# Lesson 8.3 of Exploding Dots

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All right, doing addition and subtraction with decimals
is really no different from doing ordinary
addition and subtraction in a 1←10 machine.
In fact, it is a 1←10 machine.
For example, 22.37 + 5.841.

Now I would personally go left-to-right. 2 + nothing is 2, 2 + 5 is 7. Point 3 + 8 is 11, 7 + 4 is also 11. And nothing + 1 is 1. 2.7, 11, 11, 1. OK, weird. Uh, it is technically correct.

If I drew a 1←10 machine for this— let me draw some boxes for 1←10 machine. We got two dots, two dots, three dots, seven dots. Hope it's OK if I just write numbers rather than draw dots at this point. Add to that five more dots and then 8, 4, and 1. And indeed, I do have seven dots here. I do have 11 dots here. And I do have—smudgy, smudgy, smudgy— 11 dots there.

Grand. Now, can I fix that up for society? You bet. All we have to do is do some explosions. It's a 1←10 machine, after all. So 10 dots here explode—kapow! Leaving one dot behind, making an extra dot there. 2, 8.1, 11, 1. Still mathematically fine.

Society thinks that part's weird—OK. We'll fix it up for society. Till explode—kaboom! Leaving one dot behind. Extra two dots there. Bingo. 28.211. In fact, you can also go right-to-left, the standard way, which would be the school way. Nothing + 1 is 1, 7 + 4 is 11.

But you might want to do the explosion right now, which is why the 1—they put an extra dot there. And go that way. You'll get the same answer. It's a style thing.

On subtraction, exactly the same thing for ordinary 1←10 work. Um, I would go left-to-right. 2 takeaway 1 is 1. 2 takeaway 4 is negative 2. Point 3 takeaway 5 is negative 2.. 7 takeaway 8 is negative 1. 0 takeaway 1 is negative 1. Whoa. That's a very complicated one.

All right, grand. If I draw a picture of this, here's the 1←10 machine. I guess I'll need three decimal places. Um, yes, So I have two dots and one antidot, will indeed make one actual dot. Two dots and four antidots will indeed make two antidots. Three dots and five antidots will indeed make two antidots. 7 and 8 will make one antidot. And nothing and an antidot make an antidot.

Bingo. So now the question is, can I fix up that crazy answer for society? I mean, it's mathematically fine. It's just weird. OK, the answer's yes. I guess I have to deal with this dot here, and I could unexplode it, just like we did before. This dot here is really 10 dots here. And then some annihilations will occur. For example, these two antidots will annihilate with two actual dots—pufuf.

So now I've got, what, 8, negative 2, negative 1, negative 1. And I can now do it again. One of these, making 7. Make an extra 10 there. Being annihilations makes 8 here. 1.8, negative 1, negative 1, and so on.

In fact, I'll leave the rest of the work for you, if you'd like to work it out. Or you can use the standard algorithm. You can go from right-to-left. 0 takeaway 1 is say can't do it. So what do you do? You unexplode one of these dots first, make them 10 here, and then do that. It will work out brilliantly and the same.

Because all good correct math is actually all good and all correct. It's just a style what approach you like best, what suits you—up to you. Grand. So really, no difference with addition and subtraction. All is good here.

Now I would personally go left-to-right. 2 + nothing is 2, 2 + 5 is 7. Point 3 + 8 is 11, 7 + 4 is also 11. And nothing + 1 is 1. 2.7, 11, 11, 1. OK, weird. Uh, it is technically correct.

If I drew a 1←10 machine for this— let me draw some boxes for 1←10 machine. We got two dots, two dots, three dots, seven dots. Hope it's OK if I just write numbers rather than draw dots at this point. Add to that five more dots and then 8, 4, and 1. And indeed, I do have seven dots here. I do have 11 dots here. And I do have—smudgy, smudgy, smudgy— 11 dots there.

Grand. Now, can I fix that up for society? You bet. All we have to do is do some explosions. It's a 1←10 machine, after all. So 10 dots here explode—kapow! Leaving one dot behind, making an extra dot there. 2, 8.1, 11, 1. Still mathematically fine.

Society thinks that part's weird—OK. We'll fix it up for society. Till explode—kaboom! Leaving one dot behind. Extra two dots there. Bingo. 28.211. In fact, you can also go right-to-left, the standard way, which would be the school way. Nothing + 1 is 1, 7 + 4 is 11.

But you might want to do the explosion right now, which is why the 1—they put an extra dot there. And go that way. You'll get the same answer. It's a style thing.

On subtraction, exactly the same thing for ordinary 1←10 work. Um, I would go left-to-right. 2 takeaway 1 is 1. 2 takeaway 4 is negative 2. Point 3 takeaway 5 is negative 2.. 7 takeaway 8 is negative 1. 0 takeaway 1 is negative 1. Whoa. That's a very complicated one.

All right, grand. If I draw a picture of this, here's the 1←10 machine. I guess I'll need three decimal places. Um, yes, So I have two dots and one antidot, will indeed make one actual dot. Two dots and four antidots will indeed make two antidots. Three dots and five antidots will indeed make two antidots. 7 and 8 will make one antidot. And nothing and an antidot make an antidot.

Bingo. So now the question is, can I fix up that crazy answer for society? I mean, it's mathematically fine. It's just weird. OK, the answer's yes. I guess I have to deal with this dot here, and I could unexplode it, just like we did before. This dot here is really 10 dots here. And then some annihilations will occur. For example, these two antidots will annihilate with two actual dots—pufuf.

So now I've got, what, 8, negative 2, negative 1, negative 1. And I can now do it again. One of these, making 7. Make an extra 10 there. Being annihilations makes 8 here. 1.8, negative 1, negative 1, and so on.

In fact, I'll leave the rest of the work for you, if you'd like to work it out. Or you can use the standard algorithm. You can go from right-to-left. 0 takeaway 1 is say can't do it. So what do you do? You unexplode one of these dots first, make them 10 here, and then do that. It will work out brilliantly and the same.

Because all good correct math is actually all good and all correct. It's just a style what approach you like best, what suits you—up to you. Grand. So really, no difference with addition and subtraction. All is good here.