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Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.08,0:00:05.14,Default,,0000,0000,0000,,Why does minus one times minus\None equal plus one or more
Dialogue: 0,0:00:05.14,0:00:09.79,Default,,0000,0000,0000,,generally why when we take a\Nnegative number and multiply it
Dialogue: 0,0:00:09.79,0:00:14.01,Default,,0000,0000,0000,,by another negative number, do\Nwe get a positive answer?
Dialogue: 0,0:00:15.08,0:00:19.21,Default,,0000,0000,0000,,This is a question that has\Nplagued every student of
Dialogue: 0,0:00:19.21,0:00:21.80,Default,,0000,0000,0000,,arithmetic. It disturbs us.
Dialogue: 0,0:00:22.66,0:00:27.34,Default,,0000,0000,0000,,It disturbs us because it seems\Nto lie outside our familiar
Dialogue: 0,0:00:27.34,0:00:29.46,Default,,0000,0000,0000,,experience. It doesn't seem to
Dialogue: 0,0:00:29.46,0:00:36.01,Default,,0000,0000,0000,,fit. Let me explain by reviewing\Nthe rules of arithmetic for
Dialogue: 0,0:00:36.01,0:00:38.17,Default,,0000,0000,0000,,multiplying together 2 numbers.
Dialogue: 0,0:00:38.72,0:00:43.80,Default,,0000,0000,0000,,And we should start by\Nmultiplying 2 positive numbers.
Dialogue: 0,0:00:44.61,0:00:46.69,Default,,0000,0000,0000,,Multiply positive 5.
Dialogue: 0,0:00:47.30,0:00:49.07,Default,,0000,0000,0000,,By plus 3.
Dialogue: 0,0:00:49.88,0:00:51.29,Default,,0000,0000,0000,,And we know the answer.
Dialogue: 0,0:00:51.88,0:00:53.37,Default,,0000,0000,0000,,Is 15.
Dialogue: 0,0:00:54.48,0:01:00.02,Default,,0000,0000,0000,,This. We're comfortable with it\Nmatches our experience. When,
Dialogue: 0,0:01:00.02,0:01:05.39,Default,,0000,0000,0000,,for example, were counting\Ncounting money, so we can think
Dialogue: 0,0:01:05.39,0:01:11.29,Default,,0000,0000,0000,,of 3 * 5 as representing three\Npiles, three separate piles.
Dialogue: 0,0:01:11.99,0:01:17.16,Default,,0000,0000,0000,,In each pile, there being 51\Npound coins. So in total when we
Dialogue: 0,0:01:17.16,0:01:19.55,Default,,0000,0000,0000,,have them all together, we have
Dialogue: 0,0:01:19.55,0:01:24.80,Default,,0000,0000,0000,,15. So 3 * 5 is 15\Nwere quite happy with.
Dialogue: 0,0:01:25.69,0:01:31.63,Default,,0000,0000,0000,,Next Let's see what happens when\Nwe take a negative number.
Dialogue: 0,0:01:32.15,0:01:36.90,Default,,0000,0000,0000,,Negative one for example, and\NI'll put brackets around for
Dialogue: 0,0:01:36.90,0:01:41.51,Default,,0000,0000,0000,,convenience. When we multiply\Nnegative one by one, the answer
Dialogue: 0,0:01:41.51,0:01:42.71,Default,,0000,0000,0000,,is minus one.
Dialogue: 0,0:01:43.98,0:01:50.68,Default,,0000,0000,0000,,If we then multiply negative one\Nby two, the answer is minus 2.
Dialogue: 0,0:01:51.66,0:01:56.32,Default,,0000,0000,0000,,And we can go on multiply\Nminus one by three, and the
Dialogue: 0,0:01:56.32,0:01:57.87,Default,,0000,0000,0000,,answer is minus 3.
Dialogue: 0,0:01:58.90,0:02:04.06,Default,,0000,0000,0000,,And you can see where developing\Nwhat really is a Times table for
Dialogue: 0,0:02:04.06,0:02:08.03,Default,,0000,0000,0000,,minus one, but where convertible\Nwith this, because again it
Dialogue: 0,0:02:08.03,0:02:12.80,Default,,0000,0000,0000,,matches our experience. We can\Nthink of it again in terms of
Dialogue: 0,0:02:12.80,0:02:14.78,Default,,0000,0000,0000,,our bank account when dealing
Dialogue: 0,0:02:14.78,0:02:18.00,Default,,0000,0000,0000,,with money. We can think of 1 *
Dialogue: 0,0:02:18.00,0:02:24.12,Default,,0000,0000,0000,,1. As taking £1 out of our\Naccount on one occasion only and
Dialogue: 0,0:02:24.12,0:02:27.02,Default,,0000,0000,0000,,so our account is in deficit by
Dialogue: 0,0:02:27.02,0:02:33.96,Default,,0000,0000,0000,,one pound. Two times minus one\Nwe can think of as taking £1 out
Dialogue: 0,0:02:33.96,0:02:39.11,Default,,0000,0000,0000,,of our account on two separate\Noccasions on what happens is our
Dialogue: 0,0:02:39.11,0:02:43.40,Default,,0000,0000,0000,,account is in deficit by 2\Npounds, and so on.
Dialogue: 0,0:02:44.05,0:02:48.91,Default,,0000,0000,0000,,I multiplying a positive number\Nby a negative number, giving
Dialogue: 0,0:02:48.91,0:02:52.31,Default,,0000,0000,0000,,rise to a negative answer is OK,
Dialogue: 0,0:02:52.31,0:02:57.83,Default,,0000,0000,0000,,it fits. What then when we\Nmultiply 2 negative numbers
Dialogue: 0,0:02:57.83,0:03:05.25,Default,,0000,0000,0000,,together? Minus one times minus\None, but so the answer is
Dialogue: 0,0:03:05.25,0:03:08.61,Default,,0000,0000,0000,,plus one. Why is
Dialogue: 0,0:03:08.61,0:03:11.88,Default,,0000,0000,0000,,this so? Where on earth does it
Dialogue: 0,0:03:11.88,0:03:15.45,Default,,0000,0000,0000,,come from? It didn't seem\Nto correspond to anything
Dialogue: 0,0:03:15.45,0:03:16.85,Default,,0000,0000,0000,,in our familiar experience.
Dialogue: 0,0:03:17.94,0:03:19.82,Default,,0000,0000,0000,,So what can you do?
Dialogue: 0,0:03:20.37,0:03:23.25,Default,,0000,0000,0000,,Well, you could phone a friend.\NThat's if you've got a friend
Dialogue: 0,0:03:23.25,0:03:24.45,Default,,0000,0000,0000,,who is a math teacher.
Dialogue: 0,0:03:25.18,0:03:26.87,Default,,0000,0000,0000,,Or you could ask the math
Dialogue: 0,0:03:26.87,0:03:30.90,Default,,0000,0000,0000,,teacher. And I recall doing\Nprecisely that many, many years
Dialogue: 0,0:03:30.90,0:03:35.17,Default,,0000,0000,0000,,ago. I asked him why does minus\None times minus one equals plus
Dialogue: 0,0:03:35.17,0:03:40.06,Default,,0000,0000,0000,,one. And what he said was just\Naccept it. For now. You'll
Dialogue: 0,0:03:40.06,0:03:41.44,Default,,0000,0000,0000,,understand it later on.
Dialogue: 0,0:03:43.17,0:03:46.39,Default,,0000,0000,0000,,Very unsatisfactory, I\Nthought I ask a question but
Dialogue: 0,0:03:46.39,0:03:48.18,Default,,0000,0000,0000,,I don't get an answer.
Dialogue: 0,0:03:49.32,0:03:49.80,Default,,0000,0000,0000,,But
Dialogue: 0,0:03:51.08,0:03:55.59,Default,,0000,0000,0000,,When you think of it, this\Nhappens very often in life. A
Dialogue: 0,0:03:55.59,0:03:59.73,Default,,0000,0000,0000,,question is posed, but the\Nanswer is out of reach. For
Dialogue: 0,0:03:59.73,0:04:04.62,Default,,0000,0000,0000,,example, when a small child asks\Nher parents what is a black hole
Dialogue: 0,0:04:04.62,0:04:06.87,Default,,0000,0000,0000,,or where on earth where is
Dialogue: 0,0:04:06.87,0:04:11.32,Default,,0000,0000,0000,,Infinity? The answer isn't\Nnecessarily clear. In order
Dialogue: 0,0:04:11.32,0:04:14.63,Default,,0000,0000,0000,,to appreciate the answer,\Nmore information, more
Dialogue: 0,0:04:14.63,0:04:16.04,Default,,0000,0000,0000,,knowledge is required.
Dialogue: 0,0:04:17.07,0:04:23.36,Default,,0000,0000,0000,,So let's return to minus one\Ntimes minus one equal plus one.
Dialogue: 0,0:04:24.26,0:04:29.56,Default,,0000,0000,0000,,What extra information is\Nrequired in order to
Dialogue: 0,0:04:29.56,0:04:30.89,Default,,0000,0000,0000,,understand this?
Dialogue: 0,0:04:32.19,0:04:37.94,Default,,0000,0000,0000,,It turns out that we need 2\Nextra bits of information, 2
Dialogue: 0,0:04:37.94,0:04:39.38,Default,,0000,0000,0000,,rules of arithmetic.
Dialogue: 0,0:04:40.38,0:04:45.64,Default,,0000,0000,0000,,And these rules are one the\Nrule of precedence.
Dialogue: 0,0:04:45.67,0:04:50.49,Default,,0000,0000,0000,,What is presidents? Well,\Npresidents tells us.
Dialogue: 0,0:04:51.04,0:04:53.79,Default,,0000,0000,0000,,Which operation to do first?
Dialogue: 0,0:04:54.42,0:04:56.58,Default,,0000,0000,0000,,Well, next in any given
Dialogue: 0,0:04:56.58,0:05:02.59,Default,,0000,0000,0000,,calculation. So if we look at an\Nexample with positive numbers 3.
Dialogue: 0,0:05:03.10,0:05:06.91,Default,,0000,0000,0000,,Times bracket 4 + 2. You can see
Dialogue: 0,0:05:06.91,0:05:11.31,Default,,0000,0000,0000,,that. We've got multiple\Nmultiplication to do, and we
Dialogue: 0,0:05:11.31,0:05:16.63,Default,,0000,0000,0000,,have an addition to do. Which do\Nwe do first? President says you
Dialogue: 0,0:05:16.63,0:05:21.13,Default,,0000,0000,0000,,do what's in the brackets first.\N4 + 2 is 6.
Dialogue: 0,0:05:21.64,0:05:24.42,Default,,0000,0000,0000,,3 * 6.
Dialogue: 0,0:05:24.50,0:05:28.09,Default,,0000,0000,0000,,Is 18. No problem.
Dialogue: 0,0:05:29.50,0:05:30.54,Default,,0000,0000,0000,,So that's one.
Dialogue: 0,0:05:31.52,0:05:35.26,Default,,0000,0000,0000,,Piece of information that we're\Ngoing to make yourself a second
Dialogue: 0,0:05:35.26,0:05:37.64,Default,,0000,0000,0000,,that we're going to make use of.
Dialogue: 0,0:05:38.36,0:05:42.92,Default,,0000,0000,0000,,Is the fact that multiplication\Nis distributive over addition?
Dialogue: 0,0:05:43.95,0:05:48.40,Default,,0000,0000,0000,,Now what does that mean?\NMultiplication is distributive
Dialogue: 0,0:05:48.40,0:05:53.96,Default,,0000,0000,0000,,over addition. Well, that's best\Nappreciated again by an example.
Dialogue: 0,0:05:54.55,0:05:57.58,Default,,0000,0000,0000,,And we can use the same example\Nthat we've got here.
Dialogue: 0,0:05:58.11,0:06:01.90,Default,,0000,0000,0000,,3 * 4 + 2.
Dialogue: 0,0:06:01.90,0:06:04.96,Default,,0000,0000,0000,,It involves multiplication and
Dialogue: 0,0:06:04.96,0:06:10.81,Default,,0000,0000,0000,,addition. If multiplication is\Ndistributive over addition, it
Dialogue: 0,0:06:10.81,0:06:16.81,Default,,0000,0000,0000,,means that this calculation is\Nequivalent to multiplying 3 by
Dialogue: 0,0:06:16.81,0:06:22.85,Default,,0000,0000,0000,,4. And then adding three by\Ntwo 3 * 2.
Dialogue: 0,0:06:23.40,0:06:29.97,Default,,0000,0000,0000,,And we can check\Nthat this is so.
Dialogue: 0,0:06:30.02,0:06:32.33,Default,,0000,0000,0000,,We've already worked out three.
Dialogue: 0,0:06:32.92,0:06:37.25,Default,,0000,0000,0000,,Times bracket 4 + 2 using the\Nrule of presidents and the
Dialogue: 0,0:06:37.25,0:06:42.85,Default,,0000,0000,0000,,answers 18. On the right hand\Nside with 3 * 4, which is 12.
Dialogue: 0,0:06:43.88,0:06:50.52,Default,,0000,0000,0000,,Also we've got 3 * 2 which is\N612 + 6 is 18, right outside
Dialogue: 0,0:06:50.52,0:06:51.85,Default,,0000,0000,0000,,equals left outside.
Dialogue: 0,0:06:52.37,0:06:55.53,Default,,0000,0000,0000,,So this fact that\Nmultiplication is
Dialogue: 0,0:06:55.53,0:06:59.75,Default,,0000,0000,0000,,distributive over addition\Nworks for the numbers 3,
Dialogue: 0,0:06:59.75,0:07:01.33,Default,,0000,0000,0000,,four and two.
Dialogue: 0,0:07:02.71,0:07:07.48,Default,,0000,0000,0000,,As a second example, I'm going\Nto show it works when we have a
Dialogue: 0,0:07:07.48,0:07:11.39,Default,,0000,0000,0000,,negative number. Say minus one.
Dialogue: 0,0:07:12.98,0:07:16.84,Default,,0000,0000,0000,,Times 2 + 1.
Dialogue: 0,0:07:17.59,0:07:24.28,Default,,0000,0000,0000,,Got two and one\Nboth positive numbers.
Dialogue: 0,0:07:24.79,0:07:26.87,Default,,0000,0000,0000,,And we multiply that bracket by
Dialogue: 0,0:07:26.87,0:07:32.54,Default,,0000,0000,0000,,minus one. If multiplication is\Ndistributive over addition, it
Dialogue: 0,0:07:32.54,0:07:36.04,Default,,0000,0000,0000,,means that minus 1 * 2.
Dialogue: 0,0:07:36.95,0:07:41.05,Default,,0000,0000,0000,,It means that the left hand side\Nis minus 1 * 2.
Dialogue: 0,0:07:41.99,0:07:45.93,Default,,0000,0000,0000,,Plus minus 1 * 1.
Dialogue: 0,0:07:46.58,0:07:53.40,Default,,0000,0000,0000,,There are precedents, the left\Nhand side we do what's in
Dialogue: 0,0:07:53.40,0:08:00.22,Default,,0000,0000,0000,,the brackets first 2 plus one\Nis 3 times minus one.
Dialogue: 0,0:08:00.89,0:08:01.97,Default,,0000,0000,0000,,Is minus 3.
Dialogue: 0,0:08:03.17,0:08:06.78,Default,,0000,0000,0000,,On the right hand side, minus 1\N* 2 is minus 2.
Dialogue: 0,0:08:07.41,0:08:10.13,Default,,0000,0000,0000,,Minus 1 * 1 is minus one.
Dialogue: 0,0:08:10.86,0:08:14.90,Default,,0000,0000,0000,,And you can see there on\Nthe right hand side we have
Dialogue: 0,0:08:14.90,0:08:17.94,Default,,0000,0000,0000,,minus two and minus one\Nadded together, which is
Dialogue: 0,0:08:17.94,0:08:20.97,Default,,0000,0000,0000,,minus three. So left hand\Nside equals right hand
Dialogue: 0,0:08:20.97,0:08:23.33,Default,,0000,0000,0000,,side. I multiplication is\Ndistributive over addition
Dialogue: 0,0:08:23.33,0:08:26.02,Default,,0000,0000,0000,,for the numbers minus 1,\Ntwo and one.
Dialogue: 0,0:08:27.20,0:08:29.34,Default,,0000,0000,0000,,Now the key.
Dialogue: 0,0:08:30.12,0:08:35.03,Default,,0000,0000,0000,,For understanding why minus one\Ntimes minus one equals plus one.
Dialogue: 0,0:08:35.68,0:08:39.85,Default,,0000,0000,0000,,Is that we insist that\Nmultiplication is distributive
Dialogue: 0,0:08:39.85,0:08:42.45,Default,,0000,0000,0000,,over addition for all numbers.
Dialogue: 0,0:08:43.28,0:08:49.75,Default,,0000,0000,0000,,Whether negative or positive and\Nwhat we need to consider is
Dialogue: 0,0:08:49.75,0:08:52.10,Default,,0000,0000,0000,,a particular calculation minus
Dialogue: 0,0:08:52.10,0:08:57.74,Default,,0000,0000,0000,,one times. Same as we had up\Nthere, but instead of two we put
Dialogue: 0,0:08:57.74,0:09:01.96,Default,,0000,0000,0000,,minus one. Then if\Nmultiplication is distributive
Dialogue: 0,0:09:01.96,0:09:08.62,Default,,0000,0000,0000,,over addition, this is equal to\Nminus one times minus one.
Dialogue: 0,0:09:09.28,0:09:16.87,Default,,0000,0000,0000,,Plus\NMinus one times plus one.
Dialogue: 0,0:09:18.90,0:09:23.79,Default,,0000,0000,0000,,Now for the left hand side using\Nprecedents we do what's in the
Dialogue: 0,0:09:23.79,0:09:26.80,Default,,0000,0000,0000,,brackets first minus one plus\None is 0.
Dialogue: 0,0:09:27.34,0:09:32.02,Default,,0000,0000,0000,,Anything times zero, are you\Nminus one is itself Sarah, so
Dialogue: 0,0:09:32.02,0:09:37.12,Default,,0000,0000,0000,,I left outside is 0 on the\Nright hand side. The first
Dialogue: 0,0:09:37.12,0:09:41.79,Default,,0000,0000,0000,,term is minus one times minus\None, which is what we're
Dialogue: 0,0:09:41.79,0:09:43.06,Default,,0000,0000,0000,,trying to determine.
Dialogue: 0,0:09:44.32,0:09:48.37,Default,,0000,0000,0000,,And the end term. The last term\Nminus 1 * 1 is minus one.
Dialogue: 0,0:09:49.07,0:09:55.20,Default,,0000,0000,0000,,So you can see that if we now\Ntake one to the left hand side.
Dialogue: 0,0:09:55.21,0:10:01.96,Default,,0000,0000,0000,,We have shown that minus one\Ntimes minus one is equal to plus
Dialogue: 0,0:10:01.96,0:10:07.68,Default,,0000,0000,0000,,one. Why this result follows as\Na direct consequence of these
Dialogue: 0,0:10:07.68,0:10:13.38,Default,,0000,0000,0000,,two rules of arithmetic. The\Nrule of precedence and the rule
Dialogue: 0,0:10:13.38,0:10:17.52,Default,,0000,0000,0000,,that multiplication has to be\Ndistributed over addition.
Dialogue: 0,0:10:18.62,0:10:20.01,Default,,0000,0000,0000,,So you see.
Dialogue: 0,0:10:20.58,0:10:24.69,Default,,0000,0000,0000,,But my old school teacher, Mr\NDennison, was quite right when
Dialogue: 0,0:10:24.69,0:10:28.72,Default,,0000,0000,0000,,he said. Accept it for\Nnow. Lab you will only
Dialogue: 0,0:10:28.72,0:10:30.00,Default,,0000,0000,0000,,understand it later on.