If we’re measuring a length of pi centimeters, why does it look finite but the number of digits is going on forever? It looks like that it’s moving small amount of time every time. Same with 0.5555555… or any number with infinite decimal places. I can’t wrap my head around it.
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In Ancient Greece there was a guy named Zeno and he came up with an interesting paradox:
To get from one point to another you have to cross half the distance. Then you have to cross half *that* distance. Then again. Then again.
But no amount of halvings will ever get you completely to the other point. It would take an infinite number of these half-steps to get there, and since you can’t do an infinite number of things in a finite amount of time, getting from one point to another is impossible.
I bring up the paradox to illustrate this concept of infinite halvings. Let’s convert that concept into mathematics. We take a half, 1/2. Then half of that, 1/4. Then half of that, 1/8. And so on, and add them together:
1/2 + 1/4 + 1/8 + 1/16 …
What’s great about math is we *can* do an infinite amount of things in finite time. Specifically, we can add (some) sequences of infinite numbers together. In fact, if you add the above infinite sequence the total is *exactly* 1:
1 = 1/2 + 1/4 + 1/8 + 1/16 …
What does this have to do with numbers whose decimal representations go on forever? Well, just remember that each place in a decimal representation of a number is simply some multiple of a power of ten, and the value of the number is those multiples added together:
325 = 3*10^(2) + 2*10^(1) + 5*10^(0)
And this is true for fractional pats as well:
0.325 = 3*10^(-1) + 2*10^(-2) + 5*10^(-3)
Or, written another way:
0.325 = 3/10 + 2/100 + 5/1000
Some numbers, however, can only be written this way if you have an infinite number of terms. Consider:
1/3 = 3/10 + 3/100 + 3/1000 + 3/1000…
Why? Well this is simply a quirk of our base-ten writing system. Only numbers that share a factor with 10 (2 or 5) can be written using a finite number of decimal places. Because only terms that share a factor with 10 can be evenly divided by it. All other numbers will always have a remainder that must also be divided, which itself well have a remainder that must be divided, and so forth.
But, still, at the end of the day, all of those terms sum up to a single, finite number.
You might ask, how can an infinite amount of numbers sum to a finite number?
Well, to be fair, this is not always the case. But with the examples we’ve been using, they have a special property called *convergence.* You should notice that each term is smaller than the last. So while the sum is getting bigger and bigger, it does so in smaller and smaller increments. If the increments shrink in size fast enough, instead of the sum exploding to infinity, it instead will approach a number and get continually and continually closer without ever reaching it (using only a finite amount of terms), just like Zeno.
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