Why a negative times a negative makes intuitive sense | Pre-Algebra | Khan Academy — Transcript

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1. 0:00So you, as the ancient philosopher in mathematics
2. 0:03have concluded in order for the multiplication of positive and negative numbers to be
3. 0:07consistent with everything you've been constructing so far
4. 0:10with all the other properties of multiplication that you know so far
5. 0:13that you need a negative number times a positive
6. 0:17number or a positive times a negative to give you a negative number
7. 0:20and a negative times a negative
8. 0:24to give you a positive number and so you accept
9. 0:27it's all consistent so far.. this deal does not make complete concrete
10. 0:30sense to you, you want to have a slightly deeper institution than just having to accept its
11. 0:35consistent with the distributive property and whatever else and so you try another
12. 0:40thought experiment, you say "well what is just a basic multiplication way of doing it?"
13. 0:45So if I say, two times
14. 0:47three, one way to
15. 0:51to conceptualize is basic multiplication is really repeating
16. 0:55addition, so you could view this as two threes
17. 0:58so let me write three plus three
18. 1:02and notice there are two of them, there are two of these
19. 1:05or you could view this as three twos, and so this is the same thing as
20. 1:09two plus two plus two and there are
21. 1:13three of them, and either way you can conceptualize
22. 1:17as you get the same exact answer. This is going to be equal
23. 1:20to six, fair enough!
24. 1:24Now, you knew this before you even tried to tackle negative numbers.
25. 1:27Now let's try to make one of these negatives and see what
26. 1:30happens. Let's do two
27. 1:33times negative
28. 1:35three, I want to make the negative into a different color. Two times
29. 1:42negative three.
30. 1:46Well, one way you could view this is the same analogy
31. 1:49here, it's negative three twice so it would be
32. 1:52negative.. I'll try to color code it
33. 1:56negative three and then another negative
34. 2:01three or you could say negative three minus three
35. 2:05or, and this is the interesting thing, instead of
36. 2:08over here there's a two times positive three
37. 2:11you added two, three times.
38. 2:14But since here is two times negative three
39. 2:16you could also imagine you are going to subtract two, three times
40. 2:19So instead of up here, I could
41. 2:21written two plus two plus two because this is a positive
42. 2:26two right over here, but since we're doing this over negative three
43. 2:29we could imagine subtracting two, three times, so this would be
44. 2:33subtracting two (repeated)
45. 2:37subtract another two right over here, subtract another two
46. 2:43and then you subtract another two
47. 2:46notice you did it, once again, you did it
48. 2:54three times, so this is a negative three, so essentially you are subtracting
49. 2:59two, three times. And either way, you can conceptualize
50. 3:03right over here, you are going to get negative six
51. 3:07negative six is the answer.
52. 3:10Now, so you are already starting to feel better about this part right over here
53. 3:16negative times a positive, or a positive times a negative
54. 3:18is going to give you a negative. Now lets take to the really un-intuitive one
55. 3:21and measure negative times a negative, and all of a sudden negatives kind of cancel
56. 3:24to give you a positive. Now why is that the case? Well we can just build from
57. 3:28this example right over here. Let's say we had
58. 3:30a negative two, lets say we had
59. 3:35negative two, let me do it a different color,
60. 3:38let's say we had a negative two, I already used this color
61. 3:42negative two times
62. 3:45negative three.
63. 3:48So now, we can d- actually I'll do this one first.
64. 3:53Let's do multiplying something by negative three so we'll
65. 3:57repeatedly subtract that thing three times whatever that thing is
66. 4:01so now the thing isn't a positive two so the thing over
67. 4:05here is a positive two but the thing we're going to subtract is a negative two
68. 4:08So let me make it clear, this says we are going to subtract something
69. 4:10three times, so we subtract something three times, so
70. 4:13subtracting something (repeatedly) three times
71. 4:17That's what this part right over here tells us
72. 4:20and we'll do this, exactly three times
73. 4:24Over here, it was a positive two we subtracted three times, now we're going to
74. 4:28do a negative two, now we're going to do a negative two
75. 4:32and we know from subtracting negative numbers, we already
76. 4:35built this intuition that subtracting a negative is the same thing
77. 4:40it's the same thing as adding a positive, and so this
78. 4:46this is going to be the same thing as two plus two plus two and
79. 4:50we're told once again, gives you a positive
80. 4:53six, you can same use the same logic over here, now
81. 4:56instead of adding negative three twice, really I could have written this as
82. 5:00negative three as this example
83. 5:03negative three
84. 5:05negative three, and we added it
85. 5:11we added it, now let me put a plus here to make it clear
86. 5:15over here we added it twice, we added negative three
87. 5:18two times, or here since we have a negative two, we're going to subtract
88. 5:23to negative three twice, so we're going to subtract something
89. 5:26and we're going to subtract something again, and that something is going to be
90. 5:30our negative three, it's going to be our negative three, so
91. 5:33negative, negative and put our three right over here
92. 5:37and once again, subtracting negative three is like taking away
93. 5:41someone's debt, which is essentially giving them money,
94. 5:43this is the same thing as adding three plus three which is once again six. So now
95. 5:48you, the ancient philosopher, feel pretty good. Not only this
96. 5:51all consistent with all the mathematics you know
97. 5:55the distributive property is also the property of multiplying something
98. 5:58times something all these things you already know, and now
99. 6:00this actually makes conceptual sense to you, this is actually very consistent with
100. 6:04with your notion, your original notion, or one of the possible notions
101. 6:08of multiplication which is as repeated addition