1 00:00:00,025 --> 00:00:10,189 [music] What do I get if I take one cubed plus two cubed plus three cubed plus so on 2 00:00:10,189 --> 00:00:14,158 plus K cubed? This is a truly remarkable fact it's 3 00:00:14,158 --> 00:00:19,183 called Nicomachus' Theorem, and it says that the sum of the first K perfect cubes 4 00:00:19,183 --> 00:00:23,983 is the square of just the sum of the first K whole numbers, and it's literally a 5 00:00:23,983 --> 00:00:27,943 remarkable fact that these two things are, are equal, right? 6 00:00:27,943 --> 00:00:32,841 I mean, you know, it just doesn't look like this should be true at all, but it 7 00:00:32,841 --> 00:00:35,200 is. Now, once you believe something like this 8 00:00:35,200 --> 00:00:39,022 you could rewrite this sum of cubes formula in maybe a slightly easier way. 9 00:00:39,022 --> 00:00:44,183 Right, once you've got the sum of cubes is the square of just the sum of the first k 10 00:00:44,183 --> 00:00:49,019 whole numbers, while we already know a formula for some of the first k whole 11 00:00:49,019 --> 00:00:52,568 numbers. It's k times k plus 1 over 2, and I'm just 12 00:00:52,568 --> 00:00:56,380 squaring that. So now here is a formula for the sum of 13 00:00:56,380 --> 00:01:01,152 the first k perfect cubes. It's k squared times k plus 1 squared over 14 00:01:01,152 --> 00:01:01,663 4. Right? 15 00:01:01,663 --> 00:01:07,806 I just took this formula and squared it. Again there's a slick geometric proof of 16 00:01:07,806 --> 00:01:11,026 this fact. So let's add 1 cubed plus 2 cubed plus 3 17 00:01:11,026 --> 00:01:15,738 cubed plus 4 cubed pictorially, of course the same arguments going to work in 18 00:01:15,738 --> 00:01:18,646 general, if I add up the first k perfect cubes. 19 00:01:18,646 --> 00:01:20,759 Alright, well here's 1 cubed. Alright. 20 00:01:20,760 --> 00:01:24,402 Here's 2 cubed. It's 8 dots, but I've arranged them in a, 21 00:01:24,402 --> 00:01:27,319 in a cube. Here's 3 cubed, all right? 22 00:01:27,319 --> 00:01:33,036 It's 3 times 3 times 3, 27 dots, but I've arranged them again in, in a cube shape. 23 00:01:33,036 --> 00:01:38,073 And then 4 cubed would be 64 dots, but I haven't drawn all the dots in, right? 24 00:01:38,073 --> 00:01:42,610 So I want to count all of these dots, and the trick is to rearrange these dots. 25 00:01:42,610 --> 00:01:46,620 So that's how most of these sort of pictorial arguments end up going. 26 00:01:46,620 --> 00:01:51,617 So I'm going to rearrange these dots like, like this. 27 00:01:51,618 --> 00:01:58,955 So here I've made a square, and here's the 1 cube, I added just, just 1 dot. 28 00:01:58,956 --> 00:02:03,181 Here's the 2 cube, but I've rearranged it where 1 of the layers of the 2 by 2 by 2 29 00:02:03,181 --> 00:02:06,754 cube is right here, and I've split the other layer in half. 30 00:02:06,755 --> 00:02:11,869 So if I take this vertical 2 by 1 piece and this horizontal 1 by 2 piece, I can 31 00:02:11,869 --> 00:02:16,688 rearrange them to make it 2 by 2 square that I stack on top of that one, which 32 00:02:16,688 --> 00:02:22,600 then gives me this again. Here's the 3 by 3 by 3 cube, it's 3, 3 by 33 00:02:22,600 --> 00:02:27,469 3 squares. Here's the 4 by 4 by 4 cube. 34 00:02:27,469 --> 00:02:32,240 It's 4 squares, but that top square is been cut in half. 35 00:02:32,240 --> 00:02:37,269 Right, so this piece here, and this piece here, would combine to give me another 4 36 00:02:37,269 --> 00:02:41,769 by 4 square, which I can stack on top of these three to recover the 4 by 4 by 4 37 00:02:41,769 --> 00:02:45,211 cube. Now I could put five 5 by 5 squares 38 00:02:45,211 --> 00:02:50,357 around, and then around that I could put six 6 by 6 squares, but I'd cut one of 39 00:02:50,357 --> 00:02:55,828 those squares in half just like this. So the even layers I use square that's 40 00:02:55,828 --> 00:03:00,147 been cut in half, and the odd layers, I can just stack them down. 41 00:03:00,147 --> 00:03:08,153 The, the point here is that I end up producing a square whose side length is 1 42 00:03:08,153 --> 00:03:11,590 plus 2 plus 3 plus 4. You know, and of course this is being 43 00:03:11,590 --> 00:03:13,807 general. Alright, it would be plus k. 44 00:03:13,808 --> 00:03:20,026 So this is 1 plus 2 plus 3 plus 4. I've rearranged the sum of cubes into a 45 00:03:20,026 --> 00:03:26,206 square, and that side length of that square is the sum of the first k whole 46 00:03:26,206 --> 00:03:29,924 numbers. Good news now is that I've got a formula 47 00:03:29,924 --> 00:03:35,277 for the sum of the first k whole numbers. It's k times k plus 1 over 2. 48 00:03:35,277 --> 00:03:42,495 So the resulting square has side length k times k plus 1 over 2, which then means 49 00:03:42,495 --> 00:03:48,617 that if I take the sum of the first K cubes it's that number squared. 50 00:03:48,618 --> 00:03:53,190 These kinds of formulas enable us to perform calculations that would be 51 00:03:53,190 --> 00:03:57,138 extraordinarily painful without access to these formulas. 52 00:03:57,138 --> 00:04:02,155 For instance, what if I wanted to add up the first 100 perfect cubes, right. 53 00:04:02,155 --> 00:04:09,256 That's the same thing as computing the sum, n goes from 1 to 100, of n cubed. 54 00:04:09,256 --> 00:04:16,672 Now, because I've got a formula for this, because this is, the same as, just the sum 55 00:04:16,672 --> 00:04:21,454 of The first hundred whole numbers squared, right? 56 00:04:21,455 --> 00:04:25,908 I know how to compute this, right? What's the sum of the first hundred whole 57 00:04:25,908 --> 00:04:30,880 numbers, well that's 100 times 101 over 2, and I just have to square that. 58 00:04:30,880 --> 00:04:34,519 What's a 100 times 101 over 2? What's the sum of the first hundred 59 00:04:34,519 --> 00:04:39,670 numbers? It's 5,050 and I just have to square this. 60 00:04:39,670 --> 00:04:44,770 And if I square this I get 2, 5, 5, 0, 2, 5, 0, 0. 61 00:04:44,771 --> 00:04:48,322 All right? So I get about 25 million, and this makes 62 00:04:48,322 --> 00:04:53,684 possible, you know, this calculation that would have been horrible tedious, right? 63 00:04:53,685 --> 00:04:58,509 But because I've got access to this formula, I can perform really amazing 64 00:04:58,509 --> 00:05:02,576 numeric feats, right? This would be horrible to calculate, but 65 00:05:02,576 --> 00:05:05,797 with this formula, I can just write down the answer. 66 00:05:05,798 --> 00:05:09,247 If you like these kinds of formulas, here's a challenge. 67 00:05:09,248 --> 00:05:12,998 Here's the challenge. What's the sum of fourth powers. 68 00:05:12,999 --> 00:05:18,252 The answer's going to be some polynomial in K, right, but you've gotta figure out 69 00:05:18,252 --> 00:05:23,510 which polynomial that is. One trick would be to plug in values. 70 00:05:23,510 --> 00:05:26,320 Say K equals one, K equals two, K equals three. 71 00:05:26,320 --> 00:05:31,260 And then try to find the polynomial that passes through those points. 72 00:05:31,260 --> 00:05:36,147 Now that's your challenge. And there's a long tradition here. 73 00:05:36,148 --> 00:05:43,510 Bernoulli for instance computed the sum of the first thousand tenths powers, and he 74 00:05:43,510 --> 00:05:51,164 did it 300 years ago. People have been thinking about these 75 00:05:51,164 --> 00:05:57,403 kinds of formulas for a really long time.