1 00:00:00,165 --> 00:00:02,988 Let's put all the integers together. 2 00:00:02,988 --> 00:00:09,156 [MUSIC] 3 00:00:09,156 --> 00:00:10,605 The goal here is to come up 4 00:00:10,605 --> 00:00:14,300 with a sequence that mentions every single integer. 5 00:00:14,300 --> 00:00:16,940 But before we do that, let's first try to come 6 00:00:16,940 --> 00:00:21,030 up with a sequence that mentions every non negative integer. 7 00:00:21,030 --> 00:00:28,250 It's the sequence a sub n equals n where the index n starts at zero. 8 00:00:28,250 --> 00:00:33,790 So the terms of the sequence are zero, one, two, three, and so on. 9 00:00:33,790 --> 00:00:34,480 And of course, 10 00:00:34,480 --> 00:00:36,420 I could negate that sequence to get 11 00:00:36,420 --> 00:00:40,140 a sequence that mentions every negative integer. 12 00:00:40,140 --> 00:00:42,982 So I could look at the sequence b sub 13 00:00:42,982 --> 00:00:47,770 n equals negative n, but let's start n at one. 14 00:00:47,770 --> 00:00:54,010 And if I do that, then the terms of this sequence start minus one, minus two, 15 00:00:54,010 --> 00:01:00,170 minus three, minus four and so on. But I want both, I want a single sequence 16 00:01:00,170 --> 00:01:02,880 that includes among it's terms every 17 00:01:02,880 --> 00:01:06,780 positive integer, every negative integer and zero. 18 00:01:06,780 --> 00:01:08,490 Is it even possible? 19 00:01:08,490 --> 00:01:13,110 Yes, I'll, I'll weave together the two sequences that we've already built. 20 00:01:13,110 --> 00:01:14,140 So what do I mean? 21 00:01:14,140 --> 00:01:15,890 Well let's take a look at these two sequences. 22 00:01:15,890 --> 00:01:18,840 I could put them together, right. I could weave them together. 23 00:01:18,840 --> 00:01:21,200 I could start with 0, then do minus 1, 24 00:01:21,200 --> 00:01:24,910 then 1, then minus 2, then 2, then minus 3. 25 00:01:24,910 --> 00:01:25,260 I get 26 00:01:25,260 --> 00:01:30,460 a new sequence that would end up mentioning every single integer, with 27 00:01:30,460 --> 00:01:36,020 start zero, minus one, one. Minus two, two, 28 00:01:36,020 --> 00:01:41,710 minus three, three, minus four, four, and it would keep on going like that. 29 00:01:41,710 --> 00:01:44,290 I'd like a formula for that sequence. 30 00:01:44,290 --> 00:01:50,530 Well here's a formula for the sequence. C sub n will be defined 31 00:01:50,530 --> 00:01:56,810 via this piecewise definition depending on the parity of n, whether n is odd or even. 32 00:01:56,810 --> 00:02:00,990 If n is odd, then the nth term will be negative n plus 1 over 2. 33 00:02:00,990 --> 00:02:04,750 And if n is even, then the nth term is just n over two. 34 00:02:04,750 --> 00:02:07,980 And I'll start with the index zero. 35 00:02:07,980 --> 00:02:12,638 And this will give me this sequence, right, the zero term, when I plug in 36 00:02:12,638 --> 00:02:16,350 zero, zero is even, zero over two is zero, and that gives me the zero. 37 00:02:16,350 --> 00:02:18,870 When I plug in one, one is odd, so I 38 00:02:18,870 --> 00:02:22,460 get negative one plus one over two, that's negative one. 39 00:02:22,460 --> 00:02:27,110 When I plug in two that's even, so it's two over two, that's this one here. 40 00:02:27,110 --> 00:02:30,532 When I plug in three, three's odd, so it's negative three plus one 41 00:02:30,532 --> 00:02:34,550 over two, that's negative two and it just keeps on going like that. 42 00:02:34,550 --> 00:02:36,350 There's another way to think about this. 43 00:02:36,350 --> 00:02:41,490 To say that I've got the same quantity of dots and squares. 44 00:02:41,490 --> 00:02:47,020 Is really to say that there's some method by which I can pair off the 45 00:02:47,020 --> 00:02:51,620 dots and squares so that every square gets a dot and every dot gets a square. 46 00:02:51,620 --> 00:02:53,560 And once I've paired them off like this it's very 47 00:02:53,560 --> 00:02:57,650 believable that there's the same quantity of dots and squares. 48 00:02:57,650 --> 00:03:02,098 Well something similar is going on with non-negative integers and all integers. 49 00:03:02,098 --> 00:03:06,660 If I just take a look at the non negative integers, I perhaps want 50 00:03:06,660 --> 00:03:09,640 to show others the same quantity of non 51 00:03:09,640 --> 00:03:13,860 negative integers as there are just all integers. 52 00:03:13,860 --> 00:03:16,270 And to do that, I just have to tell you some 53 00:03:16,270 --> 00:03:20,750 method for pairing off non negative integers with all the integers. 54 00:03:20,750 --> 00:03:23,470 And that's exactly what this sequence does, right? 55 00:03:23,470 --> 00:03:26,200 It assigns to zero the number zero. 56 00:03:26,200 --> 00:03:28,840 It assigns to one, the number minus one, 57 00:03:28,840 --> 00:03:31,600 it assigns to the index two, the number one, 58 00:03:31,600 --> 00:03:35,300 to the index three, the sequence assigns the number negative two. 59 00:03:35,300 --> 00:03:37,670 To the index four, it assigns the number two. 60 00:03:37,670 --> 00:03:40,630 To the index five, it assigns the number three, to the index 61 00:03:40,630 --> 00:03:43,710 six, it assigns the number three and it will keep on going. 62 00:03:43,710 --> 00:03:47,040 And eventually, right, we've ended up pairing off 63 00:03:47,040 --> 00:03:51,510 every single non negative integer with every single integer. 64 00:03:51,510 --> 00:03:57,130 And that really means that there's the same quantity of non negative integers 65 00:03:57,130 --> 00:03:59,160 as there are all integers. 66 00:03:59,160 --> 00:04:02,696 That should really give you pause, that, that sounds impossible, alright? 67 00:04:02,696 --> 00:04:07,650 Think about some physical object, some finite object like, coffee beans. 68 00:04:09,290 --> 00:04:13,020 If I've got some coffee beans, but then I take some away, now 69 00:04:13,020 --> 00:04:18,730 I've got fewer coffee beans, but the collection of all integers is different. 70 00:04:18,730 --> 00:04:22,130 If I start with all the integers and just take away the 71 00:04:22,130 --> 00:04:26,490 negative integers I've got the same quantity of things. 72 00:04:26,490 --> 00:04:28,047 Why does that work? 73 00:04:28,047 --> 00:04:31,407 Well one definition of an infinite quantity is a quantity 74 00:04:31,407 --> 00:04:35,281 that needn't get smaller, even when you take something away. 75 00:04:35,281 --> 00:04:40,115 [SOUND] 76 00:04:40,115 --> 00:04:46,370 [SOUND]