1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:03,620 It always helps me to see a lot of examples of something so I 3 00:00:03,620 --> 00:00:05,919 figured it wouldn't hurt to do more scientific 4 00:00:05,919 --> 00:00:06,980 notation examples. 5 00:00:06,980 --> 00:00:09,919 So I'm just going to write a bunch of numbers and then write 6 00:00:09,919 --> 00:00:11,390 them in scientific notation. 7 00:00:11,390 --> 00:00:14,179 And hopefully this'll cover almost every case you'll ever 8 00:00:14,179 --> 00:00:16,160 see and then at the end of this video, we'll actually do some 9 00:00:16,160 --> 00:00:18,429 computation with them to just make sure that we can 10 00:00:18,429 --> 00:00:20,960 do computation with scientific notation. 11 00:00:20,960 --> 00:00:24,835 Let me just write down a bunch of numbers. 12 00:00:24,835 --> 00:00:28,380 0.00852. 13 00:00:28,379 --> 00:00:29,640 That's my first number. 14 00:00:29,640 --> 00:00:39,445 My second number is 7012000000000. 15 00:00:39,445 --> 00:00:42,039 I'm just arbitrarily stopping the zeroes. 16 00:00:42,039 --> 00:00:50,619 The next number is 0.0000000 I'll just draw a couple more. 17 00:00:50,619 --> 00:00:55,640 If I keep saying 0, you might find that annoying. 18 00:00:55,640 --> 00:01:01,259 500 The next number -- right here, there's a 19 00:01:01,259 --> 00:01:02,579 decimal right there. 20 00:01:02,579 --> 00:01:09,090 The next number I'm going to do is the number 723. 21 00:01:09,090 --> 00:01:11,969 The next number I'll do -- I'm having a lot of 7's here. 22 00:01:11,969 --> 00:01:13,260 Let's do 0.6. 23 00:01:13,260 --> 00:01:16,300 24 00:01:16,299 --> 00:01:19,980 And then let's just do one more just for, just to make sure 25 00:01:19,980 --> 00:01:21,590 we've covered all of our bases. 26 00:01:21,590 --> 00:01:26,640 Let's say we do 823 and then let's throw some -- an 27 00:01:26,640 --> 00:01:29,640 arbitrary number of 0's there. 28 00:01:29,640 --> 00:01:33,459 So this first one, right here, what we do if we want to write 29 00:01:33,459 --> 00:01:35,909 in scientific notation, we want to figure out the largest 30 00:01:35,909 --> 00:01:38,099 exponent of 10 that fits into it. 31 00:01:38,099 --> 00:01:40,309 So we go to its first non-zero term, which 32 00:01:40,310 --> 00:01:41,230 is that right there. 33 00:01:41,230 --> 00:01:45,250 We count how many positions to the right of the decimal point 34 00:01:45,250 --> 00:01:47,000 we have including that term. 35 00:01:47,000 --> 00:01:49,069 So we have one, two, three. 36 00:01:49,069 --> 00:01:51,559 So it's going to be equal to this. 37 00:01:51,560 --> 00:01:54,230 So it's going to be equal to 8 -- that's that guy 38 00:01:54,230 --> 00:01:56,370 right there -- 0.52. 39 00:01:56,370 --> 00:01:58,219 So everything after that first term is going to 40 00:01:58,219 --> 00:01:59,189 be behind the decimal. 41 00:01:59,189 --> 00:02:03,340 So 0.52 times 10 to the number of terms we have. 42 00:02:03,340 --> 00:02:04,520 One, two, three. 43 00:02:04,519 --> 00:02:07,390 10 to the minus 3. 44 00:02:07,390 --> 00:02:09,439 Another way to think of it: this is a little bit more. 45 00:02:09,439 --> 00:02:12,289 This is like 8 1/2 thousands, right? 46 00:02:12,289 --> 00:02:13,429 Each of these is thousands. 47 00:02:13,430 --> 00:02:15,290 We have 8 1/2 of them. 48 00:02:15,289 --> 00:02:16,519 Let's do this one. 49 00:02:16,520 --> 00:02:17,900 Let's see how many 0's we have. 50 00:02:17,900 --> 00:02:23,620 We have 3, 6, 9, 12. 51 00:02:23,620 --> 00:02:26,629 So we want to do -- again, we start with our largest 52 00:02:26,629 --> 00:02:27,729 term that we have. 53 00:02:27,729 --> 00:02:29,334 Our largest non-zero term. 54 00:02:29,335 --> 00:02:30,750 In this case, it's going to be the term all 55 00:02:30,750 --> 00:02:31,430 the way to the left. 56 00:02:31,430 --> 00:02:32,900 That's our 7. 57 00:02:32,900 --> 00:02:35,569 So it's going to be 7.012. 58 00:02:35,569 --> 00:02:40,959 It's going to be equal to 7.012 times 10 to the what? 59 00:02:40,960 --> 00:02:45,349 Well it's going to be times 10 to the 1 with this many 0's. 60 00:02:45,349 --> 00:02:46,299 So how many things? 61 00:02:46,300 --> 00:02:48,300 We had a 1 here. 62 00:02:48,300 --> 00:02:56,700 Then we had 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 0's. 63 00:02:56,699 --> 00:02:57,489 I want to be very clear. 64 00:02:57,490 --> 00:02:58,600 You're not just counting the 0's. 65 00:02:58,599 --> 00:03:02,859 You're counting everything after this first 66 00:03:02,860 --> 00:03:03,730 term right there. 67 00:03:03,729 --> 00:03:07,549 So it would be equivalent to a 1 followed by 12 0's. 68 00:03:07,550 --> 00:03:11,360 So it's times 10 to the twelfth. 69 00:03:11,360 --> 00:03:12,020 Just like that. 70 00:03:12,020 --> 00:03:12,790 Not too difficult. 71 00:03:12,789 --> 00:03:14,629 Let's do this one right here. 72 00:03:14,629 --> 00:03:16,650 So we go behind our decimal point. 73 00:03:16,650 --> 00:03:18,520 We find the first non-zero number. 74 00:03:18,520 --> 00:03:19,980 That's our 5. 75 00:03:19,979 --> 00:03:21,609 It's going to be equal to 5. 76 00:03:21,610 --> 00:03:24,800 There's nothing to the right of it, so it's 5.00 if we wanted 77 00:03:24,800 --> 00:03:26,710 to add some precision to it. 78 00:03:26,710 --> 00:03:30,879 But it's 5 times and then how many numbers to the right, or 79 00:03:30,879 --> 00:03:32,789 behind to the right of the decimal will do we have? 80 00:03:32,789 --> 00:03:40,759 We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and we 81 00:03:40,759 --> 00:03:42,819 have to include this one, 14. 82 00:03:42,819 --> 00:03:47,699 5 times 10 to the minus 14th power. 83 00:03:47,699 --> 00:03:50,639 Now this number, it might be a little overkill to write this 84 00:03:50,639 --> 00:03:52,669 in scientific notation, but it never hurts to get 85 00:03:52,669 --> 00:03:53,699 the practice. 86 00:03:53,699 --> 00:03:56,489 So what's the largest 10 that goes into this? 87 00:03:56,490 --> 00:03:58,640 Well, 100 will go into this. 88 00:03:58,639 --> 00:04:02,149 And you could figure out 100 or 10 squared by saying, "OK, this 89 00:04:02,150 --> 00:04:06,990 is our largest term." And then we have two 0's behind it 90 00:04:06,990 --> 00:04:11,210 because we can say 100 will go into 723. 91 00:04:11,210 --> 00:04:17,069 So this is going to be equal to 7.23 times, we could say times 92 00:04:17,069 --> 00:04:19,995 100, but we want to stay in scientific notation, so I'll 93 00:04:19,995 --> 00:04:22,240 write times 10 squared. 94 00:04:22,240 --> 00:04:24,360 Now we have this character right here. 95 00:04:24,360 --> 00:04:25,960 What's our first non-zero term? 96 00:04:25,959 --> 00:04:29,310 It's that one right there, so it's going to be 6 times and 97 00:04:29,310 --> 00:04:31,300 then how many terms do we have to the right of the decimal? 98 00:04:31,300 --> 00:04:32,350 We have only one. 99 00:04:32,350 --> 00:04:34,300 So times 10 to the minus 1. 100 00:04:34,300 --> 00:04:36,530 That makes a lot of sense because that's essentially 101 00:04:36,529 --> 00:04:39,169 equal to 6 divided by 10 because 10 to the minus 102 00:04:39,170 --> 00:04:42,509 1 is 1/10 which is 0.6. 103 00:04:42,509 --> 00:04:44,120 One more. 104 00:04:44,120 --> 00:04:45,769 Let me throw some commas here just to make this a 105 00:04:45,769 --> 00:04:48,529 little easier to look at. 106 00:04:48,529 --> 00:04:50,609 So let's take our largest value right there. 107 00:04:50,610 --> 00:04:53,790 We have our 8. 108 00:04:53,790 --> 00:04:59,220 This is going to be 8.23 -- we don't have to add the other 109 00:04:59,220 --> 00:05:03,250 stuff because everything else is a 0 -- times 10 to the -- 110 00:05:03,250 --> 00:05:06,069 we just count how many terms are after the 8. 111 00:05:06,069 --> 00:05:12,959 So we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 112 00:05:12,959 --> 00:05:15,169 8.23 times 10 to the 10. 113 00:05:15,170 --> 00:05:16,290 I think you get the idea now. 114 00:05:16,290 --> 00:05:17,590 It's pretty straightforward. 115 00:05:17,589 --> 00:05:20,859 And more than just being able to calculate this, which is a 116 00:05:20,860 --> 00:05:23,199 good skill by itself, I want you to understand why 117 00:05:23,199 --> 00:05:23,939 this is the case. 118 00:05:23,939 --> 00:05:25,819 Hopefully that last video explained it. 119 00:05:25,819 --> 00:05:28,389 And if it doesn't, just multiply this out. 120 00:05:28,389 --> 00:05:31,430 Literally multiply 8.23 times 10 to the 10 and 121 00:05:31,430 --> 00:05:32,769 you will get this number. 122 00:05:32,769 --> 00:05:34,189 Maybe you could try it with something smaller 123 00:05:34,189 --> 00:05:35,040 than 10 to the 10. 124 00:05:35,040 --> 00:05:36,110 Maybe 10 to the fifth. 125 00:05:36,110 --> 00:05:38,139 And well, you'll get a different number but 126 00:05:38,139 --> 00:05:40,979 you'll end up with five digits after the 8. 127 00:05:40,980 --> 00:05:45,180 But anyway, let me do a couple more computation examples. 128 00:05:45,180 --> 00:05:54,790 Let's say we had the numbers -- let me just make something 129 00:05:54,790 --> 00:05:58,410 really small -- 0.0000064. 130 00:05:58,410 --> 00:06:00,060 Let me make a large number. 131 00:06:00,060 --> 00:06:03,100 Let's say I have that number and I want to multiply it. 132 00:06:03,100 --> 00:06:06,450 I want to multiply it by -- let's say I have a really large 133 00:06:06,449 --> 00:06:11,610 number -- 3 2 -- I'm just going to throw a bunch of 0's here. 134 00:06:11,610 --> 00:06:12,530 I don't know when I'm going to stop. 135 00:06:12,529 --> 00:06:13,899 Let's say I stop there. 136 00:06:13,899 --> 00:06:15,560 So this one, you can multiply out. 137 00:06:15,560 --> 00:06:18,649 But it's a little difficult. 138 00:06:18,649 --> 00:06:20,679 But let's put it into scientific notation. 139 00:06:20,680 --> 00:06:23,090 One, it'll be easier to represent these numbers and 140 00:06:23,089 --> 00:06:25,799 then hopefully you'll see that the multiplication actually 141 00:06:25,800 --> 00:06:27,720 gets simplified as well. 142 00:06:27,720 --> 00:06:30,550 So this top guy right here, how can we write him in 143 00:06:30,550 --> 00:06:31,110 scientific notation? 144 00:06:31,110 --> 00:06:37,560 It would be 6.4 times 10 to the what? 145 00:06:37,560 --> 00:06:40,430 1, 2, 3, 4, 5, 6. 146 00:06:40,430 --> 00:06:41,439 I have to include the 6. 147 00:06:41,439 --> 00:06:43,480 So times 10 to the minus 6. 148 00:06:43,480 --> 00:06:45,800 And what can this one be written as? 149 00:06:45,800 --> 00:06:47,930 This one is going to be 3.2. 150 00:06:47,930 --> 00:06:51,150 151 00:06:51,149 --> 00:06:53,560 And then you count how many digits are after the 3. 152 00:06:53,560 --> 00:06:59,329 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. 153 00:06:59,329 --> 00:07:02,750 So 3.2 times 10 to the 11th. 154 00:07:02,750 --> 00:07:06,187 So if we multiply these two things, this is equivalent to 6 155 00:07:06,187 --> 00:07:12,650 -- let me do it in a different color -- 6.4 times 10 to 156 00:07:12,649 --> 00:07:21,589 the minus 6 times 3.2 times 10 to the 11th. 157 00:07:21,589 --> 00:07:26,159 Which we saw in the last video is equivalent to 6.4 times 3.2. 158 00:07:26,160 --> 00:07:29,360 I'm just changing the order of our multiplication. 159 00:07:29,360 --> 00:07:36,560 Times 10 to the minus 6 times 10 to the 11th power. 160 00:07:36,560 --> 00:07:38,300 And now what will this be equal to? 161 00:07:38,300 --> 00:07:40,490 Well, to do this, I don't want to use a calculator. 162 00:07:40,490 --> 00:07:42,530 So let's just calculate it. 163 00:07:42,529 --> 00:07:47,829 So 6.4 times 3.2. 164 00:07:47,829 --> 00:07:49,449 Let's ignore the decimals for a second. 165 00:07:49,449 --> 00:07:50,639 We'll worry about that at the end. 166 00:07:50,639 --> 00:07:55,060 So 2 times 4 is 8, 2 times 6 is 12. 167 00:07:55,060 --> 00:07:57,769 Nowhere to carry the 1, so it's just 128. 168 00:07:57,769 --> 00:07:58,810 Put a 0 down there. 169 00:07:58,810 --> 00:08:02,290 3 times 4 is 12, carry the 1. 170 00:08:02,290 --> 00:08:04,370 3 times 6 is 18. 171 00:08:04,370 --> 00:08:08,319 You've got a 1 there, so it's 192. 172 00:08:08,319 --> 00:08:08,699 Right? 173 00:08:08,699 --> 00:08:09,509 Yeah. 174 00:08:09,509 --> 00:08:09,670 192. 175 00:08:09,670 --> 00:08:13,920 You had them up and you get 8, 4, 1 plus 9 is 10. 176 00:08:13,920 --> 00:08:14,590 Carry the 1. 177 00:08:14,589 --> 00:08:15,779 You get 2. 178 00:08:15,779 --> 00:08:17,919 Now, we just have to count the numbers behind 179 00:08:17,920 --> 00:08:19,129 the decimal point. 180 00:08:19,129 --> 00:08:21,279 We have one number there, we have another number there. 181 00:08:21,279 --> 00:08:23,044 We have two numbers behind the decimal point, 182 00:08:23,045 --> 00:08:25,120 so you count 1, 2. 183 00:08:25,120 --> 00:08:34,879 So 6.4 times 3.2 is equal to 20.48 times 10 to the -- we 184 00:08:34,879 --> 00:08:37,929 have the same base here, so we can just add the exponents. 185 00:08:37,929 --> 00:08:40,449 So what's minus 6 plus 11? 186 00:08:40,450 --> 00:08:45,830 So that's 10 to the fifth power, right? 187 00:08:45,830 --> 00:08:46,180 Right. 188 00:08:46,179 --> 00:08:47,579 Minus 6 and 11. 189 00:08:47,580 --> 00:08:49,040 10 to the fifth power. 190 00:08:49,039 --> 00:08:50,689 And so the next question, you might say, "I'm done. 191 00:08:50,690 --> 00:08:52,940 I've done the computation." And you have. 192 00:08:52,940 --> 00:08:55,030 And this is a valid answer. 193 00:08:55,029 --> 00:08:58,269 But the next question is is this in scientific notation? 194 00:08:58,269 --> 00:09:01,179 And if you wanted to be a real stickler about it, it's not in 195 00:09:01,179 --> 00:09:03,529 scientific notation because we have something here that could 196 00:09:03,529 --> 00:09:06,139 maybe be simplified a little bit. 197 00:09:06,139 --> 00:09:09,100 We could write this -- let me do it this way. 198 00:09:09,100 --> 00:09:10,960 Let me divide this by 10. 199 00:09:10,960 --> 00:09:13,950 So any number we can multiply and divide by 10. 200 00:09:13,950 --> 00:09:16,129 So we could rewrite it this way. 201 00:09:16,129 --> 00:09:19,610 We could write 1/10 on this side and then we can multiply 202 00:09:19,610 --> 00:09:21,269 times 10 on that side, right? 203 00:09:21,269 --> 00:09:22,909 That shouldn't change the number. 204 00:09:22,909 --> 00:09:24,649 You divide by 10 and multiply it by 10. 205 00:09:24,649 --> 00:09:27,769 That's just like multiplying by 1 or dividing by 1. 206 00:09:27,769 --> 00:09:33,009 So if you divide this side by 10, you get 2.048. 207 00:09:33,009 --> 00:09:36,279 You multiply that side by 10 and you get times 10 to 208 00:09:36,279 --> 00:09:38,730 the -- times 10 is just times 10 to the first. 209 00:09:38,730 --> 00:09:39,779 You can just add the exponents. 210 00:09:39,779 --> 00:09:41,199 Times 10 to the sixth. 211 00:09:41,200 --> 00:09:44,280 And now, if you're a stickler about it, this is good 212 00:09:44,279 --> 00:09:48,839 scientific notation right there. 213 00:09:48,840 --> 00:09:50,620 Now, I've done a lot of multiplication. 214 00:09:50,620 --> 00:09:54,509 Let's do some division. 215 00:09:54,509 --> 00:09:57,009 Let's divide this guy by that guy. 216 00:09:57,009 --> 00:10:04,980 So if we have 3.2 times 10 to the eleventh power divided by 217 00:10:04,980 --> 00:10:09,600 6.4 times 10 to the minus six, what is this equal to? 218 00:10:09,600 --> 00:10:13,940 Well, this is equal to 3.2 over 6.4. 219 00:10:13,940 --> 00:10:16,300 We can just separate them out because it's associative. 220 00:10:16,299 --> 00:10:23,209 So, it's this times 10 to the 11th over 10 to 221 00:10:23,210 --> 00:10:24,290 the minus six, right? 222 00:10:24,289 --> 00:10:25,529 If you multiply these two things, you'll 223 00:10:25,529 --> 00:10:26,929 get that right there. 224 00:10:26,929 --> 00:10:30,129 So 3.2 over 6.4. 225 00:10:30,129 --> 00:10:32,029 This is just equal to 0.5, right? 226 00:10:32,029 --> 00:10:36,169 32 is half of 64 or 3.2 is half of 6.4, so this 227 00:10:36,169 --> 00:10:38,139 is 0.5 right there. 228 00:10:38,139 --> 00:10:39,350 And what is this? 229 00:10:39,350 --> 00:10:43,100 This is 10 to the 11th over 10 to the minus 6. 230 00:10:43,100 --> 00:10:45,700 So when you have something in the denominator, you 231 00:10:45,700 --> 00:10:46,620 could write it this way. 232 00:10:46,620 --> 00:10:51,509 This is equivalent to 10 to the 11th over 10 to the minus 6. 233 00:10:51,509 --> 00:10:56,069 It's equal to 10 to the 11th times 10 to the 234 00:10:56,070 --> 00:10:58,800 minus 6 to the minus 1. 235 00:10:58,799 --> 00:11:03,059 Or this is equal to 10 to the 11th times 10 to the sixth. 236 00:11:03,059 --> 00:11:04,609 And what did I do just there? 237 00:11:04,610 --> 00:11:07,139 This is 1 over 10 to the minus 6. 238 00:11:07,139 --> 00:11:09,580 So 1 over something is just that something to the 239 00:11:09,580 --> 00:11:10,500 negative 1 power. 240 00:11:10,500 --> 00:11:12,279 And then I multiplied the exponents. 241 00:11:12,279 --> 00:11:14,110 You can think of it that way and so this would be equal 242 00:11:14,110 --> 00:11:19,000 to 10 to the 17th power. 243 00:11:19,000 --> 00:11:21,620 Or another way to think about it is if you have 1 -- you have 244 00:11:21,620 --> 00:11:26,360 the same bases, 10 in this case, and you're dividing them, 245 00:11:26,360 --> 00:11:29,399 you just take the 1 the numerator and you subtract the 246 00:11:29,399 --> 00:11:30,220 exponent in the denominator. 247 00:11:30,220 --> 00:11:35,009 So it's 11 minus minus 6, which is 11 plus 6, 248 00:11:35,009 --> 00:11:36,909 which is equal to 17. 249 00:11:36,909 --> 00:11:41,209 So this division problem ended up being equal to 250 00:11:41,210 --> 00:11:46,290 0.5 times 10 to the 17th. 251 00:11:46,289 --> 00:11:49,149 Which is the correct answer, but if you wanted to be a 252 00:11:49,149 --> 00:11:51,409 stickler and put it into scientific notation, we want 253 00:11:51,409 --> 00:11:54,189 something maybe greater than 1 right here. 254 00:11:54,190 --> 00:11:55,800 So the way we can do that, let's multiply 255 00:11:55,799 --> 00:11:59,419 it by 10 on this side. 256 00:11:59,419 --> 00:12:02,990 And divide by 10 on this side or multiply by 1/10. 257 00:12:02,990 --> 00:12:04,879 Remember, we're not changing the number if you multiply 258 00:12:04,879 --> 00:12:06,659 by 10 and divide by 10. 259 00:12:06,659 --> 00:12:08,949 We're just doing it to different parts of the product. 260 00:12:08,950 --> 00:12:15,629 So this side is going to become 5 -- I'll do it in pink -- 10 261 00:12:15,629 --> 00:12:20,929 times 0.5 is 5, times 10 to the 17th divided by 10. 262 00:12:20,929 --> 00:12:24,189 That's the same thing as 10 to the 17th times 10 263 00:12:24,190 --> 00:12:25,650 to the minus 1, right? 264 00:12:25,649 --> 00:12:27,120 That's 10 to the minus 1. 265 00:12:27,120 --> 00:12:29,320 So it's equal to 10 to the 16th power. 266 00:12:29,320 --> 00:12:31,930 267 00:12:31,929 --> 00:12:35,329 Which is the answer when you divide these two 268 00:12:35,330 --> 00:12:36,490 guys right there. 269 00:12:36,490 --> 00:12:39,750 So hopefully these examples have filled in all of the 270 00:12:39,750 --> 00:12:42,250 gaps or the uncertain scenarios dealing with 271 00:12:42,250 --> 00:12:43,259 scientific notation. 272 00:12:43,259 --> 00:12:46,220 If I haven't covered something, feel free to write a comment on 273 00:12:46,220 --> 00:12:48,509 this video or pop me an e-mail.