Multiplying positive and negative numbers | Pre-Algebra | Khan Academy — Transcript

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1. 0:00We know that if we were to multiply two times three,
2. 0:04that would give us positive six.
3. 0:06And so we are going to think about negative numbers in this video.
4. 0:09One way to think about it,
5. 0:10is that I have a positive number times another positive number,
6. 0:12and that gives me a positive number.
7. 0:15So if I have a positive times a positive,
8. 0:19that would give me a positive number.
9. 0:22Now it's mixed up a little bit.
10. 0:24Introduce some negative numbers.
11. 0:25So what happens if I had negative two times three?
12. 0:31Negative two times three.
13. 0:34Well, one way to think about it--
14. 0:35Now we are talking about intuition in this video
15. 0:37and in the future videos.
16. 0:38You could view this as negative two repeatedly added three times.
17. 0:42So this could be negative two plus negative two plus negative two--
18. 0:48Not negative six. Plus negative two.
19. 0:50which would be equal to--
20. 0:52well, negative two plus negative two is negative four,
21. 0:55plus another negative two is negative six.
22. 0:57This would be equal to negative six.
23. 1:02Or another way to think about it is,
24. 1:03if I had two times three, I would get six.
25. 1:05But because one of these two numbers is negative,
26. 1:09then my product is going to be negative.
27. 1:12So if I multiply, a negative times a positive,
28. 1:17I'm going to get a negative.
29. 1:21Now what if we swap the order which we multiply?
30. 1:24So if we were to multiply three times negative two,
31. 1:29it shouldn't matter.
32. 1:31The order which we multiply things don't change,
33. 1:34or shouldn't change the product.
34. 1:35When we multiply two times three, we get six.
35. 1:37When we multiply three times two, we will get six.
36. 1:40So we should have the same property here.
37. 1:42Three times negative two should give us the same result.
38. 1:45It's going to be equal to negative six.
39. 1:50And once again we say, three times two would be six.
40. 1:53One of these two numbers is negative,
41. 1:55and so our product is going to be negative.
42. 1:58So we could draw a positive times a negative
43. 2:01is also going to be a negative.
44. 2:04And both of these are just the same thing with the order
45. 2:07which we are multiplying switched around.
46. 2:10But this is one of the two numbers are negative. Exactly one.
47. 2:14So one negative, one positive number is being multiplied.
48. 2:22Then you'll get a negative product.
49. 2:24Now we'll think about the third circumstance,
50. 2:27where both of the numbers are negative.
51. 2:29So if I were to multiply--I'll just switch colors for fun here--
52. 2:32If I were to multiply negative two times negative three--
53. 2:37this might be the least intuitive for you of all,
54. 2:39and here I'm going to introduce you the rule,
55. 2:41in the future I will explore why this is,
56. 2:43and why this makes mathematics more--all fit together.
57. 2:48But this is going to be, you see, two times three would be six.
58. 2:52And I have a negative times a negative,
59. 2:54one way you can think about it is that negatives cancel out!
60. 2:57So you'll actually end up with a positive six.
61. 3:00Actually I don't have to draw a positive here.
62. 3:02But I write it here just to reemphasize.
63. 3:04This right over here is a positive six.
64. 3:06So we have another rule of thumb here.
65. 3:09If I have a negative times a negative,
66. 3:13the negatives are going to cancel out.
67. 3:16And that's going to give me a positive number.
68. 3:21Now with these out of the way, let's just do a bunch of examples.
69. 3:24I'm encouraging you to try them out before I do them.
70. 3:27Pause the video, try them out, and see if you get the same answer.
71. 3:30So let's try negative one times negative one.
72. 3:34Well, one times one would be one.
73. 3:37And we have a negative times a negative. They cancel out.
74. 3:41Negative times a negative give me a positive.
75. 3:42So this is going to be positive one.
76. 3:44I can just write one,
77. 3:45or I can literally write a plus sign there to emphasize.
78. 3:48This is a positive one.
79. 3:49What happened if I did negative one times zero?
80. 3:53Now this might seem, this doesn't fit into any of these circumstances,
81. 3:56zero is neither positive nor negative.
82. 3:59And here you just have to remember anything times zero
83. 4:02is going to be zero.
84. 4:04So negative one times zero is going to be zero.
85. 4:07Or I could've said zero times negative seven hundred and eighty-three,
86. 4:11that is also going to be zero.
87. 4:14Now what about two--let me do some interesting ones.
88. 4:18What about--I'm looking a new color.
89. 4:21Twelve times negative four.
90. 4:24Well, once again, twelve times positive four would be fourty-eight.
91. 4:29And we are in the circumstance where one of these two numbers,
92. 4:32right over here, is negative. This one right here.
93. 4:35If exactly one of the two numbers is negative,
94. 4:38then the product is going to be negative.
95. 4:41We are in this circumstance, right over here.
96. 4:45We have one negative, so the product is negative.
97. 4:47You could imagine this as repeatedly adding negative four twelve times
98. 4:52And so you will get to negative fourty-eight.
99. 4:55Let's do another one.
100. 4:56What is seven times three?
101. 5:00Well, this is a bit of a trick. There are no negative numbers here.
102. 5:03This is just going to be seven times three.
103. 5:04Positive seven times positive three. The first circumstance,
104. 5:07which you already knew how to do before this video.
105. 5:09This would just be equal to twenty-one.
106. 5:12Let's do one more.
107. 5:13So if I were to say negative five times negative ten--
108. 5:18well, once again, negative times a negative.
109. 5:20The negatives cancel out.
110. 5:21You are just left with a positive product.
111. 5:23So it's going to be five times ten. It's going to be fifty.
112. 5:26The negative and the negative cancel out.
113. 5:28Your product is going to be positive.
114. 5:30That's this situation right over there.