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Lesson 7.2 of Exploding Dots

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OK, are you ready to play with the infinite? Let's do it. Let's do it in an 1←x machine again. I'll do another polynomial division problem. But I'm going to make it a particularly simple one that's kind of deceptively, uh, tricky in the end.

What I'm going to do is take the very simple polynomial 1. Let me draw a picture of what that looks like. It's supposed to be a one dot. But let me draw the full 1←x machine. There's 1←x machine going as far left as I please. And this is just one dot.

All right, but what I want to do with this one dot is divide it by 1 - x. All right, what does 1 - x look like? Well, it's going to be—let's see. It's going to be one anti-x. So one antidot there. And one—actual one—one dot.

Here goes. 1 divided by 1 - x means I'm looking for this pattern in this picture of one. Not much to go on right now. However, we've learned life lessons in the past. If there's something in life you want, make it happen and deal with the consequences.

So I've got one dot there. I'd love to have an antidot to go with it. So make it happen and deal with the consequences. That box is still technically empty. But now I've found one of what I want. Brilliant.

Hmm. But I'm left with this dot here. I want something to go with it. I want an antidot to go with it. Make it happen. Deal with the consequences— it's lovely getting what we want.

I've got a dot here. I'd like to have an antidot next to it. Deal with the consequences—put a dot with it and another thing of what I want. In fact, I can see now, I'm going to be doing this for quite the while. In fact, this is going to go on infinitely on forever. I'm caught in an infinite loop. All right, so the answer is 11111... going on forever.

So how do I interpret that answer? Uh, we'll have to start from the left this time— maybe not. We'll actually start from the right. That is one 1 and 1x and 1x² and 1x³ and 1x to the fourth and 1x to the fifth... forever. I've actually got an infinite sum.

In fact, we've just proven something crazy. If I do this polynomial division problem right here, the answer is a sum that goes on forever. Whoa. In fact, this is a very famous formula in mathematics. But most people write it the other way around. They start with this infinite sum and say it has this answer. They switch it around.

And in most books, it's called the "Geometric Series Formula." So you might see that in a pre-calculus class or a calculus class. But it comes with the mathematics. And look, we got to the infinite just by playing with dots and boxes. I love it.

And that's only one of many infinite sums we can play with. So I invite you to play with more. Lots of them to go for. All right, have fun.

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Duration: 2 minutes and 54 seconds
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Language: English
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Posted by: jamestanton on Jun 9, 2018

See how we can play with the infinite in the story of Exploding Dots.
Global Math Week 2017 (www.theglobalmathproject.org)

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