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Lesson 8.4 of Exploding Dots

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OK, can we do multiplication with decimals? Well, yes. I mean, single digit multiplication is certainly straightforward. For example, look at 2.615 times 7.

What is this? Well, let me draw a picture of a 1←10 machine. So here's 2.615. 2, 6, 1, 5. 2 ones, 6 tenths, 1 one hundredth, and 5 thousandths. Now say multiply everything by 7. OK, I can do that.

So instead of having 2 ones, I now have 14 of them. Instead of having 6 tenths, I now have 42 of them. Instead of having 1 one hundredth, I now have 7 of them. And instead of having 5 thousandths, I now have 35 of them. The answer is 14.42 / 7/35, which is mathematically correct, just very weird for society. But we know how to fix it up for society.

For example, four explosions here—1←10 machine— exploded, leaving two behind. Extra four dots there. So we'll get 18 point—whoops—point 2 / 7/35ey. Then I can explode three and so on. Oh, actually, that one's going to be good. Explode three, you'll get ten there, which will cause another explosion! Wow. So try it out, get the answer. And grand, we can actually do very basic multiplication with decimals. No worries.

Actually, this is a good example. It speaks to something that confused me as a kid when I was going through school. Let me just clean the board for a moment, and I'll show you what I mean. All right, I'm back. Let me take the number 2.615. And this time, instead of multiplying it by 7, I'm going to multiply it by 10.

So let me draw the picture of this. So I'll draw more boxes now for my 1←10 machine. All right, so that's 2.615. And everything gets multiplied by a factor of ten. So instead of having two dots here, I now have 20 of them. Instead of having six dots here, I now have 60 of them. Instead of having one dot here, I'll have 10 of them. Instead of having five dots here, I'll have 50 of them. Grand. Now, look at this.

There's something special about multiplying by 10 in a 1←10 machine because I've got lots of groups of ten. And we know that groups of ten explode. In fact, there's two groups of ten here explode— kapow! kaboom!— to make two dots there. Six groups of ten here explode— kapow! kaboom! kathwack! kazeen! kajoup! kazup!— to make six dots there. One explosion—kaloop! One dot there. Five explosions. Kazipples, kazopples, kazubles, kadubles, kadupples. [snaps fingers] OK, bingo. And I see the answer 2, 6, 1, 5. [pen squeaking]

Wow, it's the same digits, but the decimal point, they said in my school days, moved. They said to me—in fact, they made me memorize this rule. To multiply by 10, shift the decimal point one place to the left or right. And I could never remember which one it is. But the thing is, it's not actually the decimal point that's moving.

Remember, we had 2, 6, 1, 5. When it became 20, 60, 10, 50, it's actually the numbers, because the explosion shifted over. It's not the decimal point that's moving. It's kind of the numbers that are moving. Anyhow, obviously, if I'm going to take a number that's, like, basically 2.6, like, just over two and a half, and multiply it by 10, I should get something that's about 20 and half of 10—26.

So you don't need to memorize which way the decimal point moves. Just use common sense. But the thing is I love the fact that those rules that we were often taught to memorize in grade school actually make sense in a 1←10 machine. So multiplying by 10—you can see why all the digits seem to shift. Multiply by 100, they seem to shift two places. Brilliant. Grand. All right, so that's multiplication.

Let me now talk about division a little bit with decimals. And that's a little bit awkward. So let me just clean the board again. I'll be right back. All right, I'm back. So let's try this one. Let's try, say, 5 divided by 0.3. All right, now. I'm going to be honest with you here. Every model that we come up with explaining mathematics is just a model, and has its limitations, and tends to break down. It just becomes a bit too awkward to really try to push it all the way through.

The fact is I'm a mathematician, which means I don't like hard work. And I will work very hard to avoid hard work. So if I'm given a decimal problem like this, a decimal division problem, 5 divided by 0.3, I'm going to say, OK, I could do this in 1←10 machine. I could draw a picture of five dots in the ones place. I could try to make sense of what 0.3 of a dot means. So it would be some dot that's shifted over not where I expected. And I bet I could make this work. I bet I could make this division problem work.

The trouble is it hurts my brain when I try to do that. And feel free to do it—it's kind of fun. But I think there's an easier way to think about this. In fact, a division problem is really just a fraction of it or maybe the other way around. A fraction is really the answer to a division problem.

So let me write this as a fraction. 5 divided by 0.3. Now some people might object to me having non-whole numbers for the numerator and the denominator. But that's fine, it's still a fraction. I'm good. I'm good. This is equivalent to two whole numbers. In fact, it is equivalent to two whole numbers because we're going to do the following.

I usually don't like the 0.3 on the bottom. That's what's causing me trouble, so I'm going to change it. If there's something I don't like, fix it. Make it—make it— make it be different. So what I'm going to do is multiply the bottom by 10. And a consequence of that is I have to multiply the top by 10 as well, so it stays the same number. In which case, I now see this is really the number 50 over 3. Because I just learned moments ago that multiplying by 10— well, I was told shifts the decimal point, but it's really the other way around, 50 over 3.

50 over 3. Now I'm back to one of the beginning chapters of the story. I can do that in a 1←10 machine. Or I can see the answer is actually going to be, what, 48 and 2/3? So 48 divided by 3 is 16 and 2/3. Bingo. So actually, my advice is yes, you can make division work with decimals in a 1←10 machine. But it's an awful lot of brain power.

It's fun as a logical exercise. But actually, you want to avoid hard work and turn it back to a problem from an earlier way of doing division— back to whole numbers. That seems much more natural and easier to me. I can do that in a 1←10 machine with dots and boxes I want or just see the answer. Grand.

All right, so smart thinking's the way to go. Avoid hard work. In fact, work very hard to avoid hard work. Be a mathematician.

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

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

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