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Question Number 77573 by MJS last updated on 08/Jan/20

xln x =((143851)/(40000)) solve for x ⇒ nice surprise

$${x}\mathrm{ln}\:{x}\:=\frac{\mathrm{143851}}{\mathrm{40000}} \\ $$$$\mathrm{solve}\:\mathrm{for}\:{x}\:\Rightarrow\:\mathrm{nice}\:\mathrm{surprise} \\ $$

Commented by Tony Lin last updated on 08/Jan/20

lnx=((143851)/(40000))x^(−1) e^(((143851)/(40000))x^(−1) ) =x x^(−1) e^(((143851)/(40000))x^(−1) ) =1 ((143851)/(40000))x^(−1) e^(((143851)/(40000))x^(−1) ) =((143851)/(40000)) ((143851)/(40000))x^(−1) =W(((143851)/(40000))) x=(((143851)/(40000))/(W(((143851)/(40000)))))≒((3.596275)/(1.14473))=3.141592 ≒π

$${lnx}=\frac{\mathrm{143851}}{\mathrm{40000}}{x}^{−\mathrm{1}} \\ $$$${e}^{\frac{\mathrm{143851}}{\mathrm{40000}}{x}^{−\mathrm{1}} } ={x} \\ $$$${x}^{−\mathrm{1}} {e}^{\frac{\mathrm{143851}}{\mathrm{40000}}{x}^{−\mathrm{1}} } =\mathrm{1}\: \\ $$$$\frac{\mathrm{143851}}{\mathrm{40000}}{x}^{−\mathrm{1}} {e}^{\frac{\mathrm{143851}}{\mathrm{40000}}{x}^{−\mathrm{1}} } =\frac{\mathrm{143851}}{\mathrm{40000}} \\ $$$$\frac{\mathrm{143851}}{\mathrm{40000}}{x}^{−\mathrm{1}} =\mathbb{W}\left(\frac{\mathrm{143851}}{\mathrm{40000}}\right) \\ $$$${x}=\frac{\frac{\mathrm{143851}}{\mathrm{40000}}}{\mathbb{W}\left(\frac{\mathrm{143851}}{\mathrm{40000}}\right)}\fallingdotseq\frac{\mathrm{3}.\mathrm{596275}}{\mathrm{1}.\mathrm{14473}}=\mathrm{3}.\mathrm{141592} \\ $$$$\fallingdotseq\pi \\ $$

Commented by MJS last updated on 08/Jan/20

yes. somebody tried to prove to me that π was somehow rational with these “equations” p=πln π =((143851)/(40000)) q=πln (πln π) =((100523)/(25000)) but he used a calculator with only 10 digits p=3.5962749997...≈3.596275 q=4.0209199997...≈4.02092 near enough is not good enough, snyway I I found it interesting

$$\mathrm{yes}.\:\mathrm{somebody}\:\mathrm{tried}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{to}\:\mathrm{me}\:\mathrm{that}\:\pi \\ $$$$\mathrm{was}\:\mathrm{somehow}\:\mathrm{rational}\:\mathrm{with}\:\mathrm{these}\:``\mathrm{equations}'' \\ $$$${p}=\pi\mathrm{ln}\:\pi\:=\frac{\mathrm{143851}}{\mathrm{40000}} \\ $$$${q}=\pi\mathrm{ln}\:\left(\pi\mathrm{ln}\:\pi\right)\:=\frac{\mathrm{100523}}{\mathrm{25000}} \\ $$$$\mathrm{but}\:\mathrm{he}\:\mathrm{used}\:\mathrm{a}\:\mathrm{calculator}\:\mathrm{with}\:\mathrm{only}\:\mathrm{10}\:\mathrm{digits} \\ $$$${p}=\mathrm{3}.\mathrm{5962749997}...\approx\mathrm{3}.\mathrm{596275} \\ $$$${q}=\mathrm{4}.\mathrm{0209199997}...\approx\mathrm{4}.\mathrm{02092} \\ $$$$\mathrm{near}\:\mathrm{enough}\:\mathrm{is}\:\mathrm{not}\:\mathrm{good}\:\mathrm{enough},\:\mathrm{snyway}\:\mathrm{I} \\ $$$$\mathrm{I}\:\mathrm{found}\:\mathrm{it}\:\mathrm{interesting} \\ $$

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