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Sir Roger Penrose has quantified the "very special" nature of the big bang as being "1 part in 10 to the 10 to the 243", but I do not understand what this really means - my apologies but I cannot even write the figure properly, either using notation, as my phone only copes with 1 line of superscript, or by employing solely regular numerals, as apparently even if I could fit each zero onto it's own proton there would still not be enough protons in the observable universe for me to complete it, (or have I misconstrued Sir Roger here? Presumably he at least understands the basics of scientific notation!)

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  • $\begingroup$ Is your question what does something like '10 to the 10 to the 243' mean? $\endgroup$ Commented Jul 15, 2021 at 13:48
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    $\begingroup$ MathJax: $10^{10^{243}}$ yields $10^{10^{243}}$. $\endgroup$ Commented Jul 15, 2021 at 13:50
  • $\begingroup$ Yeah, that's not scientific notation. Just a power tower. $\endgroup$
    – DKNguyen
    Commented Jul 15, 2021 at 13:51
  • $\begingroup$ Indeed. You can interpret the number as '1 with $10^{243}$' zeros behind it, where $10^{243}$ is 1 with 243 zeros behind it. It's a very big number. E.g. even $10^{10^2}$ is 1,000,000,...[94 more zeros]. $\endgroup$ Commented Jul 15, 2021 at 13:55

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You can interpret the number you mentioned, $10^{10^{243}}$, as '1 with $10^{243}$' zeros behind it, where $10^{243}$ is 1 with 243 zeros behind it. It's a very big number. E.g. even $10^{10^2}$ is 1,000,000,...[with 94 more zeros].

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  • $\begingroup$ Hi Ben, thanks for answering my question. I'm sorry that I'm a bit late getting back to you, I just had to duck-out... you know, for a few years! I think I've almost wrapped my head around it, but something still isn't quite meshing for me yet. If 10³ is 1,000, then wouldn't that be the same as 10¹⁰2? With 10² being 100, and then multiplying that by 10? Sorry, I'm sure that I was born with a natural number deficiency or something! $\endgroup$ Commented Apr 4 at 10:42
  • $\begingroup$ NB. The number is Roger Penrose's calculation for the odds that our universe just happened to have been born in the low-entropy state that it was - such that complexity, and ultimately ourselves, could exist within it. $\endgroup$ Commented Apr 4 at 10:48
  • $\begingroup$ That's an important number! For your example, think of $10^3$ as $10\times10\times10$ (Three 10s, all multiplied together). Then think of $10^{10^2}$. Since $10^2 = 100$ then $10^{10^2} = 10^{100} = 10\times10\times10\times10\times ...$. (One hundred 10s, all multiplied together). This second number is much larger than $10^3 = 1000$. In your reasoning for this same example, it looks like you want to multiply instead of raise to the power of. E.g. $10^{10^2}$ is not $10\times 10^2$, as I think you claimed. $\endgroup$ Commented Apr 5 at 15:14

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