Why a negative times a negative is a positive | Pre-Algebra | Khan Academy — Transcript

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1. 0:00Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up
2. 0:06And 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. 0:13But now you are faced with a conundrum
4. 0:15What happens when you multiply negative numbers?
5. 0:18Either when you multiply a positive number times a negative number
6. 0:21Or when you multiply two negative numbers
7. 0:23So, for example
8. 0:25You 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. 0:32What would happen if you were to multiply 5 times negative 3
10. 0:37You're not quite sure about this just yet
11. 0:39You're also not quite sure what would happen if you multiply two negative numbers.
12. 0:43So lets say negative two times negative 6
13. 0:48This is also unclear to you
14. 0:50What you do know, because you are a mathematician, is however you define this or whatever this should be
15. 0:56It should hopefully be consistant with all of the other properties of mathematics that you already know
16. 1:02And preferably all of the other properties of multiplication
17. 1:05That would make you feel comfortable that you are getting this right.
18. 1:08and later we can think about other ways to get the intuition
19. 1:11for what these might be allowed you to actually make sense
20. 1:14but to make this make consistent with the rest of mathematics
21. 1:18that you know, you go into a little bit of a thought experiment
22. 1:21you say, well, what should five times
23. 1:24three plus negative three equal
24. 1:28well you already have a philosophy of adding negative numbers
25. 1:32or adding positive numbers to negative numbers, you know negative three is
26. 1:35the opposite of three, but you add three to negative three
27. 1:39you're going to get zero, so this is going to be equal to
28. 1:42five times zero
29. 1:45based on how you already thought about adding a negative number
30. 1:52to a positive, and anything times zero is going to be
31. 1:54zero, so this expression right over here should be zero
32. 1:57but you see, I want to multiply positive and negative
33. 2:02numbers to be consistent with this distributive property so
34. 2:04I should should be able to distribute this five
35. 2:07and for math to be consistent, and math should be consistent, I should
36. 2:12get the exact same answer, so let's distribute this five so
37. 2:16we get five times three
38. 2:18is going to write out as five times three
39. 2:23let me write this multiplication sign, not this dot
40. 2:27five times three, so i distributed there
41. 2:31plus five times negative three
42. 2:35i'll do that in yellow, five times negative three
43. 2:39and this whole thing we just said should be equal to zero
44. 2:45it should be equal to zero, well five times three
45. 2:50those are two positive numbers, we should know what should should be, that is going to be fifteen
46. 2:54now we get this thing, fifteen plus
47. 2:57times whatever five times negative three is
48. 3:01needs to be equal to zero in order to be consistent with all the other mathematics
49. 3:09that we know, well what plus fifteen is going to
50. 3:12be equal to zero, well the opposite of fifteen in order for this
51. 3:16to be true, in order for this to be consistent with all the other mathematics we know
52. 3:20this right over here needs to be equal to
53. 3:23negative fifteen until you say five times negative three
54. 3:27in order to be consistent with all other mathematics we know, needs to
55. 3:31be equal to negative fifteen. That's also consistent
56. 3:34with the intuition of adding negative three repeatedly
57. 3:38five times, now look above above us
58. 3:41slightly higher so you can see ideas of multiplying
59. 3:44two negatives, but we can do the exact same product experiment.
60. 3:48We want whatever this answer to be consistent with the rest of mathematics
61. 3:52that we know so we can
62. 3:53do the same product experiment. What would negative two times
63. 3:58six plus negative six to be equal to.
64. 4:02Well, six plus negative six is going to be zero.
65. 4:05Negative two times zero, anything times zero, needs to be equal to
66. 4:09zero, but then once again, we can distribute
67. 4:12negative two times six so we get
68. 4:16negative two times six, then plus negative two times
69. 4:23negative six plus negative two
70. 4:27times negative six, then once again all of this is going to be
71. 4:30equal to zero, now based on the five experiment we just
72. 4:34did, we said "well this needs to be equal to negative twelve"
73. 4:37or we can view this as going to the six twice
74. 4:41left direction on the number line which gets us to negative twelve
75. 4:44or you could say repeatedly adding negative twos
76. 4:48times six would also get you to negative twelve and now
77. 4:52we also saw over here we want to multiply a positive and a negative
78. 4:55we got the negative so
79. 4:58this could be, you know, going to be equal to negative twelve
80. 5:02so we have negative twelve plus
81. 5:05whatever this business is
82. 5:08going to have to be equal to zero (repeated)
83. 5:13in order to be consistent with all the other mathematics that we know
84. 5:17and so what plus negative twelve is going to equal to be zero
85. 5:21Well, positive twelve plus negative twelve is going to equal to zero
86. 5:25so this needs to be equal to positive twelve in
87. 5:27order to be consistent with all the other mathematics we know
88. 5:31so there we get the idea that this is going positive
89. 5:35twelve. I'll leave you there and I'll see
90. 5:39if I can make a few other videos that can also give you a conceptual understanding
91. 5:42of why these are true