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Lesson 8.2 of Exploding Dots

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OK, let's see if we can help my sensibilities as a mathematician and bring symmetry into our lopsided story so far. Right now, we have boxes that go as far to the left as we please and we make sense of these boxes. In fact, if I tell you this is a 1←10 machine— let's go back to primary school for a moment— then we know these boxes represent, what, ones? I've got to write the numbers—ones, and tens, and a hundreds, and thousands, and so on—off to the left.

But I'm worried about it being lopsided. So I'm going to ask, as a mathematician, can we make this symmetrical? Can we have boxes going as far to the right as we please as well? Well, the answer is, sure. Just make it happen.

But the real question is, what could these boxes going infinitely far to the right actually mean? Now, if I stick with, say, 1←10 specifically right now, then I know the meaning of the boxes to the left, and I know this machine works by taking 10 dots at any one box and having them explode to become one dot, one place over. So 10 ones makes 10. 10 of these makes the next one. 10 tens is a hundred, and so on.

So if that's the game of this machine, then these boxes must work the same way as well. That is 10 dots in here, whatever they're worth— who knows what they're worth— must together explode and become one of those. So 10 of these explode. Kaboom!— messy, messy, messy—become one dot there, which is worth 1. 10 of something becomes 1. That tells me these dots here must each be worth 1/10. Ten 1/10s makes 1.

Now I'm going to see what's going on. Because 10 of these, whatever they're worth, must together explode to make one of those, 1/10. 10 whats make 1/10? 1/100. Ten 1/100s make 1/10. Ten of these makes one of thems— 1/1000. Ten 1/1000s is 100, and so on. I think I've now given meaning to the boxes off this direction as well as that direction. Beautiful.

Now, people call these decimals. Writing numbers in a symmetrical machine like this, we're writing them as a decimal. Because in English, the prefix deci- means 10. So that's language specifically for the 1←10 machine. In fact, it has become standard in society to separate the decimal part from the whole number part with something called a point, while in this case, called a decimal point in the 1←10 language. In a 1←2 machine, I guess I wouldn't use the word decimal. That means 10. Have to use the word, what, bimal or something? I don't know.

Anyhow, we'll stick with 1←10 for now. There's a decimal point. So if someone writes a number like 2.37, what they really mean is two dots in here, they mean three dots in here, and they mean seven dots in here. And how do you interpret that as an actual number? Well, what is it? It's two ones, which is 2. Plus three of these guys, 3/10, plus seven of these guys, plus 7/100.

So actually, a decimal is just the expression in terms of these particular fractions— tenths, hundredths, thousandths, and so on. Beautiful. Beautiful, beautiful, beautiful. In fact, if I wrote the fraction I know, 3/100, well then clearly, it will be just this picture— do, do, do, da, da— uh, clearing it here, messy, messy, messy— three dots in the hundredth box. And that will be the decimal, what, point 03, .03.

Some people like to write the zeros in the front of the decimal point. Some people don't. It's just a style thing. It's all good. Actually, there's one confusing point here. I should say this. Um, let's clear the box again. Some people might say, if they write 0.31— they might say this is 31 hundredths. Listen to what I just said, 31 hundredths, which is actually a little strange at first thought. Because 0.31 is literally this, it's 0.3, three dots, and one dot. So technically, this is 3/10 and 1/100.

Yet people will say this is 31/100. Different language there. The question is, are they saying the correct things? Well, of course they are. But why are these two different ways of thinking actually equivalent? So is 3/10 and 1/100 the same as 31/100? Well, actually yes.

Because if I unexplode these dots—kapow! No, sorry. Wrong sound effect. Shlp. One, two, three, four, five, six, seven, eight, nine, ten. Shlp—one, two, three, four, five, six, seven, eight, nine, ten. Shlp—one, two, three, four, five, six, seven, eight, nine, ten. All right, bingo. Three dots here, one dot here is indeed equivalent to 31 dots here. It literally is 31/100, so saying 31/100 is fine. Saying 3/10 and 1/100 is fine. All is good. Grand.

So I love this work. This is really super. So we've got decimals become fractions. Getting fractions into decimals is a little more delicate. Let me just, uh, make some space here on the board. Excuse me one moment. I will clean some space. All right, I'm back. I've made a little bit of space up here. So I want to get some fractions now into decimals.

So I've learned how to get decimals back to fractions—other way around. Um, let me start with a fraction like 1/2. Now, this is actually a bit delicate here because decimals are all set up to think in terms of tenths, hundredths, thousandths, and so forth. So the challenge is, can I think of this fraction 1/2 in terms of tenths, hundredths, or thousandths—whatever?

Hmm. So what I'm going to do is actually— OK, the denominator of a 2 is annoying. I'd like to have one of these denominators, please. So what I could do there is multiply the bottom by 5, which, I guess, I'd better multiply the top by 5 as well. Doing that gives me a bottom of 10 and a top of 5. So actually, I now see that 1/2 is the same as 5/10—five of these guys. Whoops, five of them. So actually, I can write it as 0.5. So that's the trick.

So some fractions are going to be amenable to doing this. Because I think 3/4 is also a good one. Any way you can think about getting a denominator of either a 10, or a 100, or a 1000 from that? I can. I can think of multiplying by 25 on the top and the bottom— keeps the fractions the same— but I now see 75/100. So it's really 75 of these guys.

And if you like, I can do some explosions— 10 explode, 10 explode, 10 explode, 10 explode, 10 explode, 10 explode, 10 explode— leaving 5 behind. Seven dots there is actually 0.75, before I run off the edge of the board. All right, so some fractions are amenable to writing as decimals. And other fractions, I have a feeling, are going to be a bit awkward, like 3/13.

Huh. That seems like a topic for another lesson. I bet there's a way to handle even those ones. That ought to be cool.

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Duration: 6 minutes and 39 seconds
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Language: English
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Posted by: jamestanton on Jun 9, 2018

This first lesson of chapter 8 explains decimals via Exploding Dots for Global Math Week 2017 (www.theglobalmathproject.org).

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