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← Multiplying and dividing negative numbers

Multiplying and dividing negative numbers

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Showing Revision 12 created 06/11/2012 by michal.stevove.

  1. Welcome to the presentation
    on multiplying and
  2. dividing negative numbers.
  3. Let's get started.
  4. I think you're going to find
    that multiplying and dividing
  5. negative numbers are a lot
    easier than it might
  6. look initially. You just have to
    remember a couple of rules.
  7. And I am going to teach probably
    in the future like I'm actually going
  8. to give you more intuition on
    why there rules work.
  9. So the basic rules are when you
    multiply two negative numbers,
  10. so let's say I had negative
    2 times negative 2.
  11. First you just look at each
    of the numbers as if there
  12. was no negative sign.
  13. Well you say well, 2
    times 2 that equals 4.
  14. And it turns out that if you
    have a negative times a
  15. negative, that that
    equals a positive.
  16. So let's write that
    first rule down.
  17. A negative times a negative
    equals a positive.

  18. What if it was negative
    2 times positive 2?
  19. Well in this case, let's
    first of all look at the
  20. two numbers without signs.
  21. We know that 2 times 2 is 4.
  22. But here we have a negative
    times a positive 2, and it
  23. turns out that when you
    multiply a negative times a
  24. positive you get a negative.
  25. So that's another rule.
  26. Negative times positive
    is equal to negative.
  27. What happens if you have a
    positive 2 times a negative 2?
  28. I think you'll probably guess
    this one right, as you can tell
  29. that these two are pretty much
    the same thing by, I believe
  30. it's the transitive property --
    no, no I think it's the
  31. communicative property.
  32. I have to remember that.
  33. But 2 times negative 2, this
    also equals negative 4.
  34. So we have the final rule that
    a positive times a negative
  35. also equals the negative.

  36. And actually these second
    two rules, they're kind
  37. of the same thing.
  38. A negative times a positive
    is a negative, or a positive
  39. times a negative is negative.
  40. You could also say that as when
    the signs are different and
  41. you multiply the two numbers,
    you get a negative number.
  42. And of course, you already know
    what happens when you have a
  43. positive times a positive.
  44. Well that's just a positive.
  45. So let's review.
  46. Negative times a
    negative is a positive.
  47. A negative times a
    positive is a negative.
  48. A positive times a
    negative is a negative.
  49. And positive times each
    other equals positive.
  50. I think that last little bit
    completely confused you.
  51. Maybe I can simplify
    it for you.
  52. What if I just told you if when
    you're multiplying and they're
  53. the same signs that gets
    you a positive result.

  54. And different signs gets
    you a negative result.
  55. So that would be either, let's
    say a 1 times 1 is equal to 1,
  56. or if I said negative 1 times
    negative 1 is equal to
  57. positive 1 as well.
  58. Or if I said 1 times negative
    1 is equal to negative 1, or
  59. negative 1 times 1 is
    equal to negative 1.
  60. You see how on the bottom two
    problems I had two different
  61. signs, positive 1
    and negative 1?
  62. And the top two problems,
    this one right here
  63. both 1s are positive.
  64. And this one right here
    both 1s are negative.
  65. So let's do a bunch of problems
    now, and hopefully it'll hit
  66. the point home, and you also
    could try to do along the
  67. practice problems and also give
    the hints and give you what rules to you so that should help you as well
  68. So if I said negative 4 times
    positive 3, well 4 times
  69. 3 is 12, and we have a
    negative and a positive.
  70. So different signs
    mean negative.
  71. So negative 4 times
    3 is a negative 12.
  72. That makes sense because we're
    essentially saying what's
  73. negative 4 times itself three
    times, so it's like negative 4
  74. plus negative 4 plus negative
    4, which is negative 12.
  75. If you've seen the video on
    adding and subtracting negative
  76. numbers, you probably
    should watch first.
  77. Let's do another one.
  78. What if I said minus
    2 times minus 7.
  79. And you might want to pause the
    video at any time to see if you
  80. know how to do it and
    then restart it to see
  81. what the answer is.
  82. Well, 2 times 7 is 14, and we
    have the same sign here, so
  83. it's a positive 14 -- normally
    you wouldn't have to write the
  84. positive but that makes it a
    little bit more explicit.
  85. And what if I had -- let me
    think -- 9 times negative 5.
  86. Well, 9 times 5 is 45.
  87. And once again, the signs are
    different so it's a negative.
  88. And then finally what if it I
    had -- let me think of some
  89. good numbers -- minus
    6 times minus 11.
  90. Well, 6 times 11 is 66 and
    then it's a negative and
  91. negative, it's a positive.
  92. Let me give you a
    trick problem.
  93. What is 0 times negative 12?
  94. Well, you might say that the
    signs are different, but
  95. 0 is actually neither
    positive nor negative.
  96. And 0 times anything
    is still 0.
  97. It doesn't matter if the thing
    you multiply it by is a
  98. negative number or
    a positive number.
  99. 0 times anything is still 0.
  100. So let's see if we can apply
    these same rules to division.
  101. It actually turns out that
    the same rules apply.
  102. If I have 9 divided
    by negative 3.
  103. Well, first we say
    what's 9 divided by 3?
  104. Well that's 3.
  105. And they have different signs,
    positive 9, negative 3.
  106. So different signs
    means a negative.
  107. 9 divided by negative 3
    is equal to negative 3.
  108. What is minus 16 divided by 8?
  109. Well, once again, 16
    divided by 8 is 2, but
  110. the signs are different.
  111. Negative 16 divided by positive
    8, that equals negative 2.
  112. Remember, different signs will
    get you a negative result.
  113. What is minus 54
    divided by minus 6?
  114. Well, 54 divided by 6 is 9.
  115. And since both terms, the
    divisor and the dividend, are
  116. both negative -- negative 54
    and negative 6 -- it turns out
  117. that the answer is positive.
    Remember, same signs
  118. result in a positive sign.
  119. Let's do one more.
  120. Obviously, 0 divided by
    anything is still 0.
  121. That's pretty straightforward.
  122. And of course, you can't
    divide anything by 0
  123. -- that's undefined.
  124. Let's do one more.
  125. What is -- I'm just going to
    think of random numbers --
  126. 4 divided by negative 1?
  127. Well, 4 divided by 1 is 4,
    but the signs are different.
  128. So it's negative 4.
  129. I hope that helps.
  130. Now what I want you to do is
    actually try
  131. as many of these multiplying and dividing
    negative numbers as you can.
  132. And you click on hints
  133. and it'll remind you of which rule to use.
  134. In your own time you might want
    to actually think about
  135. why these rules apply and what it
    means
  136. to multiply a negative number times a positive number.
  137. And even more interesting, what
    it means
  138. to multiply a negative number times a negative number.
  139. But I think at this point,
  140. hopefully, you are ready to start doing some problems.
  141. Good luck.