What is 10 divided by 49?

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Well? I'm sure you've got access to a calculator program. Plug it in and see if you spot anything odd.

So why might this be happening? Well, it all has to do with a little thing called a geometric series.

What is a series?

A sequence of numbers is just a bunch of numbers one after the other. A series is, without getting too technical, what you get when you come up with some rule for generating numbers (say, each number is the previous number plus two), use that rule to generate a sequence of numbers off to infinity (imagine the sequence [1, 3, 5, 7, 9...] extending on forever, adding two each time), and add up all the numbers. A geometric series, specifically, is what you get when you add up all the numbers in a sequence generated by specific rule. What is that rule, you might reasonably ask at this point! Other sensible questions might include "how do you add together an infinite number of numbers?", "why don't you just get infinity if you're adding together an infinite number of numbers?", and "what is this nonsense, my brain hurts". Well, that last one isn't a question, but read on and maybe it'll stop hurting. Or hurt more. That's just a gamble you'll have to take.

So, the infinity problem. Math around infinity can get super weird, and the idea that an infinite pile of numbers could be added up into a finite value was only established in the 17th century. Indeed, an ancient Greek fellow called Zeno of Elea divided a finite distance into an infinite number of smaller distances and concluded that since covering an infinite number of distances is impossible, motion itself cannot occur. Plenty of people thought this argument was wrong, but the mathematical technology to solidly demonstrate WHY wouldn't exist for another 2,000 years! As you can see, it's no simple matter. So I'm not gonna cover the exact specifics until Part 2, which means luckily for you we're just going to skip to the result for now.

Geometric Series

So as I mentioned earlier, a geometric series is a special type of series which is produced by a specific rule. The most basic version of this rule is to suppose we start with any two numbers; call them a and r The first number in our sequence is a. We then generate the rest of the sequence by multiplying the previous number by r. Simple, right? So if we started with an a of 2 and an r of 2, we would get the sequence (2, 4, 8, 16, 32, 64...) and so on. Look familiar? Anyways, a geometric series is simply the value we get by adding up any sequence that can be produced by that rule (known, appropriately enough, as a geometric sequence). Importantly, though, r doesn't have to be a whole number; it can be anything you want. If we start with a = 1 and r = ¹⁄₂, for instance, we instead get (1, ¹⁄₂, ¹⁄₄, ¹⁄₈, ¹⁄₁₆...); brush up on your fraction multiplication if you're unsure.

So that's geometric series in a nutshell. The last piece we're missing is the idea that a series can converge. This just means that when you add up the entire infinite sequence of numbers, you get a finite result instead of infinity. We're skipping over a lot here (you'll want to check out part 2 if you're interested in why this is true), but one of the reasons geometric series are neat is that if r is less than 1 and greater than -1 (although we won't be worrying about anything negative today; good vibes only), the series ALWAYS converges! Better yet, we have an incredibly simple formula to tell us what finite number you get by adding up the entire infinite sequence:

Yep, that's it. You find out what 1-r is, you divide a by that number, and there's your answer. Let's look at our a = 1 and r = ¹⁄₂ example again. Replacing the letters in the formula with our chosen values, we're simply dividing 1 by (1 - ¹⁄₂), so we're dividing 1 by ¹⁄₂. This is the same as multiplying it by ²⁄₁, or just 2; dividing by a fraction being the same as multiplying by the same fraction but upside down. So the answer is 2! And sure enough, if you keep just adding together the numbers from our sequence (1, ¹⁄₂, ¹⁄₄, ¹⁄₈, ¹⁄₁₆...), your result will keep getting closer and closer to 2, but never quite make it there because you've only added up a finite number of them.

That's great, but what does this have to do with ¹⁰⁄₄₉?

Well, let's think about that result we got for a second, waaay back at the top. 0.204081632...Can we think of an easy way to construct this by adding up a bunch of other numbers?

As you can see, each number just translates directly downward into the final product. What's slightly less obvious is that we can generate each number pretty easily from the one above by simply doubling it and slapping two more zeroes in there. In other words, we start with 0.2, then for each subsequent number we multiply by 2 (doubling) and divide by 100 (pushing the decimal point two places to the left, or adding in two zeroes). That's starting to sound like a geometric sequence! We simply set a = 0.2 (or ²⁄₁₀) and r = 0.02 (or ²⁄₁₀₀) and it generates the sequence (0.2, 0.004, 0.00008, 0.0000016...) for us. So we know that sequence adds up to our result there if we just take the first five numbers, but what if we try our infinite addition formula from up above? It might get a little messy, but:

We can now slap in our chosen values for a and r:

Ew. Let's take a quick break and sort out that mess on the bottom.

Any number divided by itself is 1, which means we can do this:

And that's a much easier subtraction than the previous one:

Great! So that makes the original formula turn into:

Still kinda yucky. Fortunately, as we saw earlier dividing by a fraction is the same as multiplying by that fraction upside down, so we can instead just do:

We brushed up on our fraction multiplication earlier, so we can just multiply these together like this:

Which gives us:

Now it's simplification time. If we divide both parts of the fraction by the same number, then its value stays the same; it's like dividing it by 1, which just gives you the same number back. You should be able to see why based on some of the stuff from above! First, let's divide by 10 since that's as easy as knocking a zero off the end of both:

And now, since we already know what value we were going for, let's divide by 2 to finish it up!

And that's why dividing 10 by 49 produces a decimal result that looks like powers of 2; it's the sum of an infinite geometric sequence constructed just so in order to plop them all down into place in the final product.

Epilogue

You might have noticed a few things along the way here. Let me just bring up those decimals again, but with a few more added on:

Ack! What happened to our lovely pattern? Well, unfortunately if you go past 32 the numbers start getting into triple digits which evidently messes up the pattern. Mighty inconsiderate, if you ask me. But this is fixable! With some minor tweaks to the a and r values, you can both give the numbers more room to breathe and have sequences of entirely different numbers instead. I'll leave it to you to experiment; you might need a calculator with a lot of decimal places, but if you mess around you can more or less adjust the idea however you like!

That just leaves one loose end: where did that formula we used even come from? Check out Part 2 for more information about that.