[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:03.18,Default,,0000,0000,0000,,So you, as the ancient philosopher in mathematics
Dialogue: 0,0:00:03.18,0:00:07.18,Default,,0000,0000,0000,,have concluded in order for the multiplication of positive and negative numbers to be
Dialogue: 0,0:00:07.18,0:00:10.18,Default,,0000,0000,0000,,consistent with everything you've been constructing so far
Dialogue: 0,0:00:10.18,0:00:13.50,Default,,0000,0000,0000,,with all the other properties of multiplication that you know so far
Dialogue: 0,0:00:13.50,0:00:17.18,Default,,0000,0000,0000,,that you need a negative number times a positive
Dialogue: 0,0:00:17.18,0:00:20.77,Default,,0000,0000,0000,,number or a positive times a negative to give you a negative number
Dialogue: 0,0:00:20.77,0:00:24.18,Default,,0000,0000,0000,,and a negative times a negative
Dialogue: 0,0:00:24.18,0:00:27.85,Default,,0000,0000,0000,,to give you a positive number and so you accept
Dialogue: 0,0:00:27.85,0:00:30.84,Default,,0000,0000,0000,,it's all consistent so far.. this deal does not make complete concrete
Dialogue: 0,0:00:30.84,0:00:35.77,Default,,0000,0000,0000,,sense to you, you want to have a slightly deeper institution than just having to accept its
Dialogue: 0,0:00:35.77,0:00:40.37,Default,,0000,0000,0000,,consistent with the distributive property and whatever else and so you try another
Dialogue: 0,0:00:40.37,0:00:45.38,Default,,0000,0000,0000,,thought experiment, you say "well what is just a basic multiplication way of doing it?"
Dialogue: 0,0:00:45.38,0:00:47.37,Default,,0000,0000,0000,,So if I say, two times
Dialogue: 0,0:00:47.37,0:00:51.18,Default,,0000,0000,0000,,three, one way to
Dialogue: 0,0:00:51.18,0:00:55.10,Default,,0000,0000,0000,,to conceptualize is basic multiplication is really repeating
Dialogue: 0,0:00:55.10,0:00:58.04,Default,,0000,0000,0000,,addition, so you could view this as two threes
Dialogue: 0,0:00:58.04,0:01:02.35,Default,,0000,0000,0000,,so let me write three plus three
Dialogue: 0,0:01:02.35,0:01:05.70,Default,,0000,0000,0000,,and notice there are two of them, there are two of these
Dialogue: 0,0:01:05.70,0:01:09.77,Default,,0000,0000,0000,,or you could view this as three twos, and so this is the same thing as
Dialogue: 0,0:01:09.77,0:01:13.43,Default,,0000,0000,0000,,two plus two plus two and there are
Dialogue: 0,0:01:13.43,0:01:17.10,Default,,0000,0000,0000,,three of them, and either way you can conceptualize
Dialogue: 0,0:01:17.10,0:01:20.60,Default,,0000,0000,0000,,as you get the same exact answer. This is going to be equal
Dialogue: 0,0:01:20.60,0:01:24.60,Default,,0000,0000,0000,,to six, fair enough!
Dialogue: 0,0:01:24.60,0:01:27.57,Default,,0000,0000,0000,,Now, you knew this before you even tried to tackle negative numbers.
Dialogue: 0,0:01:27.57,0:01:30.52,Default,,0000,0000,0000,,Now let's try to make one of these negatives and see what
Dialogue: 0,0:01:30.52,0:01:33.37,Default,,0000,0000,0000,,happens. Let's do two
Dialogue: 0,0:01:33.37,0:01:35.52,Default,,0000,0000,0000,,times negative
Dialogue: 0,0:01:35.52,0:01:42.10,Default,,0000,0000,0000,,three, I want to make the negative into a different color. Two times
Dialogue: 0,0:01:42.10,0:01:46.31,Default,,0000,0000,0000,,negative three.
Dialogue: 0,0:01:46.31,0:01:49.85,Default,,0000,0000,0000,,Well, one way you could view this is the same analogy
Dialogue: 0,0:01:49.85,0:01:52.85,Default,,0000,0000,0000,,here, it's negative three twice so it would be
Dialogue: 0,0:01:52.85,0:01:56.77,Default,,0000,0000,0000,,negative.. I'll try to color code it
Dialogue: 0,0:01:56.77,0:02:01.04,Default,,0000,0000,0000,,negative three and then another negative
Dialogue: 0,0:02:01.04,0:02:05.18,Default,,0000,0000,0000,,three or you could say negative three minus three
Dialogue: 0,0:02:05.18,0:02:08.57,Default,,0000,0000,0000,,or, and this is the interesting thing, instead of
Dialogue: 0,0:02:08.57,0:02:11.04,Default,,0000,0000,0000,,over here there's a two times positive three
Dialogue: 0,0:02:11.04,0:02:14.31,Default,,0000,0000,0000,,you added two, three times.
Dialogue: 0,0:02:14.31,0:02:16.26,Default,,0000,0000,0000,,But since here is two times negative three
Dialogue: 0,0:02:16.26,0:02:19.04,Default,,0000,0000,0000,,you could also imagine you are going to subtract two, three times
Dialogue: 0,0:02:19.04,0:02:21.70,Default,,0000,0000,0000,,So instead of up here, I could
Dialogue: 0,0:02:21.70,0:02:26.52,Default,,0000,0000,0000,,written two plus two plus two because this is a positive
Dialogue: 0,0:02:26.52,0:02:29.44,Default,,0000,0000,0000,,two right over here, but since we're doing this over negative three
Dialogue: 0,0:02:29.44,0:02:33.70,Default,,0000,0000,0000,,we could imagine subtracting two, three times, so this would be
Dialogue: 0,0:02:33.70,0:02:37.93,Default,,0000,0000,0000,,subtracting two (repeated)
Dialogue: 0,0:02:37.93,0:02:43.19,Default,,0000,0000,0000,,subtract another two right over here, subtract another two
Dialogue: 0,0:02:43.19,0:02:46.10,Default,,0000,0000,0000,,and then you subtract another two
Dialogue: 0,0:02:46.10,0:02:54.94,Default,,0000,0000,0000,,notice you did it, once again, you did it
Dialogue: 0,0:02:54.94,0:02:59.93,Default,,0000,0000,0000,,three times, so this is a negative three, so essentially you are subtracting
Dialogue: 0,0:02:59.93,0:03:03.77,Default,,0000,0000,0000,,two, three times. And either way, you can conceptualize
Dialogue: 0,0:03:03.77,0:03:07.26,Default,,0000,0000,0000,,right over here, you are going to get negative six
Dialogue: 0,0:03:07.26,0:03:10.19,Default,,0000,0000,0000,,negative six is the answer.
Dialogue: 0,0:03:10.19,0:03:16.26,Default,,0000,0000,0000,,Now, so you are already starting to feel better about this part right over here
Dialogue: 0,0:03:16.26,0:03:18.37,Default,,0000,0000,0000,,negative times a positive, or a positive times a negative
Dialogue: 0,0:03:18.37,0:03:21.60,Default,,0000,0000,0000,,is going to give you a negative. Now lets take to the really un-intuitive one
Dialogue: 0,0:03:21.60,0:03:24.68,Default,,0000,0000,0000,,and measure negative times a negative, and all of a sudden negatives kind of cancel
Dialogue: 0,0:03:24.68,0:03:28.04,Default,,0000,0000,0000,,to give you a positive. Now why is that the case? Well we can just build from
Dialogue: 0,0:03:28.04,0:03:30.97,Default,,0000,0000,0000,,this example right over here. Let's say we had
Dialogue: 0,0:03:30.97,0:03:35.93,Default,,0000,0000,0000,,a negative two, lets say we had
Dialogue: 0,0:03:35.93,0:03:38.10,Default,,0000,0000,0000,,negative two, let me do it a different color,
Dialogue: 0,0:03:38.10,0:03:42.85,Default,,0000,0000,0000,,let's say we had a negative two, I already used this color
Dialogue: 0,0:03:42.85,0:03:45.24,Default,,0000,0000,0000,,negative two times
Dialogue: 0,0:03:45.24,0:03:48.70,Default,,0000,0000,0000,,negative three.
Dialogue: 0,0:03:48.70,0:03:53.98,Default,,0000,0000,0000,,So now, we can d- actually I'll do this one first.
Dialogue: 0,0:03:53.98,0:03:57.78,Default,,0000,0000,0000,,Let's do multiplying something by negative three so we'll
Dialogue: 0,0:03:57.78,0:04:01.23,Default,,0000,0000,0000,,repeatedly subtract that thing three times whatever that thing is
Dialogue: 0,0:04:01.23,0:04:05.77,Default,,0000,0000,0000,,so now the thing isn't a positive two so the thing over
Dialogue: 0,0:04:05.77,0:04:08.64,Default,,0000,0000,0000,,here is a positive two but the thing we're going to subtract is a negative two
Dialogue: 0,0:04:08.64,0:04:10.90,Default,,0000,0000,0000,,So let me make it clear, this says we are going to subtract something
Dialogue: 0,0:04:10.90,0:04:13.97,Default,,0000,0000,0000,,three times, so we subtract something three times, so
Dialogue: 0,0:04:13.97,0:04:17.10,Default,,0000,0000,0000,,subtracting something (repeatedly) three times
Dialogue: 0,0:04:17.10,0:04:20.64,Default,,0000,0000,0000,,That's what this part right over here tells us
Dialogue: 0,0:04:20.64,0:04:24.30,Default,,0000,0000,0000,,and we'll do this, exactly three times
Dialogue: 0,0:04:24.30,0:04:28.37,Default,,0000,0000,0000,,Over here, it was a positive two we subtracted three times, now we're going to
Dialogue: 0,0:04:28.37,0:04:32.27,Default,,0000,0000,0000,,do a negative two, now we're going to do a negative two
Dialogue: 0,0:04:32.27,0:04:35.77,Default,,0000,0000,0000,,and we know from subtracting negative numbers, we already
Dialogue: 0,0:04:35.77,0:04:39.43,Default,,0000,0000,0000,,built this intuition that subtracting a negative is the same thing
Dialogue: 0,0:04:40.37,0:04:46.10,Default,,0000,0000,0000,,it's the same thing as adding a positive, and so this
Dialogue: 0,0:04:46.10,0:04:50.37,Default,,0000,0000,0000,,this is going to be the same thing as two plus two plus two and
Dialogue: 0,0:04:50.37,0:04:53.61,Default,,0000,0000,0000,,we're told once again, gives you a positive
Dialogue: 0,0:04:53.61,0:04:56.90,Default,,0000,0000,0000,,six, you can same use the same logic over here, now
Dialogue: 0,0:04:56.90,0:05:00.30,Default,,0000,0000,0000,,instead of adding negative three twice, really I could have written this as
Dialogue: 0,0:05:00.30,0:05:03.85,Default,,0000,0000,0000,,negative three as this example
Dialogue: 0,0:05:03.85,0:05:05.57,Default,,0000,0000,0000,,negative three
Dialogue: 0,0:05:05.57,0:05:11.52,Default,,0000,0000,0000,,negative three, and we added it
Dialogue: 0,0:05:11.52,0:05:15.30,Default,,0000,0000,0000,,we added it, now let me put a plus here to make it clear
Dialogue: 0,0:05:15.30,0:05:18.60,Default,,0000,0000,0000,,over here we added it twice, we added negative three
Dialogue: 0,0:05:18.60,0:05:23.35,Default,,0000,0000,0000,,two times, or here since we have a negative two, we're going to subtract
Dialogue: 0,0:05:23.35,0:05:26.18,Default,,0000,0000,0000,,to negative three twice, so we're going to subtract something
Dialogue: 0,0:05:26.18,0:05:30.10,Default,,0000,0000,0000,,and we're going to subtract something again, and that something is going to be
Dialogue: 0,0:05:30.10,0:05:33.44,Default,,0000,0000,0000,,our negative three, it's going to be our negative three, so
Dialogue: 0,0:05:33.44,0:05:37.04,Default,,0000,0000,0000,,negative, negative and put our three right over here
Dialogue: 0,0:05:37.04,0:05:41.24,Default,,0000,0000,0000,,and once again, subtracting negative three is like taking away
Dialogue: 0,0:05:41.24,0:05:43.18,Default,,0000,0000,0000,,someone's debt, which is essentially giving them money,
Dialogue: 0,0:05:43.18,0:05:48.27,Default,,0000,0000,0000,,this is the same thing as adding three plus three which is once again six. So now
Dialogue: 0,0:05:48.27,0:05:51.44,Default,,0000,0000,0000,,you, the ancient philosopher, feel pretty good. Not only this
Dialogue: 0,0:05:51.44,0:05:55.10,Default,,0000,0000,0000,,all consistent with all the mathematics you know
Dialogue: 0,0:05:55.10,0:05:58.37,Default,,0000,0000,0000,,the distributive property is also the property of multiplying something
Dialogue: 0,0:05:58.37,0:06:00.77,Default,,0000,0000,0000,,times something all these things you already know, and now
Dialogue: 0,0:06:00.77,0:06:04.77,Default,,0000,0000,0000,,this actually makes conceptual sense to you, this is actually very consistent with
Dialogue: 0,0:06:04.77,0:06:08.11,Default,,0000,0000,0000,,with your notations, your original notations, or one of the positive notations
Dialogue: 0,0:06:08.11,0:06:12.11,Default,,0000,0000,0000,,of multiplication which is as repeated addition