Humans don’t really get numbers. Some isolated communities function without having numbers bigger than 4, which incidentally is about the highest number of ‘things’ we can reliably count with a glance.
1 thing. No worries.
2 things. Yep. Got it.
3 things. Indeed. At a glance, I’m all over this.
4 things. That’s 4 things, I’m pretty sure. Yep, definitely 4.
5+ things. Um. That’s lots. Wait while I count. Hang on, did I count that one already?
We are good at comparative estimation, though. We can look at a couple of fruit trees, and go, yes, that tree has more. Let’s climb that one.
Or, there are more of us than them; I reckon we can take them.
These are practical, useful skills to have, and have served us well, getting the species to survive for the last 100,000 years or so. But these days, we find numbers that are so very much bigger, the estimation trick is no longer workable. And if we’re looking at a number written down, generally we’ll need to look at something greater than 9,999,999 very carefully (and perhaps count the numbers back from the end to work out whether it’s millions or billions… or more).
We hear numbers like 1 million, or even 1 billion, and have some sense of what that means, kind of. But we have difficulty applying that to specific situations. Take the lottery, for example (please do, then give me some).
In Australia, the chance of winning the Division 1 Powerball lottery is 1 in 76,767,600.
Yes, Rory, that’s a big number, but what’s your point?
OK, for context, there are only 3,153,600 seconds in a year. This means the chances of winning the Division 1 prize are the same as picking a single winning second out of a greater than 24-year span.
This is a bit more than the chances of rolling a 6 with a single 6-sided dice 10 times in a row.
6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 = 60,466,176
Ok, there’s around a 16.5 million difference… but the same ball park.
But if we add an 11th roll of the dice, we leave the ball park entirely ( 1 in 362,797,056). 12 rolls, and we now have a completely different frame of reference: 1 in 2,176,782,336, which means rolling 12 consecutive 6s is 28 times less likely than winning the Division 1 lottery. (Or the same odds of picking the right random second out of a 690-year span.) However, you’ll note this very large number is written with only 10 digits.
It really didn’t take that much for our simple dice rolls to spiral the odds into the stratosphere. This is an example of exponential growth. When numbers start to get really big (and I mean really big), we can’t really write them down any more, because:
- There’s not a big enough piece of paper.
- There isn’t enough time in the history of the universe to write this down.
I’m not being melodramatic. Wait till we get to a googolplex.
When numbers get big, we can use a shortened form of notation.
1,000,000 = 10 ^ 6 [I’m using the ^ symbol to indicate that the number that follows should be superscript. This example is 10 to the power of 6; or 10 x 10 x 10 x 10 x 10 x 10.]
You’ll see that 10 to the power of 6 is a 1 with 6 zeros behind it.
10 ^ 9 is 1 billion (1,000,000,000), which we have already mentioned.
10 ^ 12 is 1 trillion. This is a big number. Humans have existed for around 100,000 years, or 3 trillion seconds.
10 ^ 15 is 1 quadrillion. There are about this many ants on Earth. (Don’t make me count them again.) Note, this is 1000 times bigger than a trillion.
10 ^ 17 is 100 quadrillion. Approximately this many seconds have elapsed since the Big Bang. (Do you feel small yet? These numbers are going to get much bigger.)
10 ^ 19 is 1 quintillion.
10 ^ 21 is 1 sextillion.
10 ^ 24 is 1 septillion
10 ^ 27 is 1 octillion
10 ^ 30 is a common ‘strength’ (dilution) for homeopathic ‘medicine’.[Did you know that this means: at 10 ^ 30 there is only a 1 in 10,000,000 chance that there would be even a single molecule of the active ingredient remaining?]
And back to the numbers!
10 ^ 100 is 1 googol.
How big is that?
From Wait But Why:
It’s the number of grains of sand that could fit in the universe, times 10 billion. So picture the universe jam-packed with small grains of sand—for tens of billions of light years above the Earth, below it, in front of it, behind it, just sand. Endless sand. You could fly a plane for trillions of years in any direction at full speed through it, and you’d never get to the end of the sand. Lots and lots and lots of sand.
Now imagine that you stop the plane at some point, reach out the window, and grab one grain of sand to look at under a powerful microscope—and what you see is that it’s actually not a single grain, but 10 billion microscopic grains wrapped in a membrane, all of which together is the size of a normal grain of sand. If that were the case for every single grain of sand in this hypothetical—if each were actually a bundle of 10 billion tinier grains—the total number of those microscopic grains would be a googol.
1 googol ^ googol is a googolplex.
And the numbers get even bigger. Much bigger than I can comprehend. Much bigger than I thought when I started researching this post.
This is the true meaning of awesome.
Blow your mind by reading Wait But Why’s superlative number posts: