[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.49,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.49,0:00:05.43,Default,,0000,0000,0000,,What number sets does\Nthe number 3.4028
Dialogue: 0,0:00:05.43,0:00:07.33,Default,,0000,0000,0000,,repeating belong to?
Dialogue: 0,0:00:07.33,0:00:09.15,Default,,0000,0000,0000,,And before even answering the\Nquestion, let's just think
Dialogue: 0,0:00:09.15,0:00:10.69,Default,,0000,0000,0000,,about what this represents.
Dialogue: 0,0:00:10.69,0:00:13.00,Default,,0000,0000,0000,,And especially what this\Nline on top means.
Dialogue: 0,0:00:13.00,0:00:15.77,Default,,0000,0000,0000,,So this line on top means\Nthat the 28 just
Dialogue: 0,0:00:15.77,0:00:17.42,Default,,0000,0000,0000,,keep repeating forever.
Dialogue: 0,0:00:17.42,0:00:25.09,Default,,0000,0000,0000,,So I could express this number\Nas 3.4028, but the 28 just
Dialogue: 0,0:00:25.09,0:00:26.11,Default,,0000,0000,0000,,keep repeating.
Dialogue: 0,0:00:26.11,0:00:29.74,Default,,0000,0000,0000,,Just keep repeating on and\Non and on forever.
Dialogue: 0,0:00:29.74,0:00:32.30,Default,,0000,0000,0000,,I could just keep writing\Nthem forever and ever.
Dialogue: 0,0:00:32.30,0:00:35.21,Default,,0000,0000,0000,,And obviously, it's just easier\Nto write this line over
Dialogue: 0,0:00:35.21,0:00:37.62,Default,,0000,0000,0000,,the 28 to say that it\Nrepeats forever.
Dialogue: 0,0:00:37.62,0:00:41.29,Default,,0000,0000,0000,,Now let's think about what\Nnumber sets it belongs to.
Dialogue: 0,0:00:41.29,0:00:44.60,Default,,0000,0000,0000,,Well, the broadest number set\Nwe've dealt with so far is the
Dialogue: 0,0:00:44.60,0:00:45.33,Default,,0000,0000,0000,,real numbers.
Dialogue: 0,0:00:45.33,0:00:48.42,Default,,0000,0000,0000,,And this definitely belongs\Nto the real numbers.
Dialogue: 0,0:00:48.42,0:00:50.30,Default,,0000,0000,0000,,The real numbers is essentially\Nthe entire number
Dialogue: 0,0:00:50.30,0:00:51.99,Default,,0000,0000,0000,,line that we're used to using.
Dialogue: 0,0:00:51.99,0:00:55.66,Default,,0000,0000,0000,,And 3.4028 repeating sits\Nsomeplace over here.
Dialogue: 0,0:00:55.66,0:01:01.34,Default,,0000,0000,0000,,If this is negative 1, this\Nis 0, 1, 2, 3, 4.
Dialogue: 0,0:01:01.34,0:01:04.73,Default,,0000,0000,0000,,3.4028 is a little bit more\Nthan 3.4, a little
Dialogue: 0,0:01:04.73,0:01:06.49,Default,,0000,0000,0000,,bit less than 3.41.
Dialogue: 0,0:01:06.49,0:01:07.76,Default,,0000,0000,0000,,It would sit right over there.
Dialogue: 0,0:01:07.76,0:01:09.45,Default,,0000,0000,0000,,So it definitely sits\Non the number line.
Dialogue: 0,0:01:09.45,0:01:11.09,Default,,0000,0000,0000,,It's a real number.
Dialogue: 0,0:01:11.09,0:01:13.87,Default,,0000,0000,0000,,So it definitely is real.
Dialogue: 0,0:01:13.87,0:01:16.37,Default,,0000,0000,0000,,It definitely is\Na real number.
Dialogue: 0,0:01:16.37,0:01:19.08,Default,,0000,0000,0000,,But the not so obvious question\Nis whether it is a
Dialogue: 0,0:01:19.08,0:01:20.18,Default,,0000,0000,0000,,rational number.
Dialogue: 0,0:01:20.18,0:01:25.04,Default,,0000,0000,0000,,Remember, a rational number is\None that can be expressed as a
Dialogue: 0,0:01:25.04,0:01:26.89,Default,,0000,0000,0000,,rational expression\Nor as a fraction.
Dialogue: 0,0:01:26.89,0:01:34.39,Default,,0000,0000,0000,,If I were to tell you that p is\Nrational, that means that p
Dialogue: 0,0:01:34.39,0:01:37.84,Default,,0000,0000,0000,,can be expressed as the\Nratio of two integers.
Dialogue: 0,0:01:37.84,0:01:45.62,Default,,0000,0000,0000,,That means that p can be\Nexpressed as the ratio of two
Dialogue: 0,0:01:45.62,0:01:47.90,Default,,0000,0000,0000,,integers, m/n.
Dialogue: 0,0:01:47.90,0:01:50.96,Default,,0000,0000,0000,,So the question is, can I\Nexpress this as the ratio of
Dialogue: 0,0:01:50.96,0:01:51.41,Default,,0000,0000,0000,,two integers?
Dialogue: 0,0:01:51.41,0:01:52.41,Default,,0000,0000,0000,,Or another way to think\Nof it, can I
Dialogue: 0,0:01:52.41,0:01:53.99,Default,,0000,0000,0000,,express this as a fraction?
Dialogue: 0,0:01:53.99,0:01:58.51,Default,,0000,0000,0000,,And to do that, let's actually\Nexpress it as a fraction.
Dialogue: 0,0:01:58.51,0:02:01.31,Default,,0000,0000,0000,,Let's define x as being\Nequal to this number.
Dialogue: 0,0:02:01.31,0:02:09.96,Default,,0000,0000,0000,,So x is equal to 3.4028\Nrepeating.
Dialogue: 0,0:02:09.96,0:02:12.65,Default,,0000,0000,0000,,Let's think about\Nwhat 10,000x is.
Dialogue: 0,0:02:12.65,0:02:14.47,Default,,0000,0000,0000,,And the only reason why I want\N10,000x is because I want to
Dialogue: 0,0:02:14.47,0:02:16.96,Default,,0000,0000,0000,,move the decimal point all the\Nway to the right over here.
Dialogue: 0,0:02:16.96,0:02:21.71,Default,,0000,0000,0000,,So 10,000x.
Dialogue: 0,0:02:21.71,0:02:23.38,Default,,0000,0000,0000,,What is that going\Nto be equal to?
Dialogue: 0,0:02:23.38,0:02:26.35,Default,,0000,0000,0000,,Well every time you multiply\Nby a power of 10, you shift
Dialogue: 0,0:02:26.35,0:02:27.42,Default,,0000,0000,0000,,the decimal one to the right.
Dialogue: 0,0:02:27.42,0:02:29.79,Default,,0000,0000,0000,,10,000 is 10 to the\Nfourth power.
Dialogue: 0,0:02:29.79,0:02:31.78,Default,,0000,0000,0000,,So it's like shifting\Nthe decimal over to
Dialogue: 0,0:02:31.78,0:02:32.83,Default,,0000,0000,0000,,the right four spaces.
Dialogue: 0,0:02:32.83,0:02:36.40,Default,,0000,0000,0000,,1, 2, 3, 4.
Dialogue: 0,0:02:36.40,0:02:40.58,Default,,0000,0000,0000,,So it'll be 34,028.
Dialogue: 0,0:02:40.58,0:02:42.70,Default,,0000,0000,0000,,But these 28's just\Nkeep repeating.
Dialogue: 0,0:02:42.70,0:02:45.82,Default,,0000,0000,0000,,So you'll still have the 28's\Ngo on and on, and on and on,
Dialogue: 0,0:02:45.82,0:02:46.72,Default,,0000,0000,0000,,and on after that.
Dialogue: 0,0:02:46.72,0:02:49.55,Default,,0000,0000,0000,,They just all got shifted to the\Nleft of the decimal point
Dialogue: 0,0:02:49.55,0:02:50.43,Default,,0000,0000,0000,,by five spaces.
Dialogue: 0,0:02:50.43,0:02:51.07,Default,,0000,0000,0000,,You can view it that way.
Dialogue: 0,0:02:51.07,0:02:53.14,Default,,0000,0000,0000,,That makes sense.
Dialogue: 0,0:02:53.14,0:02:54.67,Default,,0000,0000,0000,,It's nearly 3 and 1/2.
Dialogue: 0,0:02:54.67,0:02:57.81,Default,,0000,0000,0000,,If you multiply by 10,000,\Nyou get almost 35,000.
Dialogue: 0,0:02:57.81,0:02:59.49,Default,,0000,0000,0000,,So that's 10,000x.
Dialogue: 0,0:02:59.49,0:03:00.97,Default,,0000,0000,0000,,Now, let's also think\Nabout 100x.
Dialogue: 0,0:03:00.97,0:03:04.34,Default,,0000,0000,0000,,And my whole exercise here is\NI want to get two numbers
Dialogue: 0,0:03:04.34,0:03:06.59,Default,,0000,0000,0000,,that, when I subtract them and\Nthey're in terms of x, the
Dialogue: 0,0:03:06.59,0:03:08.13,Default,,0000,0000,0000,,repeating part disappears.
Dialogue: 0,0:03:08.13,0:03:10.97,Default,,0000,0000,0000,,And then we can just treat them\Nas traditional numbers.
Dialogue: 0,0:03:10.97,0:03:13.26,Default,,0000,0000,0000,,So let's think about\Nwhat 100x is.
Dialogue: 0,0:03:13.26,0:03:15.53,Default,,0000,0000,0000,,100x.
Dialogue: 0,0:03:15.53,0:03:17.01,Default,,0000,0000,0000,,That moves this decimal point.
Dialogue: 0,0:03:17.01,0:03:18.37,Default,,0000,0000,0000,,Remember, the decimal point\Nwas here originally.
Dialogue: 0,0:03:18.37,0:03:20.86,Default,,0000,0000,0000,,It moves it over to the\Nright two spaces.
Dialogue: 0,0:03:20.86,0:03:24.83,Default,,0000,0000,0000,,So 100x would be 300-- Let\Nme write it like this.
Dialogue: 0,0:03:24.83,0:03:30.75,Default,,0000,0000,0000,,It would be 340.28 repeating.
Dialogue: 0,0:03:30.75,0:03:32.22,Default,,0000,0000,0000,,We could have put the 28\Nrepeating here, but it
Dialogue: 0,0:03:32.22,0:03:33.01,Default,,0000,0000,0000,,wouldn't have made\Nas much sense.
Dialogue: 0,0:03:33.01,0:03:34.67,Default,,0000,0000,0000,,You always want to write it\Nafter the decimal point.
Dialogue: 0,0:03:34.67,0:03:37.34,Default,,0000,0000,0000,,So we have to write 28 again to\Nshow that it is repeating.
Dialogue: 0,0:03:37.34,0:03:39.71,Default,,0000,0000,0000,,Now something interesting\Nis going on.
Dialogue: 0,0:03:39.71,0:03:42.40,Default,,0000,0000,0000,,These two numbers, they're\Njust multiples of x.
Dialogue: 0,0:03:42.40,0:03:45.79,Default,,0000,0000,0000,,And if I subtract the bottom one\Nfrom the top one, what's
Dialogue: 0,0:03:45.79,0:03:46.71,Default,,0000,0000,0000,,going to happen?
Dialogue: 0,0:03:46.71,0:03:48.53,Default,,0000,0000,0000,,Well the repeating part\Nis going to disappear.
Dialogue: 0,0:03:48.53,0:03:49.17,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:03:49.17,0:03:52.28,Default,,0000,0000,0000,,Let's do that on both sides\Nof this equation.
Dialogue: 0,0:03:52.28,0:03:53.23,Default,,0000,0000,0000,,Let's do it.
Dialogue: 0,0:03:53.23,0:03:58.21,Default,,0000,0000,0000,,So on the left-hand side of this\Nequation, 10,000x minus
Dialogue: 0,0:03:58.21,0:04:03.62,Default,,0000,0000,0000,,100x is going to be 9,900x.
Dialogue: 0,0:04:03.62,0:04:06.96,Default,,0000,0000,0000,,And on the right-hand side,\Nlet's see-- The decimal part
Dialogue: 0,0:04:06.96,0:04:08.23,Default,,0000,0000,0000,,will cancel out.
Dialogue: 0,0:04:08.23,0:04:12.03,Default,,0000,0000,0000,,And we just have to figure out\Nwhat 34,028 minus 340 is.
Dialogue: 0,0:04:12.03,0:04:14.12,Default,,0000,0000,0000,,So let's just figure this out.
Dialogue: 0,0:04:14.12,0:04:16.01,Default,,0000,0000,0000,,8 is larger than 0, so we\Nwon't have to do any
Dialogue: 0,0:04:16.01,0:04:16.65,Default,,0000,0000,0000,,regrouping there.
Dialogue: 0,0:04:16.65,0:04:19.77,Default,,0000,0000,0000,,2 is less than 4.
Dialogue: 0,0:04:19.77,0:04:22.20,Default,,0000,0000,0000,,So we will have to do some\Nregrouping, but we can't
Dialogue: 0,0:04:22.20,0:04:25.51,Default,,0000,0000,0000,,borrow yet because we\Nhave a 0 over there.
Dialogue: 0,0:04:25.51,0:04:27.71,Default,,0000,0000,0000,,And 0 is less than 3, so we\Nhave to do some regrouping
Dialogue: 0,0:04:27.71,0:04:29.00,Default,,0000,0000,0000,,there or some borrowing.
Dialogue: 0,0:04:29.00,0:04:31.77,Default,,0000,0000,0000,,So let's borrow from\Nthe 4 first.
Dialogue: 0,0:04:31.77,0:04:36.59,Default,,0000,0000,0000,,So if we borrow from the 4, this\Nbecomes a 3 and then this
Dialogue: 0,0:04:36.59,0:04:38.14,Default,,0000,0000,0000,,becomes a 10.
Dialogue: 0,0:04:38.14,0:04:40.46,Default,,0000,0000,0000,,And then the 2 can now\Nborrow from the 10.
Dialogue: 0,0:04:40.46,0:04:44.09,Default,,0000,0000,0000,,This becomes a 9 and\Nthis becomes a 12.
Dialogue: 0,0:04:44.09,0:04:45.82,Default,,0000,0000,0000,,And now we can do\Nthe subtraction.
Dialogue: 0,0:04:45.82,0:04:48.39,Default,,0000,0000,0000,,8 minus 0 is 8.
Dialogue: 0,0:04:48.39,0:04:51.11,Default,,0000,0000,0000,,12 minus 4 is 8.
Dialogue: 0,0:04:51.11,0:04:53.88,Default,,0000,0000,0000,,9 minus 3 is 6.
Dialogue: 0,0:04:53.88,0:04:55.92,Default,,0000,0000,0000,,3 minus nothing is 3.
Dialogue: 0,0:04:55.92,0:04:57.95,Default,,0000,0000,0000,,3 minus nothing is 3.
Dialogue: 0,0:04:57.95,0:05:05.32,Default,,0000,0000,0000,,So 9,900x is equal to 33,688.
Dialogue: 0,0:05:05.32,0:05:09.18,Default,,0000,0000,0000,,We just subtracted 340\Nfrom this up here.
Dialogue: 0,0:05:09.18,0:05:13.11,Default,,0000,0000,0000,,So we get 33,688.
Dialogue: 0,0:05:13.11,0:05:15.71,Default,,0000,0000,0000,,Now, if we want to solve\Nfor x, we just divide
Dialogue: 0,0:05:15.71,0:05:21.61,Default,,0000,0000,0000,,both sides by 9,900.
Dialogue: 0,0:05:21.61,0:05:23.99,Default,,0000,0000,0000,,Divide the left by 9,900.
Dialogue: 0,0:05:23.99,0:05:26.90,Default,,0000,0000,0000,,Divide the right by 9,900.
Dialogue: 0,0:05:26.90,0:05:28.00,Default,,0000,0000,0000,,And then, what are\Nwe left with?
Dialogue: 0,0:05:28.00,0:05:36.85,Default,,0000,0000,0000,,We're left with x is equal\Nto 33,688 over 9,900.
Dialogue: 0,0:05:36.85,0:05:38.55,Default,,0000,0000,0000,,Now what's the big\Ndeal about this?
Dialogue: 0,0:05:38.55,0:05:41.90,Default,,0000,0000,0000,,Well, x was this number. x was\Nthis number that we started
Dialogue: 0,0:05:41.90,0:05:44.58,Default,,0000,0000,0000,,off with, this number that\Njust kept on repeating.
Dialogue: 0,0:05:44.58,0:05:47.50,Default,,0000,0000,0000,,And by doing a little bit of\Nalgebraic manipulation and
Dialogue: 0,0:05:47.50,0:05:49.66,Default,,0000,0000,0000,,subtracting one multiple of it\Nfrom another, we're able to
Dialogue: 0,0:05:49.66,0:05:52.53,Default,,0000,0000,0000,,express that same exact\Nx as a fraction.
Dialogue: 0,0:05:52.53,0:05:55.78,Default,,0000,0000,0000,,Now this isn't in simplest\Nterms. I mean they're both
Dialogue: 0,0:05:55.78,0:05:58.90,Default,,0000,0000,0000,,definitely divisible by 2\Nand it looks like by 4.
Dialogue: 0,0:05:58.90,0:06:01.96,Default,,0000,0000,0000,,So you could put this in lowest\Ncommon form, but we
Dialogue: 0,0:06:01.96,0:06:02.91,Default,,0000,0000,0000,,don't care about that.
Dialogue: 0,0:06:02.91,0:06:05.06,Default,,0000,0000,0000,,All we care about is the fact\Nthat we were able to represent
Dialogue: 0,0:06:05.06,0:06:09.05,Default,,0000,0000,0000,,x, we were able to represent\Nthis number, as a fraction.
Dialogue: 0,0:06:09.05,0:06:11.62,Default,,0000,0000,0000,,As the ratio of two integers.
Dialogue: 0,0:06:11.62,0:06:14.72,Default,,0000,0000,0000,,So the number is\Nalso rational.
Dialogue: 0,0:06:14.72,0:06:16.55,Default,,0000,0000,0000,,It is also rational.
Dialogue: 0,0:06:16.55,0:06:19.01,Default,,0000,0000,0000,,And this technique we\Ndid, it doesn't only
Dialogue: 0,0:06:19.01,0:06:20.70,Default,,0000,0000,0000,,apply to this number.
Dialogue: 0,0:06:20.70,0:06:24.37,Default,,0000,0000,0000,,Any time you have a number that\Nhas repeating digits, you
Dialogue: 0,0:06:24.37,0:06:25.00,Default,,0000,0000,0000,,could do this.
Dialogue: 0,0:06:25.00,0:06:27.53,Default,,0000,0000,0000,,So in general, repeating\Ndigits are rational.
Dialogue: 0,0:06:27.53,0:06:30.09,Default,,0000,0000,0000,,The ones that are irrational are\Nthe ones that never, ever,
Dialogue: 0,0:06:30.09,0:06:32.86,Default,,0000,0000,0000,,ever repeat, like pi.
Dialogue: 0,0:06:32.86,0:06:34.59,Default,,0000,0000,0000,,And so the other things, I think\Nit's pretty obvious,
Dialogue: 0,0:06:34.59,0:06:35.81,Default,,0000,0000,0000,,this isn't an integer.
Dialogue: 0,0:06:35.81,0:06:37.41,Default,,0000,0000,0000,,The integers are the\Nwhole numbers that
Dialogue: 0,0:06:37.41,0:06:38.02,Default,,0000,0000,0000,,we're dealing with.
Dialogue: 0,0:06:38.02,0:06:40.39,Default,,0000,0000,0000,,So this is someplace in\Nbetween the integers.
Dialogue: 0,0:06:40.39,0:06:43.36,Default,,0000,0000,0000,,It's not a natural number or a\Nwhole number, which depending
Dialogue: 0,0:06:43.36,0:06:46.24,Default,,0000,0000,0000,,on the context are viewed\Nas subsets of integers.
Dialogue: 0,0:06:46.24,0:06:47.36,Default,,0000,0000,0000,,So it's definitely\Nnone of those.
Dialogue: 0,0:06:47.36,0:06:49.11,Default,,0000,0000,0000,,So it is real and\Nit is rational.
Dialogue: 0,0:06:49.11,0:06:51.46,Default,,0000,0000,0000,,That's all we can\Nsay about it.