1 00:00:00,730 --> 00:00:03,457 In this video I want to familiarize ourselves 2 00:00:03,457 --> 00:00:04,768 with negative numbers. 3 00:00:04,768 --> 00:00:07,564 And also learn a bit of how do we add and subtract them. 4 00:00:07,564 --> 00:00:09,326 And when you first encounter them, 5 00:00:09,326 --> 00:00:11,612 they look like this deep and mysterious thing. 6 00:00:11,612 --> 00:00:14,769 When we first count things, we're counting positive numbers. 7 00:00:14,769 --> 00:00:17,346 What does a negative number even mean? 8 00:00:17,346 --> 00:00:20,835 But when we think about it, you probably have encountered 9 00:00:20,835 --> 00:00:22,905 negative numbers in your everyday life. 10 00:00:22,905 --> 00:00:25,195 And let me just give you a few examples. 11 00:00:25,195 --> 00:00:26,412 So before I give the example, the general idea is 12 00:00:26,412 --> 00:00:30,674 a negative number is any number less than zero. 13 00:00:30,674 --> 00:00:34,839 Less than zero. 14 00:00:34,839 --> 00:00:37,191 And if that sounds strange and abstract to you, 15 00:00:37,191 --> 00:00:39,851 let's just think about it in a couple of different contexts. 16 00:00:39,851 --> 00:00:45,251 If I have... if we're measuring the temperature ... 17 00:00:45,251 --> 00:00:47,114 (and it can be in Celsius or Farenheit, 18 00:00:47,114 --> 00:00:49,881 but let's just say we're measuring it in Celsius), 19 00:00:49,881 --> 00:00:51,878 and so let me draw a little scale 20 00:00:51,878 --> 00:00:53,707 that we can measure the temperature on. 21 00:00:53,707 --> 00:00:57,099 So let's say that this is 0° Celsius, 22 00:00:57,099 --> 00:01:02,909 that is 1° Celsius, 2° Celsius, 3° Celsius. 23 00:01:02,924 --> 00:01:05,817 Now, let's say it's a pretty chilly day 24 00:01:05,817 --> 00:01:10,296 and it's currently 3° Celsius. 25 00:01:10,296 --> 00:01:12,004 And someone who predicts the future 26 00:01:12,004 --> 00:01:16,976 tells you that it is going to get 4° colder the next day. 27 00:01:16,976 --> 00:01:21,502 So how cold will it be? How can you represent that coldness? 28 00:01:21,502 --> 00:01:25,126 Well, if it only got 1° colder it would be at 2°, 29 00:01:25,126 --> 00:01:26,874 but we know we have to go 4° colder. 30 00:01:26,874 --> 00:01:32,068 If we got 2° colder, we would be at 1°. 31 00:01:32,068 --> 00:01:35,311 If we got 3° colder, we would be at 0°. 32 00:01:35,311 --> 00:01:38,453 But 3° isn't enough, we have to get 4° colder, 33 00:01:38,453 --> 00:01:43,965 so we actually have to go one more below zero. 34 00:01:43,965 --> 00:01:50,416 And that 1 below 0 we call that "negative 1". 35 00:01:50,416 --> 00:01:52,907 And so you can kind of see that the number line, 36 00:01:52,907 --> 00:01:57,029 as you go to the right of zero increases in positive values, 37 00:01:57,029 --> 00:02:04,207 as you go to the left of zero you're going to have -1, -2, -3. 38 00:02:04,207 --> 00:02:07,301 And you're going to have-depending on how you think about it- 39 00:02:07,301 --> 00:02:09,948 you're going to have larger negative numbers. 40 00:02:09,948 --> 00:02:15,372 But I want to make it very clear: -3 is LESS than -1. 41 00:02:15,372 --> 00:02:19,488 There is less heat in the air at -3° than at -1°. 42 00:02:19,488 --> 00:02:23,329 It is colder---there is less temperature there. 43 00:02:23,329 --> 00:02:39,824 So let me just make it very clear: -100 is much smaller than -1. 44 00:02:39,824 --> 00:02:41,913 You might look at 100 and you might look at 1 and 45 00:02:41,913 --> 00:02:45,094 your gut reaction might be that 100 is much larger than 1. 46 00:02:45,094 --> 00:02:46,272 But when you think about it, 47 00:02:46,272 --> 00:02:48,849 -100 means there is a lack of something. 48 00:02:48,849 --> 00:02:52,304 -100: if it's -100° there is a lack of heat, 49 00:02:52,304 --> 00:02:55,737 so there is much less heat here than if we had -1°. 50 00:02:55,737 --> 00:02:57,493 Let me give you another example. 51 00:02:57,493 --> 00:03:11,264 Let's say in my bank account today I have $10. 52 00:03:11,264 --> 00:03:13,114 Now, let's say I go out there 53 00:03:13,114 --> 00:03:14,949 (because I feel good about my $10), 54 00:03:14,949 --> 00:03:21,426 and let's say I go and spend $30. 55 00:03:21,426 --> 00:03:22,735 And, for the sake of argument 56 00:03:22,735 --> 00:03:24,275 let's say I have a very flexible bank, 57 00:03:24,275 --> 00:03:26,861 one that lets me spend more money than I have 58 00:03:26,861 --> 00:03:28,354 (and these actually exist!). 59 00:03:28,354 --> 00:03:30,422 So I spend $30. 60 00:03:30,422 --> 00:03:32,907 So what's my bank account going to look like? 61 00:03:32,907 --> 00:03:35,564 So let me draw a number line here. 62 00:03:35,564 --> 00:03:37,839 And you may already have an intuitive response: 63 00:03:37,839 --> 00:03:43,284 I will owe the bank some money. 64 00:03:43,284 --> 00:03:47,130 Tomorrow, what is my bank account? 65 00:03:47,130 --> 00:03:51,642 You might immediately say, "if I have $10 and I spend $30, 66 00:03:51,642 --> 00:03:54,219 there's $20 that had to come from some place." 67 00:03:54,219 --> 00:03:56,489 And that $20 is coming from the bank. 68 00:03:56,489 --> 00:03:59,119 So I'm going to owe the bank $20. 69 00:03:59,119 --> 00:04:00,789 And so, in my bank account, 70 00:04:00,789 --> 00:04:07,199 to show how much I have, I could say $10 - $30 is -$20. 71 00:04:07,199 --> 00:04:13,345 So in my bank account tomorrow, I'm going to have -$20. 72 00:04:13,345 --> 00:04:18,075 So, if I say I have -$20, that means that I owe the bank. 73 00:04:18,075 --> 00:04:19,357 ---I don't even have it. 74 00:04:19,357 --> 00:04:20,946 Not only do I have nothing, I owe something. 75 00:04:20,946 --> 00:04:22,535 It's going in reverse. 76 00:04:22,535 --> 00:04:25,589 Here, I have something to spend... 77 00:04:25,589 --> 00:04:29,228 if my $10 in my bank account means the bank owes me $10. 78 00:04:29,228 --> 00:04:31,817 I have $10 that I can use to go spend. 79 00:04:31,817 --> 00:04:33,390 Now, all of a sudden I owe the bank. 80 00:04:33,390 --> 00:04:34,963 I've gone the other direction. 81 00:04:34,963 --> 00:04:36,778 If we use a number line here 82 00:04:36,778 --> 00:04:38,593 it should hopefully make more sense. 83 00:04:38,593 --> 00:04:39,832 So that is 0. 84 00:04:39,832 --> 00:04:43,347 I'm starting off with $10, 85 00:04:43,347 --> 00:04:47,417 and spending $30 means I'm moving 30 spaces to the left. 86 00:04:47,417 --> 00:04:49,848 So if I move 10 spaces to the left--- 87 00:04:49,848 --> 00:04:52,804 if I only spend $10 I'll be back at $0. 88 00:04:52,804 --> 00:04:56,829 If I spend another $10, I'll be at -$10. 89 00:04:56,829 --> 00:05:02,208 If I spend another $10 after that, I will be at -$20. 90 00:05:02,208 --> 00:05:04,483 So, each of these distances, I spent $10 I'd be at $0. 91 00:05:04,483 --> 00:05:06,759 Another $10 I'd be at -$10. 92 00:05:06,759 --> 00:05:09,754 Another $10 I would be at -$20. 93 00:05:09,754 --> 00:05:13,144 So this whole distance here is how much I spent. 94 00:05:13,144 --> 00:05:17,270 I spent $30. 95 00:05:17,270 --> 00:05:20,398 So the general idea when you spend or if you subtract, 96 00:05:20,398 --> 00:05:23,327 or getting colder, you would move to the left. 97 00:05:23,327 --> 00:05:24,803 The numbers would get smaller. 98 00:05:24,803 --> 00:05:27,229 And we now know they can get even smaller than 0. 99 00:05:27,229 --> 00:05:31,604 They can go to -1, -2---they can even go to -1.5, -1.6. 100 00:05:31,604 --> 00:05:34,948 The more and more negative, the more you lose. 101 00:05:34,948 --> 00:05:37,268 If you're adding, if I go and get my paycheck, 102 00:05:37,268 --> 00:05:41,389 I will move to the right of the number line. 103 00:05:41,389 --> 00:05:43,058 Now, with that out of the way, 104 00:05:43,058 --> 00:05:46,082 let's just do a couple more pure math problems. 105 00:05:46,082 --> 00:05:52,317 What it means, if we were to say... 106 00:05:52,317 --> 00:05:56,474 Let's say, 3 - 4. 107 00:05:56,474 --> 00:05:57,710 So once again, 108 00:05:57,710 --> 00:06:00,193 this is exactly the situation we did with the temperature. 109 00:06:00,193 --> 00:06:01,944 We're starting with 3 and we're subtracting 4, 110 00:06:01,944 --> 00:06:03,695 so we're going to move 4 to the left. 111 00:06:03,695 --> 00:06:05,854 We go 1, 2, 3, 4. 112 00:06:05,854 --> 00:06:10,417 That gets us to -1. 113 00:06:10,417 --> 00:06:11,767 And when you're starting to do this 114 00:06:11,767 --> 00:06:13,934 you really understand what a negative number means. 115 00:06:13,934 --> 00:06:15,843 I really encourage you to visualize the number line 116 00:06:15,843 --> 00:06:17,752 and really move along it depending on 117 00:06:17,752 --> 00:06:19,961 whether you're adding or subtracting. 118 00:06:19,961 --> 00:06:21,236 Let's do a couple more. 119 00:06:21,236 --> 00:06:27,848 Let's say I have 2 - 8 120 00:06:27,848 --> 00:06:30,552 (and we'll think about more ways to do this in future videos), 121 00:06:30,552 --> 00:06:33,612 but once again, you just want to do the number line. 122 00:06:33,612 --> 00:06:35,083 You have a 0 here. 123 00:06:35,083 --> 00:06:38,729 We're at (let me draw the spacing a little bit). 124 00:06:38,729 --> 00:06:42,032 We have 0 here...we're at 1... 2. 125 00:06:42,032 --> 00:06:43,891 If we're subtracting 8, 126 00:06:43,891 --> 00:06:47,043 that means we're going to move 8 to the left. 127 00:06:47,043 --> 00:06:50,993 So we're going to go 1 to the left, 2 to the left. 128 00:06:50,993 --> 00:06:53,036 So, we've gone 2 to the left to get to 0. 129 00:06:53,036 --> 00:06:55,080 We have to move how many more to the left? 130 00:06:55,080 --> 00:06:56,961 We've already moved 2 to the left, 131 00:06:56,961 --> 00:06:59,010 to get to 8, we have to move 6 more to the left. 132 00:06:59,010 --> 00:07:07,396 So we're going to have to move 1-2-3-4-5-6 more to the left. 133 00:07:07,396 --> 00:07:09,106 Well, where is that going to put us? 134 00:07:09,106 --> 00:07:10,074 Well, we were at 0. 135 00:07:10,074 --> 00:07:18,706 This is -1, -2, -3, -4, -5, -6. 136 00:07:18,706 --> 00:07:24,310 So, 2 - 8 is -6. 137 00:07:24,310 --> 00:07:26,540 2-2 would be 0. 138 00:07:26,540 --> 00:07:29,633 When you're subtracting 8 you're subtracting another 6. 139 00:07:29,633 --> 00:07:33,366 So we go to -6, we go 6 below 0. 140 00:07:33,366 --> 00:07:34,994 Let me do one more example. 141 00:07:34,994 --> 00:07:36,562 (and this will be a little less conventional 142 00:07:36,562 --> 00:07:38,807 but hopefully it will make sense). 143 00:07:38,807 --> 00:07:43,976 Let us take... (and I'll do this in a new color)... 144 00:07:43,976 --> 00:07:47,870 Let me take -4 - 2. 145 00:07:47,870 --> 00:07:49,439 So we're starting at a negative number 146 00:07:49,439 --> 00:07:51,009 and then we're subtracting from that. 147 00:07:51,009 --> 00:07:56,808 Now, if this seems confusing just remember the number line! 148 00:07:56,838 --> 00:07:59,680 So this is 0 right here. 149 00:07:59,680 --> 00:08:07,015 This is -1, -2, -3, -4. So that's where we're starting. 150 00:08:07,015 --> 00:08:09,279 Now we're going to subtract 2 from -4, 151 00:08:09,279 --> 00:08:11,543 so we're going to move 2 to the left. 152 00:08:11,543 --> 00:08:15,387 So if we subtract 1 we'll be at -5. 153 00:08:15,387 --> 00:08:21,723 If we subtract another 1 we are going to be at -6. 154 00:08:21,723 --> 00:08:23,332 So this is -6. 155 00:08:23,332 --> 00:08:24,748 Let's do another interesting thing. 156 00:08:24,748 --> 00:08:30,266 Let's start at -3 ... let's say we have -3. 157 00:08:30,266 --> 00:08:34,261 Instead of subtracting something from that, let's add 2. 158 00:08:34,261 --> 00:08:36,562 So where would this put us on the number line? 159 00:08:36,562 --> 00:08:39,389 So we're starting at -3 and we're adding 2. 160 00:08:39,389 --> 00:08:42,217 So we're going to move to the right. 161 00:08:42,217 --> 00:08:44,750 So you add 1, you become -2 162 00:08:44,750 --> 00:08:46,760 But if you add another 1 (which we have to do), 163 00:08:46,760 --> 00:08:47,969 you become -1. 164 00:08:47,969 --> 00:08:49,394 You move 2 to the right. 165 00:08:49,394 --> 00:08:54,395 So, -3 + 2 is -1. 166 00:08:54,395 --> 00:08:55,957 And you can see for yourself, 167 00:08:55,957 --> 00:08:59,934 this all fits our traditional notion of adding and subtracting. 168 00:08:59,934 --> 00:09:05,399 If we start at -1 and we subtract 2, we should get -3. 169 00:09:05,399 --> 00:09:07,506 Kind of reverses this thing up here. 170 00:09:07,506 --> 00:09:09,637 -3 +2 gets us there. 171 00:09:09,637 --> 00:09:11,229 And if we start there and we subtract 2 172 00:09:11,229 --> 00:09:12,514 we should get back to -3. 173 00:09:12,514 --> 00:09:13,429 And we see that happens. 174 00:09:13,429 --> 00:09:17,634 If you start at -1, right over here, 175 00:09:17,634 --> 00:09:20,469 and you subtract 2, you move 2 to the left. 176 00:09:20,469 --> 00:09:22,106 You get back to -3. 177 00:09:22,106 --> 00:09:25,322 So hopefully this starts to give you a sense of what it means 178 00:09:25,322 --> 00:09:29,028 to deal with or add and subtract negative numbers. 179 00:09:29,028 --> 00:09:31,337 But we are going to give a lot more examples in the next video. 180 00:09:31,337 --> 00:09:32,475 And we're actually going to see what it means 181 00:09:32,475 --> 00:09:35,337 to subtract a negative number.