1 00:00:00,300 --> 00:00:06,635 Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up 2 00:00:06,635 --> 00:00:13,783 And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers 3 00:00:13,783 --> 00:00:15,767 But now you are faced with a conundrum 4 00:00:15,767 --> 00:00:18,501 What happens when you multiply negative numbers? 5 00:00:18,501 --> 00:00:21,233 Either when you multiply a positive number times a negative number 6 00:00:21,233 --> 00:00:23,900 Or when you multiply two negative numbers 7 00:00:23,900 --> 00:00:25,167 So, for example 8 00:00:25,167 --> 00:00:32,502 You aren't quite sure what should happen if you were to multiply (and im just picking two numbers where one is positive and one is negative) 9 00:00:32,502 --> 00:00:37,081 What would happen if you were to multiply 5 times negative 3 10 00:00:37,081 --> 00:00:39,773 You're not quite sure about this just yet 11 00:00:39,773 --> 00:00:43,702 You're also not quite sure what would happen if you multiply two negative numbers. 12 00:00:43,702 --> 00:00:48,669 So lets say negative two times negative 6 13 00:00:48,669 --> 00:00:50,835 This is also unclear to you 14 00:00:50,835 --> 00:00:56,569 What you do know, because you are a mathematician, is however you define this or whatever this should be 15 00:00:56,569 --> 00:01:02,410 It should hopefully be consistant with all of the other properties of mathematics that you already know 16 00:01:02,410 --> 00:01:05,233 And preferably all of the other properties of multiplication 17 00:01:05,233 --> 00:01:08,233 That would make you feel comfortable that you are getting this right. 18 00:01:08,233 --> 00:01:11,233 and later we can think about other ways to get the intitution 19 00:01:11,233 --> 00:01:14,608 of what these might be allowed you to actually make sense 20 00:01:14,608 --> 00:01:18,034 but to make this make consistent with the rest of mathematics 21 00:01:18,034 --> 00:01:21,300 that you know, you go into a little bit of a thought experiment 22 00:01:21,300 --> 00:01:24,700 you say, well, what should five times 23 00:01:24,700 --> 00:01:28,524 three plus negative three equal 24 00:01:28,524 --> 00:01:32,681 well you already have a philosophy of adding negative numbers 25 00:01:32,681 --> 00:01:35,551 or adding positive numbers to negative numbers, you know negative three is 26 00:01:35,551 --> 00:01:39,298 the opposite of three, but you add three to negative three 27 00:01:39,298 --> 00:01:42,423 you're going to get zero, so this is going to be equal to 28 00:01:42,423 --> 00:01:45,520 five times zero 29 00:01:45,520 --> 00:01:52,102 based on how you already thought about adding a negative number 30 00:01:52,102 --> 00:01:54,369 to a positive, and anything times zero is going to be 31 00:01:54,369 --> 00:01:57,900 zero, so this expression right over here should be zero 32 00:01:57,900 --> 00:02:02,283 but you see, I want to multiply positive and negative 33 00:02:02,283 --> 00:02:04,697 numbers to be consistent with this distributive property so 34 00:02:04,697 --> 00:02:07,367 I should should be able to distribute this five 35 00:02:07,367 --> 00:02:12,943 and for math to be consistent, and math should be consistent, I should 36 00:02:12,943 --> 00:02:16,548 get the exact same answer, so let's distribute this five so 37 00:02:16,548 --> 00:02:18,103 we get five times three 38 00:02:18,103 --> 00:02:23,417 is going to write out as five times three 39 00:02:23,417 --> 00:02:27,020 let me write this multiplication sign, not this dot 40 00:02:27,020 --> 00:02:31,434 five times three, so i distributed there 41 00:02:31,434 --> 00:02:35,925 plus five times negative three 42 00:02:35,925 --> 00:02:39,358 i'll do that in yellow, five times negative three 43 00:02:39,358 --> 00:02:45,843 and this whole thing we just said should be equal to zero 44 00:02:45,843 --> 00:02:50,700 it should be equal to zero, well five times three 45 00:02:50,700 --> 00:02:54,843 those are two positive numbers, we should know what should should be, that is going to be fifteen 46 00:02:54,843 --> 00:02:57,825 now we get this thing, fifteen plus 47 00:02:57,825 --> 00:03:01,506 times whatever five times negative three is 48 00:03:01,506 --> 00:03:09,433 needs to be equal to zero in order to be consistent with all the other mathematics 49 00:03:09,433 --> 00:03:12,806 that we know, well what plus fifteen is going to 50 00:03:12,806 --> 00:03:16,498 be equal to zero, well the opposite of fifteen in order for this 51 00:03:16,498 --> 00:03:20,000 to be true, in order for this to be consistent with all the other mathematics we know 52 00:03:20,000 --> 00:03:23,588 this right over here needs to be equal to 53 00:03:23,588 --> 00:03:27,690 negative fifteen until you say five times negative three 54 00:03:27,690 --> 00:03:31,170 in order to be consistent with all other mathematics we know, needs to 55 00:03:31,170 --> 00:03:34,433 be equal to negative fifteen. That's also consistent 56 00:03:34,433 --> 00:03:38,251 with the intuition of adding negative three repeatedly 57 00:03:38,251 --> 00:03:41,302 five times, now look above above us 58 00:03:41,302 --> 00:03:44,769 slightly higher so you can see ideas of multiplying 59 00:03:44,769 --> 00:03:48,272 two negatives, but we can do the exact same product experiment. 60 00:03:48,272 --> 00:03:52,033 We want whatever this answer to be consistent with the rest of mathematics 61 00:03:52,033 --> 00:03:53,769 that we know so we can 62 00:03:53,769 --> 00:03:58,347 do the same product experiment. What would negative two times 63 00:03:58,347 --> 00:04:02,035 six plus negative six to be equal to. 64 00:04:02,035 --> 00:04:05,012 Well, six plus negative six is going to be zero. 65 00:04:05,012 --> 00:04:09,145 Negative two times zero, anything times zero, needs to be equal to 66 00:04:09,145 --> 00:04:12,367 zero, but then once again, we can distribute 67 00:04:12,367 --> 00:04:16,300 negative two times six so we get 68 00:04:16,300 --> 00:04:23,621 negative two times six, then plus negative two times 69 00:04:23,621 --> 00:04:27,102 negative six plus negative two 70 00:04:27,102 --> 00:04:30,767 times negative six, then once again all of this is going to be 71 00:04:30,767 --> 00:04:34,634 equal to zero, now based on the five experiment we just 72 00:04:34,634 --> 00:04:37,569 did, we said "well this needs to be equal to negative twelve" 73 00:04:37,569 --> 00:04:41,415 or we can view this as going to the six twice 74 00:04:41,415 --> 00:04:44,501 left direction on the number line which gets us to negative twelve 75 00:04:44,501 --> 00:04:48,433 or you could say repeatedly adding negative twos 76 00:04:48,433 --> 00:04:52,003 times six would also get you to negative twelve and now 77 00:04:52,003 --> 00:04:55,147 we also saw over here we want to multiply a positive and a negative 78 00:04:55,147 --> 00:04:58,769 we got the negative so 79 00:04:58,769 --> 00:05:02,501 this could be, you know, going to be equal to negative twelve 80 00:05:02,501 --> 00:05:05,648 so we have negative twelve plus 81 00:05:05,648 --> 00:05:08,234 whatever this business is 82 00:05:08,234 --> 00:05:13,833 going to have to be equal to zero (repeated) 83 00:05:13,833 --> 00:05:17,369 in order to be consistent with all the other mathematics that we know 84 00:05:17,369 --> 00:05:21,621 and so what plus negative twelve is going to equal to be zero 85 00:05:21,621 --> 00:05:25,101 Well, positive twelve plus negative twelve is going to equal to zero 86 00:05:25,101 --> 00:05:27,835 so this needs to be equal to positive twelve in 87 00:05:27,835 --> 00:05:31,564 order to be consistent with all the other mathematics we know 88 00:05:31,564 --> 00:05:35,609 so there we get the idea that this is going positive 89 00:05:35,609 --> 00:05:39,153 twelve. I'll leave you there and I'll see 90 00:05:39,153 --> 00:05:42,569 if I can make a few other videos that can also give you a conceptual understanding 91 00:05:42,569 --> 00:05:46,569 of why these are true