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What number sets does[br]the number 3.4028

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repeating belong to?

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And before even answering the[br]question, let's just think

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about what this represents.

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And especially what this[br]line on top means.

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So this line on top means[br]that the 28 just

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keep repeating forever.

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So I could express this number[br]as 3.4028, but the 28 just

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keep repeating.

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Just keep repeating on and[br]on and on forever.

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I could just keep writing[br]them forever and ever.

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And obviously, it's just easier[br]to write this line over

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the 28 to say that it[br]repeats forever.

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Now let's think about what[br]number sets it belongs to.

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Well, the broadest number set[br]we've dealt with so far is the

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real numbers.

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And this definitely belongs[br]to the real numbers.

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The real numbers is essentially[br]the entire number

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line that we're used to using.

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And 3.4028 repeating sits[br]someplace over here.

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If this is negative 1, this[br]is 0, 1, 2, 3, 4.

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3.4028 is a little bit more[br]than 3.4, a little

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bit less than 3.41.

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It would sit right over there.

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So it definitely sits[br]on the number line.

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It's a real number.

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So it definitely is real.

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It definitely is[br]a real number.

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But the not so obvious question[br]is whether it is a

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rational number.

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Remember, a rational number is[br]one that can be expressed as a

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rational expression[br]or as a fraction.

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If I were to tell you that p is[br]rational, that means that p

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can be expressed as the[br]ratio of two integers.

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That means that p can be[br]expressed as the ratio of two

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integers, m/n.

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So the question is, can I[br]express this as the ratio of

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two integers?

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Or another way to think[br]of it, can I

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express this as a fraction?

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And to do that, let's actually[br]express it as a fraction.

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Let's define x as being[br]equal to this number.

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So x is equal to 3.4028[br]repeating.

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Let's think about[br]what 10,000x is.

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And the only reason why I want[br]10,000x is because I want to

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move the decimal point all the[br]way to the right over here.

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So 10,000x.

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What is that going[br]to be equal to?

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Well every time you multiply[br]by a power of 10, you shift

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the decimal one to the right.

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10,000 is 10 to the[br]fourth power.

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So it's like shifting[br]the decimal over to

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the right four spaces.

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1, 2, 3, 4.

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So it'll be 34,028.

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But these 28's just[br]keep repeating.

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So you'll still have the 28's[br]go on and on, and on and on,

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and on after that.

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They just all got shifted to the[br]left of the decimal point

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by five spaces.

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You can view it that way.

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That makes sense.

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It's nearly 3 and 1/2.

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If you multiply by 10,000,[br]you get almost 35,000.

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So that's 10,000x.

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Now, let's also think[br]about 100x.

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And my whole exercise here is[br]I want to get two numbers

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that, when I subtract them and[br]they're in terms of x, the

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repeating part disappears.

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And then we can just treat them[br]as traditional numbers.

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So let's think about[br]what 100x is.

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100x.

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That moves this decimal point.

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Remember, the decimal point[br]was here originally.

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It moves it over to the[br]right two spaces.

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So 100x would be 300-- Let[br]me write it like this.

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It would be 340.28 repeating.

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We could have put the 28[br]repeating here, but it

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wouldn't have made[br]as much sense.

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You always want to write it[br]after the decimal point.

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So we have to write 28 again to[br]show that it is repeating.

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Now something interesting[br]is going on.

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These two numbers, they're[br]just multiples of x.

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And if I subtract the bottom one[br]from the top one, what's

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going to happen?

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Well the repeating part[br]is going to disappear.

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So let's do that.

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Let's do that on both sides[br]of this equation.

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Let's do it.

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So on the left-hand side of this[br]equation, 10,000x minus

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100x is going to be 9,900x.

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And on the right-hand side,[br]let's see-- The decimal part

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will cancel out.

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And we just have to figure out[br]what 34,028 minus 340 is.

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So let's just figure this out.

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8 is larger than 0, so we[br]won't have to do any

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regrouping there.

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2 is less than 4.

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So we will have to do some[br]regrouping, but we can't

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borrow yet because we[br]have a 0 over there.

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And 0 is less than 3, so we[br]have to do some regrouping

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there or some borrowing.

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So let's borrow from[br]the 4 first.

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So if we borrow from the 4, this[br]becomes a 3 and then this

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becomes a 10.

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And then the 2 can now[br]borrow from the 10.

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This becomes a 9 and[br]this becomes a 12.

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And now we can do[br]the subtraction.

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8 minus 0 is 8.

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12 minus 4 is 8.

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9 minus 3 is 6.

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3 minus nothing is 3.

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3 minus nothing is 3.

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So 9,900x is equal to 33,688.

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We just subtracted 340[br]from this up here.

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So we get 33,688.

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Now, if we want to solve[br]for x, we just divide

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both sides by 9,900.

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Divide the left by 9,900.

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Divide the right by 9,900.

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And then, what are[br]we left with?

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We're left with x is equal[br]to 33,688 over 9,900.

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Now what's the big[br]deal about this?

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Well, x was this number. x was[br]this number that we started

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off with, this number that[br]just kept on repeating.

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And by doing a little bit of[br]algebraic manipulation and

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subtracting one multiple of it[br]from another, we're able to

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express that same exact[br]x as a fraction.

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Now this isn't in simplest[br]terms. I mean they're both

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definitely divisible by 2[br]and it looks like by 4.

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So you could put this in lowest[br]common form, but we

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don't care about that.

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All we care about is the fact[br]that we were able to represent

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x, we were able to represent[br]this number, as a fraction.

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As the ratio of two integers.

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So the number is[br]also rational.

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It is also rational.

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And this technique we[br]did, it doesn't only

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apply to this number.

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Any time you have a number that[br]has repeating digits, you

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could do this.

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So in general, repeating[br]digits are rational.

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The ones that are irrational are[br]the ones that never, ever,

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ever repeat, like pi.

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And so the other things, I think[br]it's pretty obvious,

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this isn't an integer.

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The integers are the[br]whole numbers that

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we're dealing with.

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So this is someplace in[br]between the integers.

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It's not a natural number or a[br]whole number, which depending

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on the context are viewed[br]as subsets of integers.

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So it's definitely[br]none of those.

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So it is real and[br]it is rational.

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That's all we can[br]say about it.