1 00:00:00,042 --> 00:00:02,557 Before we get into the meat of algebra, 2 00:00:02,557 --> 00:00:03,895 I want to give you a quote 3 00:00:03,895 --> 00:00:05,879 from one of the greatest minds in human history, 4 00:00:05,879 --> 00:00:09,591 Galileo Galilei, because I think this quote encapsulates 5 00:00:09,591 --> 00:00:13,907 the true point of algebra and really mathematics in general. 6 00:00:13,907 --> 00:00:16,991 He said : Philosophy is written in that great book 7 00:00:16,991 --> 00:00:19,324 which ever lies before our eyes 8 00:00:19,324 --> 00:00:20,958 -- I mean the universe -- 9 00:00:20,958 --> 00:00:22,944 but we cannot understand it 10 00:00:22,944 --> 00:00:23,444 if we do not first learn the language 11 00:00:23,444 --> 00:00:27,601 and grasp the symbols in which it is written. 12 00:00:27,601 --> 00:00:28,101 This book is written in the mathematical language... 13 00:00:28,101 --> 00:00:35,823 without which one wanders in vain through a dark labyrinth. 14 00:00:35,823 --> 00:00:38,298 So very dramatic but very deep 15 00:00:38,298 --> 00:00:38,798 and this really is the point of mathematics 16 00:00:38,798 --> 00:00:42,470 and what we'll see as we start 17 00:00:42,470 --> 00:00:44,874 getting deeper and deeper into algebra 18 00:00:44,874 --> 00:00:46,918 is that we're gonna start abstracting things 19 00:00:46,918 --> 00:00:48,915 and we're going to start getting to core ideas 20 00:00:48,915 --> 00:00:51,861 that start explaining really how the universe is structured. 21 00:00:51,861 --> 00:00:54,021 Sure these ideas can be applied to things 22 00:00:54,021 --> 00:00:56,923 like economics and finance and physics and chemistry 23 00:00:56,923 --> 00:00:58,933 but at their core they're the same idea. 24 00:00:58,933 --> 00:01:02,368 And so they're more fundamental, more pure, 25 00:01:02,368 --> 00:01:05,076 than any one of those applications. 26 00:01:05,076 --> 00:01:08,103 And to see what I mean by getting down to the root idea. 27 00:01:08,103 --> 00:01:10,580 Let's go with a...I guess we could, 28 00:01:10,580 --> 00:01:13,357 we started with the very grand philosophy of the universe 29 00:01:13,357 --> 00:01:14,536 as written in mathematics. 30 00:01:14,536 --> 00:01:17,863 but let's start with a very concrete simple idea. 31 00:01:17,863 --> 00:01:21,067 but we'll keep abstracting and see how the same idea connects 32 00:01:21,067 --> 00:01:24,401 across many domains in our universe. 33 00:01:24,401 --> 00:01:26,898 So lets just say we're at the store 34 00:01:26,898 --> 00:01:30,024 and we're going to buy something and there is a sale. 35 00:01:30,024 --> 00:01:36,200 The sale says that it is 30% off and I'm interested -- 36 00:01:36,200 --> 00:01:38,871 and I don't shop at too fancy stores -- 37 00:01:38,871 --> 00:01:40,810 so let's say I'm interested in a pair of pants 38 00:01:40,810 --> 00:01:43,375 and the pair of pants before the sale event is about $20. 39 00:01:43,375 --> 00:01:46,500 That is about how much I spend on my pants. 40 00:01:46,500 --> 00:01:49,625 so I'm interested in a $20 pair of pants but it's even better. 41 00:01:49,625 --> 00:01:54,267 there is a thirty percent off sale on these pants. 42 00:01:54,267 --> 00:01:57,702 How would I think about how much I'm going to get off of that $20 ? 43 00:01:57,702 --> 00:01:59,200 And this isn't algebra yet 44 00:01:59,200 --> 00:02:01,169 this is something that you have probably had exposure to. 45 00:02:01,169 --> 00:02:04,800 You would multiply thirty percent times the twenty dollars. 46 00:02:04,800 --> 00:02:09,974 so you would say your discount --your discount-- is equal to, 47 00:02:09,974 --> 00:02:14,115 you could write is as 30% times $20. 48 00:02:14,115 --> 00:02:16,533 We'll write twenty dollars in purple. 49 00:02:16,533 --> 00:02:19,200 Or you could write it --if you wanted to write-- this is a decimal 50 00:02:19,200 --> 00:02:24,200 you could write it as 0.30*20 dollars 51 00:02:24,200 --> 00:02:25,693 And if you were to do the maths 52 00:02:25,693 --> 00:02:27,821 you would get 6 dollars. 53 00:02:27,821 --> 00:02:30,067 So nothing, nothing new over there. 54 00:02:30,067 --> 00:02:32,354 But what if I wanted to generalize a little bit ? 55 00:02:32,354 --> 00:02:35,400 That's the discount on this particular pair of pants, 56 00:02:35,400 --> 00:02:37,400 but what if I wanted to know the discount 57 00:02:37,400 --> 00:02:39,333 on anything in the store ? 58 00:02:39,333 --> 00:02:43,133 Well then I could say, let x be the price 59 00:02:43,133 --> 00:02:44,486 -- let's do it in another color -- 60 00:02:44,486 --> 00:02:46,221 I'm just gonna make a symbol. 61 00:02:46,221 --> 00:02:51,169 Let x be the price of the product I want to buy, 62 00:02:51,169 --> 00:03:00,837 the nondiscount price of the product in the store. 63 00:03:00,837 --> 00:03:06,142 So now all of a sudden, we can say our discount, 64 00:03:06,142 --> 00:03:15,267 is equal to 30%, 30% times x, 65 00:03:15,267 --> 00:03:17,465 or if we wanted to write it as a decimal, 66 00:03:17,465 --> 00:03:19,411 we could write 30% as a decimal. 67 00:03:19,411 --> 00:03:25,329 we could write 0.30 times x --times x--. 68 00:03:25,329 --> 00:03:27,351 Now this is interesting. 69 00:03:27,351 --> 00:03:29,841 Now you give me the price of any product in the store 70 00:03:29,841 --> 00:03:31,608 and I can substitute it in for x, 71 00:03:31,608 --> 00:03:34,395 and then I can essentially multiply it 0.3 times that 72 00:03:34,395 --> 00:03:36,225 and I would get the discount. 73 00:03:36,225 --> 00:03:39,533 Now we're starting to, very slowly, 74 00:03:39,533 --> 00:03:42,331 starting to get into the abstraction of algebra. 75 00:03:42,331 --> 00:03:44,867 We'll see that they would get much more nuanced and deep 76 00:03:44,867 --> 00:03:46,591 and frankly more beautiful 77 00:03:46,591 --> 00:03:52,409 as we start studying more and more algebraic ideas. 78 00:03:52,409 --> 00:03:54,080 But we aren't done here. 79 00:03:54,080 --> 00:03:56,077 We can abstract this even more. 80 00:03:56,077 --> 00:03:57,643 Over here we've said 81 00:03:57,643 --> 00:03:59,067 we've generalized this for any product, 82 00:03:59,067 --> 00:04:01,400 we're not just saying for this 20 dollar product. 83 00:04:01,400 --> 00:04:02,795 If there's a 10 dollar product, 84 00:04:02,795 --> 00:04:05,737 we can put that 10 dollar product in here for x, 85 00:04:05,737 --> 00:04:07,966 and then we would say 0.30 times 10. 86 00:04:07,966 --> 00:04:10,667 The discount would be 3 dollars. 87 00:04:10,667 --> 00:04:12,795 It might be a $100 product, 88 00:04:12,795 --> 00:04:14,303 then the discount would be 30 dollars. 89 00:04:14,303 --> 00:04:16,071 But let's generalize even more. 90 00:04:16,071 --> 00:04:19,576 Let's say : "What is the discount for any given sale, 91 00:04:19,576 --> 00:04:21,133 --when the sale has a certain percentage-- ?" 92 00:04:21,133 --> 00:04:22,919 Now we can say that the discount 93 00:04:22,919 --> 00:04:25,759 --let me define a variable-- 94 00:04:25,759 --> 00:04:32,290 Let's let m = ... I'll say p, just so it makes sense, 95 00:04:32,290 --> 00:04:41,200 p is equal to the percentage off. 96 00:04:41,200 --> 00:04:42,667 Now what can we do ? 97 00:04:42,667 --> 00:04:47,046 Now we can say that the discount 98 00:04:47,046 --> 00:04:50,405 is equal to the percentage off. 99 00:04:50,405 --> 00:04:52,733 In these other examples we were picking 30 percent, 100 00:04:52,733 --> 00:04:55,729 but now we can say it's p, it's the percentage off. 101 00:04:55,729 --> 00:04:56,852 It's p. 102 00:04:56,852 --> 00:05:01,200 That's the percentage off times the product in question, 103 00:05:01,200 --> 00:05:04,262 times the price, the non discount price of the product in question. 104 00:05:04,262 --> 00:05:06,138 Well, that was x. 105 00:05:06,138 --> 00:05:08,667 The discount is equal to p times x. 106 00:05:08,667 --> 00:05:10,364 Now this is really interesting, 107 00:05:10,364 --> 00:05:13,268 now we have a general way of calculating a discount 108 00:05:13,268 --> 00:05:19,243 for any given percentage off and any given product x. 109 00:05:19,243 --> 00:05:22,333 And we didn't have to use these words and these letters. 110 00:05:22,333 --> 00:05:25,402 We could have said "Let y equal the discount" 111 00:05:25,402 --> 00:05:28,998 Let y is equal to the discount. 112 00:05:28,998 --> 00:05:32,373 then we could have written the same underlying idea 113 00:05:32,373 --> 00:05:34,400 instead of writing discount, we could have written 114 00:05:34,400 --> 00:05:40,200 y is equal to the percentage off p 115 00:05:40,200 --> 00:05:44,067 times the non discount price of the product 116 00:05:44,067 --> 00:05:45,667 --times x. 117 00:05:45,667 --> 00:05:47,779 And you could have defined these letters anyway you wanted. 118 00:05:47,779 --> 00:05:49,513 Instead of writing y there, 119 00:05:49,513 --> 00:05:52,267 you could have written a greek letter, 120 00:05:52,267 --> 00:05:53,779 as long as you can keep track of the fact 121 00:05:53,779 --> 00:05:56,729 that the symbol represents the actual dollar discount. 122 00:05:56,729 --> 00:05:58,911 But now things get really interesting, 123 00:05:58,911 --> 00:06:02,215 because we can use this type of a relationship... 124 00:06:02,215 --> 00:06:03,741 which is an equation, you're equating 125 00:06:03,741 --> 00:06:05,600 y to this thing right over here, 126 00:06:05,600 --> 00:06:07,734 that's why we call it an equation. 127 00:06:07,734 --> 00:06:09,406 This can be used for things 128 00:06:09,406 --> 00:06:10,985 that are completely unrelated 129 00:06:10,985 --> 00:06:14,231 to the discount price at the store over here. 130 00:06:14,231 --> 00:06:15,810 You might have... 131 00:06:15,810 --> 00:06:17,431 so in physics you will see 132 00:06:17,431 --> 00:06:21,600 that force is equal to mass times acceleration. 133 00:06:21,600 --> 00:06:23,000 The letters are different 134 00:06:23,000 --> 00:06:24,933 but these are fundamentally the same idea. 135 00:06:24,933 --> 00:06:31,754 We could have let y is equal to force 136 00:06:31,754 --> 00:06:36,067 and m is equal to... or mass is equal to p. 137 00:06:36,067 --> 00:06:39,200 So let me write p is equal to mass 138 00:06:39,200 --> 00:06:42,333 -- and this wouldn't be an intuitive way to define it -- 139 00:06:42,333 --> 00:06:45,646 but I wanna show you that the same idea, the same relationship 140 00:06:45,646 --> 00:06:48,323 that's being applied to 2 different things. 141 00:06:48,323 --> 00:06:51,314 And we could say x is equal to acceleration. 142 00:06:51,333 --> 00:06:56,961 We could say x is equal to acceleration. 143 00:06:56,961 --> 00:07:01,698 Well then the famous "force is equal to mass times acceleration" 144 00:07:01,698 --> 00:07:03,795 can be rewritten and it is really the same exact idea 145 00:07:03,800 --> 00:07:07,533 as y -- which we've defined has force -- 146 00:07:07,533 --> 00:07:08,964 can be equal to mass 147 00:07:08,964 --> 00:07:10,210 -- for which we are going to use the symbol p -- 148 00:07:10,210 --> 00:07:13,482 which is equal to p times acceleration. 149 00:07:13,482 --> 00:07:15,579 We are just going to happen to use the letter x here 150 00:07:15,579 --> 00:07:17,298 -- times x. 151 00:07:17,298 --> 00:07:19,200 Well this is the exact same equation. 152 00:07:19,200 --> 00:07:21,400 This is the exact same equation. 153 00:07:21,400 --> 00:07:23,200 We can see that we can take this equation 154 00:07:23,200 --> 00:07:28,287 and it can apply to things in economics, 155 00:07:28,287 --> 00:07:30,864 or it can apply to things in finance 156 00:07:30,864 --> 00:07:33,019 or it can apply to things in computer science 157 00:07:33,019 --> 00:07:35,123 or logic or electrical engineering 158 00:07:35,123 --> 00:07:36,338 or anything -- accounting -- 159 00:07:36,338 --> 00:07:38,400 There is an infinite number of applications 160 00:07:38,400 --> 00:07:40,200 of this one equation. 161 00:07:40,200 --> 00:07:42,054 And what's neat about mathematics, 162 00:07:42,054 --> 00:07:44,902 and what's neat about algebra in particular is 163 00:07:44,902 --> 00:07:47,087 we can focus on this abstraction. 164 00:07:47,087 --> 00:07:49,975 we can focus on the abstract here, 165 00:07:49,975 --> 00:07:52,185 we can manipulate the abstract here, 166 00:07:52,185 --> 00:07:55,462 and what we discover from these ideas, from these manipulations 167 00:07:55,462 --> 00:07:58,533 can then go and be reapplied 168 00:07:58,533 --> 00:08:02,000 to all of these other applications, to all of them. 169 00:08:02,000 --> 00:08:03,733 And even neater, it's kind of telling us 170 00:08:03,733 --> 00:08:06,063 the true structure of the universe 171 00:08:06,063 --> 00:08:08,718 if you were to strip away all of these human definitions 172 00:08:08,733 --> 00:08:10,429 and all of those human applications. 173 00:08:10,429 --> 00:08:11,729 So for example we can say : 174 00:08:11,729 --> 00:08:14,237 "Look, if y is equal to p times x". 175 00:08:14,237 --> 00:08:15,723 So litterally if someone said 176 00:08:15,723 --> 00:08:17,267 "Hey, this is y." 177 00:08:17,267 --> 00:08:19,692 and someone says on the other hand "I have p times x." 178 00:08:19,692 --> 00:08:22,482 I can say "Well, you have the same thing in both of your hands." 179 00:08:22,482 --> 00:08:25,438 And if your were to divide one of them by a number 180 00:08:25,438 --> 00:08:27,533 and you wanted them to still be equal, 181 00:08:27,533 --> 00:08:30,200 you'd divide the other one by that number. 182 00:08:30,200 --> 00:08:31,652 So for example, 183 00:08:31,652 --> 00:08:34,577 let's say -- we know that y is equal to p times x -- 184 00:08:34,577 --> 00:08:38,575 what if you want to have them both be equal. 185 00:08:38,575 --> 00:08:41,595 What is y divided by x gonna be equal to ? 186 00:08:41,595 --> 00:08:43,467 Well y was equal to p times x 187 00:08:43,467 --> 00:08:45,073 so y divided by x 188 00:08:45,073 --> 00:08:48,731 is gonna be the same as p times x divided by x 189 00:08:48,733 --> 00:08:50,321 But now this is interesting, 190 00:08:50,321 --> 00:08:52,544 because p times x divided by x 191 00:08:52,544 --> 00:08:53,711 -- well if you multiply by something 192 00:08:53,711 --> 00:08:55,568 and divide by that something 193 00:08:55,568 --> 00:08:57,333 you're gonna get the original number. 194 00:08:57,333 --> 00:08:59,467 If you multiply by 5 and divide by 5, 195 00:08:59,467 --> 00:09:01,373 you're just gonna get p, 196 00:09:01,373 --> 00:09:02,867 or whatever this number is. 197 00:09:02,867 --> 00:09:04,333 So those would cancel out. 198 00:09:04,333 --> 00:09:06,333 But we were able to manipulate the abstraction here, 199 00:09:06,333 --> 00:09:11,159 and get y over x is equal to p 200 00:09:11,159 --> 00:09:12,380 -- we make that x green -- 201 00:09:12,380 --> 00:09:16,324 y over x is equal to p 202 00:09:16,324 --> 00:09:18,667 and now this has implications, 203 00:09:18,667 --> 00:09:21,067 this has implications for every one of these ideas. 204 00:09:21,067 --> 00:09:24,805 One is telling us a fundamental truth about the universe, 205 00:09:24,805 --> 00:09:26,637 almost devoid of any of those applications, 206 00:09:26,637 --> 00:09:28,467 but now we can go and take them back 207 00:09:28,467 --> 00:09:30,400 to any place where we applied it. 208 00:09:30,400 --> 00:09:32,100 The really interesting thing is we're gonna find new... 209 00:09:32,100 --> 00:09:33,831 -- there are an infinite number of applications 210 00:09:33,831 --> 00:09:36,400 and we don't even know, frankly, most of them. 211 00:09:36,400 --> 00:09:38,386 We're gonna discover new ones for them 212 00:09:38,386 --> 00:09:40,290 in a thousand years. 213 00:09:40,290 --> 00:09:41,785 And so hopefully this gives you a sense 214 00:09:41,785 --> 00:09:44,933 for why Galileo said what he said about 215 00:09:44,933 --> 00:09:48,133 mathematics is really the language 216 00:09:48,133 --> 00:09:51,959 with which we can understand the philosophy of the universe. 217 00:09:51,959 --> 00:09:53,567 That's why people will tell us 218 00:09:53,567 --> 00:09:56,714 that if a completely alien life form would contact humans, 219 00:09:56,714 --> 00:10:00,067 mathematics would probably be our first common ground, 220 00:10:00,067 --> 00:10:04,108 the place that we can start to form a basis 221 00:10:04,108 --> 00:10:07,108 that we can start to communicate from.