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Lesson 8.5 of Exploding Dots

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All right, we've left one question hanging, how to write all decimals as fractions. Now let me start off with something familiar, like 1/4. The thing to realize here, that a fraction is really an answer to a division problem, that this one here is really the answer to 1 divided by 4. And I can literally do that in a 1←10 machine.

In fact, let me do it now. So here's a 1←10 machine. In fact, I have a feeling I'm going to go into decimals. So let's do this part of it here. I'm looking for a 1—one dot divided by 4. What does 4 look like? Well, it looks like dot, dot, dot, dot.

OK, so the question is, can I find any groups of four in the picture I have up here? That's what division is. And the answer is, well, not right now. But we know what to do. In fact, I can unexplode this—shlp— and have 10 dots appear here.

So I'm getting tired of drawing dots. So I'm just going to write numbers, if that's OK. All right, the question is, do I see any 4s in there? Well yeah, I see one group of four. I see a second group of four. That's two groups of four. That's eight of them. And that'll leave two behind. So I'll see two at that level. And there will be two dots left behind.

Great. Now, those two dots there—what can I do with them? Well, I could unexplode them—shlp, shlp. And that would make, what, 20 dots here. Do I see any groups of four amongst those 20? You bet. I would see five of them. And I believe that leaves none behind. So actually, when I do conduct this division problem, 1 divided by 4, I see I get the answer nothing point 25, nothing, nothing, nothing, nothing, nothing. So actually, the answer is 0.25.

Now, most people don't bother writing the extra zeros because there's a whole lot of them, infinitely many of them. They just stop at 0.25. Or if you want to write a whole bunch of zeros, feel free. All good. In fact, I'm going to choose to write a bunch of zeros. Because I see those keep going—0, 0, 0. Nice repeating pattern.

All right, because you know I was hinting at something right now. 1/4 was nice, because you know some decimals do give you repeating patterns. For example, the classic one you might be thinking of right now is one threeth—1/3. That is, 1 divided by 3. So if I do that one in a 1←10 machine, let's see what happens.

One dot—bingo. I'm looking for 3. What does 3 look like? Well, it would be three dots. I don't see any right now. So what I'm going to do is I'm going to unexplode it—shlp and have 10 dots there. Can I find any 3s in that picture? You bet. I'll find three groups of three with one dot left behind. So it will be three groups of three, one dot left behind. Oh, one dot here. Unexplode—shlp. Will I find any 3s in there? You bet I will. I'll find three groups of three with one dot left behind. Ah, unexplode—shlp. 10 dots there. Will I find any groups of three? You bet. Three groups of three, one left behind. I seem to be in a loop and I'll be doing this forever.

So I'm now going to say that 1/3 is 0.3333... forever. Again, another repeating decimal pattern. So 1/4, here we'll say the finite decimal. Or if you want to say it is actually a repeating pattern, whole bunch of repeating zeros eventually. Grand. 1/3 is also a repeating pattern. Let's do a more awkward fraction next.

Let me work out the decimal representation of 6/7. That's a really awkward number. It's going to take us a little while. We're going to do it. But I need to clean the board, so I'll be right back. All right, I'm back. Let's do the decimal expansion of the fraction 6/7. That is, let's work out the division problem 6 divided by 7. It's going to be awkward. It's going to be long. But let's actually do it.

All right, here goes. So let me draw my 1←10 machine. I'm going to need a lot of boxes, methinks. Whoa, maybe that's enough. Who knows? Uh, six dots. Bing there, six dots. Divide by 7. So I'm looking for groups of seven and a picture of six dots. Right now, I don't have any. But we know what to do. We'll unexplode. So unexplode—shlp, shlp—six times will make 60 dots there.

Now the question, will I find any 7s amongst 60? You bet. I'll find eight 7s amongst there. So I guess I get eight groups of seven, that's 56, leaving what? Four dots behind. All right. Now what happens? Uh, let's see. So four dots unexplode, makes 40 dots here. Any 7s amongst that? Uh, 7 times 5 is 35. Five groups of seven, leaving five behind.

So far so good. Five dots left behind. Don't see any 7s there. But if I unexplode, I get 50 dots there. 50 dots. Any groups of 7 amongst that? Uh, 7 times 7 is 49. Yes. So it will be seven groups of seven there, leaving one behind.

[sighs] OK, still going. No 7s there, but its unexplode makes 10 dots. Any 7s in there? Yep, one group of seven leaving what? Three behind. I'm going to be here for a while. OK, three unexplode makes 30. Um—whoops, didn't mean to cross that out—30. Four groups of seven, 28. Four groups of seven. Uh, leaving what, 2 behind.

Oh, golly gee. Uh, all right. Unexplode makes 20. Uh, 14— that's two groups of seven leaving six behind. Six behind. Unexplode makes 60. Uh, 56—that's eight groups of seven, leaving four behind. I feel like I'm doing this all again.

In fact, we started with 6 here. I went through this chain, chain, chain, chain, chain. And I went back to 6 again. So I must be repeating what I just did, which must come what be 8, 5, 7, 1, 4, 2. And I must have another remainder of 6, getting me back to 8, 5, 7, 1, 4, 2— I'm in an infinite loop. That 6/7 is going to be 0.857142, 857142, 857142... forever.

And some people like to use a horizontal bar to denote a repeating pattern. It's called a vinculum, a very cool word. That means that group of digits gets repeated over and over again. So even 6/7 is a repeating decimal.

OK, so we have right now that 1/4 is 0.250000 forever, that it repeats... eventually. Has a little hiccup first. 1/3 is 0.3 forever, 0.33333. And 6/7 is repeating as well. All the fractions I pulled out so far have a repeating decimal expansion. In fact, if you play with more examples, I bet you'll find your examples, too, have a repeating decimal expansion. Eventually, they might have a little hiccup first and then fall into a repeating pattern.

Curious. That begs a big question. Is it indeed true that every fraction as a decimal is a repeating decimal? Might be repeating zeros, or it might be a repeating block like this. I wonder if it's always repeating. So that's the topic of the next lesson.

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Duration: 6 minutes and 49 seconds
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Language: English
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Posted by: jamestanton on Jun 9, 2018

This is the fourth lesson on Decimals in the story of Exploding Dots for Global Math Week 2017 (www.theglobalmathproject.org).

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