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In some areas you can sometimes encounter numbers so big or so small that they are nothing more than just a bunch of symbols that don't say much. In this situation knowing another extreme number somewhat close to what you're reading can be a great tool in really understanding what you're dealing with.

A good example is a lecture I was watching on YouTube about search algorithms for games like chess. After a few calculations the professor figured out that if you were to explore every single route to the end of a typical chess game you'd have about $10^{120}$ of what are usually called "leaf nodes" at the end of the algorithm to reach and analyze.

This number doesn't say much until you start comparing it to other useful numbers like the professor did, if you want to watch him explain it yourself here's the video https://youtu.be/STjW3eH0Cik?t=726.

This is what he did:

  • $10^{80}$ number of atoms in the observable universe
  • $\pi * 10^7$ seconds in a year
  • $10^9$ nanoseconds in a second
  • $10^{10}$ years since the beginning of the universe

$10^{106}$ nanoseconds since the beginning of the universe multiplied by the number of atoms in the universe.

And with this we can easily figure out that if every nanosecond each atom in the universe were to analyze one leaf node since the beginning of time we would still be way short.

Daniel
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  • I wouldn't say "we're way short" in a situation where we'd done just $1/10^{14}$ of the work. I'd call that "practically not even started yet." – David K Apr 18 '17 at 23:01

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Off the top of my head,

Grains of sand on earth $\approx 7.5 \times 10^{18}$

Atoms in 12 grams of $C_{12} \approx 6.022 \times 10^{23}$

awright96
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I think this depends entirely upon the disciplines in which you're interested. For instance:

Economics

  • World gross domestic product (equivalent): $\$108,000,000,000,000$ US

Physics

  • Number of particles in universe: $10^{80}$
  • Planck time: $5.4 \times 10^{-44} s$
  • Planck density: $5.1 × 10^{96}\ kg/m^3$
  • Mass/energy of universe: $4 \times 10^{69} J$
  • Stars in the universe: $7 \times 10^{22}$

Mathematics

  • Size of Monster group: $8 \times 10^{53}$
  • Skewe's number: $e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}$
  • Graham's number, the largest number whose value mattered in a mathematical proof: too big to even write using exponential notation

Biology

  • Cells in human body: $37,200,000,000,000$
David G. Stork
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