# How does nested powers work?

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!)

• Is your question what does something like '10 to the 10 to the 243' mean? Commented Jul 15, 2021 at 13:48
• MathJax: $10^{10^{243}}$ yields $10^{10^{243}}$. Commented Jul 15, 2021 at 13:50
• Yeah, that's not scientific notation. Just a power tower. Commented Jul 15, 2021 at 13:51
• 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]. Commented Jul 15, 2021 at 13:55

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].
• 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. Commented Apr 5 at 15:14