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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><b>From:</b> extropy-chat [mailto:extropy-chat-bounces@lists.extropy.org] <b>On Behalf Of </b>John Clark<br><br><o:p></o:p></p><div><div><div><div><p class=MsoNormal><o:p> </o:p></p></div><div><div><p class=MsoNormal><span style='font-family:"Arial",sans-serif'>>…</span><span style='font-size:13.5pt;font-family:"Arial",sans-serif'>I know this is a bit heretical but perhaps irrational numbers really do have a last digit. If the computational resources of the entire universe is insufficient to calculate the 10^100^100^100 digit of PI, and given that there are only about 10^81</span><span style='font-family:"Arial",sans-serif'> </span><span style='font-size:13.5pt;font-family:"Arial",sans-serif'>atoms in the observable universe that seems like a reasonable assumption, could the </span><span style='font-family:"Arial",sans-serif'><o:p></o:p></span></p></div><p class=MsoNormal><span style='font-size:13.5pt;font-family:"Arial",sans-serif'>10^100^100^100 digit of PI even be said to exist? John K Clark</span><span style='font-family:"Arial",sans-serif'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:13.5pt;font-family:"Arial",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Sure can John. We don’t even need to resort to the definition of a rational number to show that pi is irrational, with a definition: if there is a last digit then you can write it as a fraction, which makes it rational.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>The Taylor series expansion of pi doesn’t know or care how many atoms there are:<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><img width=367 height=59 style='width:3.8229in;height:.6145in' id="Picture_x0020_3" src="cid:image001.jpg@01D2FA34.97B52440"><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>We could even estimate the number of terms we need to get to your 10^100^100^100<sup>th</sup> digit, but we don’t need to for we can already see how the Taylor series is at least 17 different kinds of cool, with wicked being one of them. <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Wouldn’t it have been cool to have been the guy who discovered that expansion? Imagine it: Hey cool, check this guys! This thing just goes on and on, each fraction getting smaller! It goes on… forever! We can’t even possibly write pi as a fraction! <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Oh for a thousand lifetimes to be able to dig deeper and wider.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>spike<o:p></o:p></p></div><div><p class=MsoNormal><span style='font-family:"Arial",sans-serif'><o:p> </o:p></span></p><p class=MsoNormal><o:p> </o:p></p></div></div></div></div></div></body></html>