{\pi }=2\notin \mathbb{T}.$$$$$$$$$$$$$$

Commented by Rasheed Soomro last updated on 27/Dec/15

$${\mathcal{EX}}_{\mathcal{PLANATION}}^{\mathcal{CELLENT}}!!!$$

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