The Quest to Ground the Googol, Part I

What’s the biggest number you can visualize? The biggest number you know? The biggest number you can attribute to a physical phenomenon or quantity? The human mind and mathematics offer humanity really beautiful and powerful tools when trying to understand the universe, mainly because the natural world is filled with numbers and concepts that would be incomprehensible without them. The best way to realize this is by asking those three questions. What are your answers?

Even this first question can throw you for a loop. What’s the largest number you can visualize? Is it in the thousands? Hundreds? Think of those carnival games where you see a jar of jelly beans, and you have to guess how many are in there…how much of a struggle is it? With your powers of observation alone, it quickly becomes difficult to imagine large numbers in any physical, meaningful way. To get to these normally unreachable numbers, we need to start using an abstract means of understanding the enormously large (or enormously small). Enter: mathematics.

With scientific notation, we become able to work with and understand large or small numbers without having to visualize them. Scientific notation is aptly used to accomplish the goal of quickly and understandably abstracting numbers for use. When I throw the number 1,000,000,000,000 at you, you might start to struggle with counting all 12 zeroes. But if I say it’s 10^12, it’s more quickly apparently that I mean one trillion. As such, in many scientific calculations (especially astronomy and physics), you have to work with crazy numbers all the time! The mass of the Sun is roughly 10^30 kilograms, the size of an atom is roughly 10^-10 meters, and the Milky Way Galaxy is roughly 10^21 meters across. These numbers may be unfathomable to comprehend with our senses alone, but we can abstract them to calculate other important quantities.

Okay, maybe this discussion isn’t too enlightening, but let’s move on to the other questions I posed earlier. What’s the biggest number you know of? Well, you might think of a googol. A googol is a huge number defined as a 1 with 100 zeroes after it (or 10^100 …see, that’s way more manageable looking!). Perhaps you’ve heard of a googolplex, which is 10 to the googol power, or 10^(10^100). That number might make your brain hurt if you think about it for too long. Sure, there is no largest number since infinity goes on forever, but at least a googolplex has a name. And everyone knows of Google, whose company is named after this number, so lets just stick with a googol for now.

This exercise is all well and good, but before when we discussed big numbers like the mass of the Sun, these numbers were grounded in a physical quantity. We could say well, it’s hard to visualize, but if you cut up the Sun into chunks 1 kilogram in size, you’ll get 10^30 of them. So that brings us to the last question. Now we have to try and tie the number googol to a physical phenomenon or quantity. When we take physical things, it becomes a lot harder to ground these obscenely large numbers…how far up can we actually climb this number ladder?

 

I prefer the physical analog of a number ladder you climb (instead of a number line) where one side is the number and the other side a physical quantity to represent it.

I prefer the physical analog of a number ladder you climb (instead of a number line) where one side is the number and the other side a physical quantity to represent it.

You can liken this discussion to when America kept its money tethered to the gold standard–at some point, we could only print as much money as we had in gold, so each dollar had a specific value grounded in a physical quantity. Printing any additional money was meaningless. We can count numbers as high as we want, but after a certain point, there’s no point in going any higher because those numbers are physically meaningless when dealing with nature or science in general. For the sake of argument, I’m going to say to ground a number physically, we’re going to use the standard SI units of measurement, meters, kilograms, etc. so we avoid the semantics of “well, 5 meters is the same as 5 million micrometers.”

So where’s a good place to start? How about stars, the building blocks of galaxies–how many stars are there in our observable universe? Well, there is about 100 billion (10^11) stars in our Milky Way galaxy, and there are about 100 billion (10^11) galaxies in our observable universe, so if you multiply the two together, we get a meager 10^22 stars in our Universe. Okay, that’s a big number, but it’s not that big. After all, we already said the mass of our Sun is roughly 10^30 kilograms, so that’s already a bigger number.

Okay, we failed there…how about the building blocks of matter, protons? How many protons are there in the entire universe? Our current estimate is pretty large, but it still falls short of a googol, topping out at roughly 10^80 protons. That’s a full 20 orders of magnitude smaller than a googol. Well, maybe instead of counting objects like protons, we can count units of distance? After all, the observable universe itself is a big place…what’s its volume? Well, we are doomed to fail again, as the current estimate is roughly 10^80 meters, similar to that of the total number of protons! Darn.

At this point, you might start to panic…protons and subatomic particles are as small as you can go for the “stuff” of the universe, and the universe is actually a bit too small for size to work (there’s a good joke hidden in there somewhere). If we want to try and ground the number googol into a physical quantity, the technique of just counting tangible “stuff” is not going to work, because there just isn’t enough “stuff” to count!

So far we’ve done well–grounding a number as high as 10^80, twice! But if we want to ground the Googol, we’re going to need a new approach…stay tuned for part II!

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