Valentine’s Day Math
Jason Marshall, PhD
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Valentine’s Day Math
Valentine’s Day—that oh-so-special day when roses, heart-shaped thingies, saccharine candy, and overpriced romantic dinners are all the rage. But we in the Math Dude household don’t go for these easy gestures of our affection. Nope, we’re trend-buckers…we prefer poetry.
And so it is, math fans, that I present to you this piece of St. Valentine’s-inspired mathematical poetry:
Roses are red
Violets are blue
All of the sand grains on all of the Earth
Are too few to enumerate how much I love you.
Roses are red
Violets are blue
All of the stars the universe wide
Are too few to count how much I love you.
Roses are red
Violets are blue
All of the atoms that make up our planet
Are still too few to add up to how much I love you.
Even if you multiply these numbers together
Raise one to the power of the next and then the other
Or get crazy and do both in rapid succession
You’ll find love can’t be quantified by mathematical expression.
So, just how much affection am I talking about here? In other words: How many grains of sand are on Earth’s surface? How many stars are there in the entire universe? How many atoms does the whole planet contain? And if these numbers aren’t big enough, how can we use math to express how much we love somebody? Those are exactly the questions we’ll be answering today..
How Many Grains of Sand Are on Earth?
Let’s lead off our list of big numbers that are still not big enough to express my love with the number of grains of sand on the surface of the Earth. Long time math fans may recall estimating the number of grains of sand on all of Earth’s beaches long ago. Through a series of reasonable assumptions and guesstimates, we arrived at the conclusion that Earth’s beaches must contain something like 5,000 billion billion—about 5×1021—grains of sand.
But instead of just Earth’s beaches, what about the entire surface of the planet—including deserts, ocean floors, and every other sandy place? Can we come up with an estimate of this number? Of course we can.
There are perhaps as many as 1024 grains of sand on Earth.
To estimate the sand on Earth’s beaches before, we multiplied our estimate of the number of grains per cubic centimeter by our estimate of the number of cubic centimeters of beach. This time around, all we need to do is multiply that same estimated number of grains per cubic centimeter by an estimate of the total volume of Earth’s sandy surface.
How can we estimate this new volume? By making the very simple assumption that the entire Earth is covered by a uniform 1 meter deep layer of sand. After all, the ocean floors—which are 70% of the planet—are sandy, so we won’t be too far off by assuming that pretty much the whole thing is sandy. Of course, this isn’t actually true, but it’s a decent assumption for the level of accuracy we’re looking for.
Why 1 meter deep? Well, why not? We could have said 2, or 10, or any other reasonable number of meters…but 1 meter seems like a good (not to mention convenient) choice.
If you do the math, you’ll find that the volume of a 1 meter deep layer of sand all the way around the Earth is about 5×1014 cubic meters. If we multiply this by our assumption from last time that a cubic meter of sand contains roughly 8 billion sand grains (and do a bit of rounding to account for the large uncertainties in our estimates), we come to the conclusion that there are perhaps as many as 1024 grains of sand on Earth.
It’s certainly a huge number, but it’s not nearly enough to measure my love.
How Many Stars Are in the Universe?
How about the number of stars in the universe? Is that number perhaps big enough to quantify my love?
Thankfully, this calculation is a bit simpler. To estimate the number of stars in the universe we just need to multiply the average number of stars in a galaxy by an estimate of the total number of galaxies in the universe. An average galaxy like the Milky Way contains roughly 400 billion stars. And there are estimated to be roughly 400 billion galaxies in the visible universe. Which, multiplying these numbers together, tells us that there are something like 1023 stars in the universe!
See also: Is it a Star or a Planet?
Yes, it’s another really big number. In fact, our estimate says that there are probably more stars in the universe than there are grains of sand on all of Earth’s beaches. But there may very well be more grains of sand on the Earth as a whole than stars in the universe…although the numbers are close.
Regardless, this still isn’t enough to measure my love……
How Many Atoms Are on Earth?
Perhaps we can find an even bigger number by estimating the number of atoms that the Earth contains. This one is another fairly easy estimate if we assume that the Earth’s mass is mostly (or at least on the average) made from atoms with a weight similar to silicon or oxygen (which are components of its silicate rocks). There are lots and lots of heavier atoms like iron in Earth’s core, but it’s not a terrible estimate to say that the whole thing is on average made of stuff with the weight of silicon.
See also: Atomic Bonds – The Ties That Bind
The Earth is made up of roughly 1050 atoms.
If we divide the total mass of the Earth by the mass of a single silicon atom, we find that the Earth is made up of roughly 1050 atoms. That’s way, way more than the number of grains of sand on Earth (which should be obvious since each of those grains of sand are themselves composed of a lot of atoms) and it’s also way more than the number of stars in the universe.
But in no way is it enough to measure my love.
How Much Do I Love Thee?
Multiply these three huge numbers together and you get 1024 • 1023 • 1050 = 1097. It’s definitely an enormous number—but, of course, it’s minuscule compared to my love.
Raise one to the power of the next and then the other, as the poem says, and you’ll get a number that has over 1074 decimal digits—an unfathomably large number. Yet, as you know by now, it’s still a number that’s dwarfed by my love. As the poet said:
Because although numbers may get bigger, out to infinity and beyond
My love will always be greater, equal to infinity plus one.
Wrap Up
OK, 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!
Heart curve from Wolfram Alpha.
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