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Lesson 8.6 of Exploding Dots

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Okay, with a very interesting philosophical point right now, the question at hand is, is it true that every fraction, when converted to a decimal, falls into a repeating pattern? The answer turns out to be yes. And being kind of see why right now. Here's the examples of 6/7 we've had in the previous lesson. And we saw that we started with number 6 We have all these different numbers unexploding with these different remainders which got unexploded. Now remainders. We've got remainders of 4. We've got remainders of 5, remainder of 1, remainder of 3, remainder of 2, remainder of 6, remainder of 4, remainder of 5, so there are many of this stuff repeating. As soon as the remainder stop repeating, We know we're in a repeating pattern, our answer will repeat as well. So the question is, is it true, when I do a fraction problem could be a decimal, that I'll get repeating remainders? Now think about the remainders. When I'm dividing by 7, I will never see a remainder of 7 because that would mean there's a group of 7 I should've collected. I'll never see a remainder of 8, I'll never see a remainder of 9 for the only remainders I'll ever possibly see when divided by 7 are remainder by the 1, or 2, or 3, or 4, or 5, or 6 not 7 or 8 and so on. I might see a remainder of zero I didn't happen to this time but there's only seven possible remainders I could see when dividing by 7: 0,1,2,3,4,5, and 6. Grand! So that means as I do my division process, there's only a finite number of remainders and I'm trying for this process for as long as I can I must eventually repeat a remainder. In fact we repeated the 6th right off the bat. So with our repeating remainder, that means I'm in a cycle. I've got a repeating decimal pattern Okay. Now when we did one quarter, we actually got a repeat remainder of 0. We got 0.25, we figured that out. Then a remainder of 0, which led to a remainder of 0, which led to a remainder of 0 It fell on this small repeating cycle. So there was repeating remainders as well. See, even 1 quarter is repeating decimal, repeating zeros after a while. When we did 1/3, it fell under the same remainder right away. remainder 3, remainder 3, remainder 3. So when it falls repeat remainder ones, it's in a cycle. If I did a number like 3/13 When we draw it out be even worse than this one but we can think a way through it. As I draw 3 dots that equips the 13 I know the remainders that I possibly see could be 0,2,3,4,5,6,7,8,9,10,11, or 12 but we'll not see a remainder 13, we'll not see a remainder 15, we'll not see a remainder 15 and so on. There's only a finite number of possible remainders I will get to. So as I play this game, I have to eventually repeat the remainder. I can't keep doing different remainders. There's only 13 possible remainders. I've got to especially hit the same remainder like before and as soon as I hit the same remainder, that means I'm in a cycle. We've just proven every fraction, when converted to a decimal has a repeating pattern. It maybe a repeating zeros or repeating pattern like this but every fraction has repeating decimal expansion. Woah! This is deep. It's so deep I'm going to write it in words. Every fraction has a repeating -you can see my atrocious writing- a repeating decimal expansion. Alright. It could be repeating zeros like a quarter or a half or 8th or something or it could be repeating blocks like this but has repeating decimal expansion. This is deep. Because what it says now, suppose I gave you a decimal that didn't have a repeating decimal expansion, did not have a repeating pattern of any kind, then that means, that number cannot be a fraction. Now, people often call fractions rational numbers like a ratio of 2 whoel numbers So numbers not a fraction will be called irrational and I'm going to write for you, write down for you right now an example of an irrational number. We can now prove at this very moment, that irrational numbers actually do exist, here's one. It's .10110111011110, and so on. This number, now it has a pattern, So there's a pattern. You know what this pattern is. You could figure out what the millionth digits gonna be or the 2 millionth digit gonna be You know what this number actually is. You could figure it out. But it doesn't have a repeating pattern. There's no repeating cycle going on. In which case, this is some number, it's on the number line. It's just bigger than 1/10, cannot be a fraction. It had no repeating pattern therefore it's not a fraction This is our very first example of an irrational number. There's one thing that drives me crazy as a teacher. A lot of books say that we all know that pie, for example, is an irrational number. the squareroot of 2 is an irrational number. We don't actually prove these things as hard. Some books might prove the square root of 2 is actually irrational. It doesn't have repeating decimal but not pie. Pie was very hard to prove. that took mathematicians 2 millenia to figure out how to prove that pie actually was an irrational number is not at all straight board but what I love about what we've just done now, dots and boxes, we could see every fraction must have repeating decimal expansion. Therefore, any number you write down that does not have a repeating decimal expansion must be an example of a number that's not a fraction. We have a first example here of an actual irrational number that we could see and own for ourselves. Brilliant! In fact, your turn. Write down another example of an irrational number. Isn't this grand?

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Duration: 5 minutes and 29 seconds
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Posted by: jamestanton on Jun 9, 2018

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

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