How to Think About Small Numbers, Part 2
Now that we know how to think about size scale as small as 0.000001 meter, it’s time to shrink ourselves down even more and think about really, really small sizes. How small? Keep on reading to find out.
Jason Marshall, PhD
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How to Think About Small Numbers, Part 2
In the last episode, we began an imaginary journey down to the tiniest size scales of the universe. Why? Because we’re on a mission to understand really tiny numbers. And the easiest way that I know to do that is to think about really, really (really) tiny sizes.
To explore these small size scales (and thus to come to an understanding of small numbers), we decided to shrink ourselves down and see what we can find “swimming around” in an otherwise boring puddle of water. But to make all of this easier to think about and picture in our minds, we’ve actually been imaging enlarging—which, if you think about it, has the same effect as zooming in on—a one meter wide puddle of water. Thus far, we’ve been zooming in on that puddle until it’s as big as a lake or a sea, and today we’re going to go even further.
What will we find if we keep on looking at tinier and tinier size scales? And what does this all tell us about the meaning of almost unimaginably small numbers? Those are exactly the questions we’ll be answering today.
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Recap: Millionths, Micrometers, and Microbes
Here’s a recap of the journey we’ve taken so far: We started by imagining staring at a puddle of water 1 meter across. We then zoomed in on this puddle 1,000 times, turning it into a 1 kilometer wide lake. What did we see at this level of magnification? Not much besides the water.
So we decided to zoom in another 1,000 times, at which point our once 1-meter-wide puddle became a 1,000-kilometer-wide sea about one-third the size of the Mediterranean! Being zoomed in a total of 1,000,000 times, something that is actually one-millionth of a meter in length in real life appears to you (a person swimming in the 1,000 kilometer wide sea) to be 1 meter across. And that means that lots and lots of millionth of a meter sized microbes that are normally too small to see have suddenly become visible to you.
So that’s what a millionth of a meter looks like. It’s a really, really small size. And thus, we can see that 1/1,000,000 = 0.000001 is a really, really small number. But as we’re about to find out, it’s still huge compared to many much smaller numbers.
Billionths, Nanometers, and Atoms
To see this, let’s continue our journey into the puddle of water by imagining we zoom in another 1,000 times. At this level of magnification, our once 1-meter-wide puddle has expanded into a 1,000,000,000-meter- or 1,000,000-kilometer-wide ocean. In truth, this is much, much bigger than an ocean—it’s 2.5 times the distance from the Earth to the Moon, or a little less than 3/4 the diameter of the Sun. That’s a pretty big puddle of water!
Atoms are unbelievably tiny…to see them we had to expand our puddle to be as big as the Sun!
And here’s the crazy part: If you were swimming in this nearly-Sun sized “puddle” of water, the hydrogen and oxygen atoms that make up the water would be roughly 0.1 meter or 10 centimeters in size. Equally crazy, those zoomed-in 1 meter wide microbes that you were swimming with in the 1,000 kilometer wide sea would be something like 1 kilometer across microbes at this level of magnification. And you’d hardly be able to tell that they were microbes at all since you’d be able to see all the frenetically moving 10-centimeter-wide atoms of which they’re made.
All of which tells you that 1/1,000,000,000 is a really tiny number. And it also tells you that atoms—which are about one-tenth of 1/1,000,000,000 of a meter in size (a distance known as 0.1 nanometer)—are unbelievably tiny. After all, in order to see them individually, we had to expand our puddle to be as big as the Sun!
Definitely pretty wild, right? Sure, but not wild enough for me…let’s go smaller.
Quadrillionths, Femtometers, and Atomic Nuclei
In fact, let’s now zoom in another factor of 1,000 times on our now nearly Sun-sized puddle of water until it’s 1,000,000,000 km in size. How big is this? It’s almost 7 times the distance from the Sun to Earth. Put another way, at this level of magnification, if our once 1-meter-wide puddle of water were centered on the position of the Sun, it’s edge would extend out beyond Mercury, Venus, Earth, Mars, the asteroid belt, all the way past the orbit of Jupiter.
And what would you see if you were swimming in this nearly solar system sized puddle? Well, individual atoms would now be about 100 meters across—as wide as a football field is long. But besides these huge football field sized atoms, you wouldn’t see much else. In particular, there’d still be no sign of the nuclei of these atoms. We’re still simply not dealing with small enough size scales and tiny enough numbers.
…the numbers will happily keep on getting smaller and smaller.
So let’s zoom in another factor of 1,000 so that the puddle of water now extends into the middle of the extremely distant Oort cloud of icy bodies orbiting the Sun way out beyond Pluto in the outskirts of the solar system. If you were to jump into this rather ridiculous pool, the protons of the hydrogen and oxygen atoms making up the water would now be roughly 1 meter across.
So yeah, 1/1,000,000,000,000 (that’s one trillionth) and 1/1,000,000,000,000,000 (that’s one quadrillionth) are extremely small numbers, as are the nuclei of atoms.
But of course, if we wanted to, we could go further. In fact, this is all just the tip of the iceberg, because the numbers will happily keep on getting smaller and smaller. But I’m content to stop right here for a while and contemplate the meaning of some of the tiniest size scales in the universe.
Wrap Up
Okay, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Water image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.