WEBVTT - What are octonions? 0:00:08.520 --> 0:00:11.600 Hey, Daniel, what are your favorite numbers in physics? 0:00:11.920 --> 0:00:14.600 Oh? There are so many good ones. I mean I 0:00:14.640 --> 0:00:17.960 love planks constant, which tells us about like when things 0:00:17.960 --> 0:00:20.919 become quantum. I love the speed of light that tells 0:00:21.000 --> 0:00:24.640 us all about relativity. Too many to choose from. There 0:00:24.680 --> 0:00:25.919 are a number of favorites there. 0:00:26.880 --> 0:00:29.960 But have you ever wondered why numbers are so important 0:00:30.000 --> 0:00:30.680 in the universe? 0:00:30.800 --> 0:00:34.000 I mean, numbers are like the currency of physics. We're 0:00:34.000 --> 0:00:37.360 predicting things happening at times and places, and those are 0:00:37.479 --> 0:00:38.080 just numbers. 0:00:38.280 --> 0:00:40.519 Wow. So does that mean you get paid in numbers? 0:00:42.200 --> 0:00:43.880 I get a number of dollars every year. 0:00:44.440 --> 0:00:46.519 I guess we all get paid in numbers if there's 0:00:46.560 --> 0:00:49.320 just numbers in our bank account. If you're so lucky, 0:00:50.080 --> 0:00:52.960 the whole universe is just numbers. I think you said 0:00:52.960 --> 0:00:56.600 that a number of times. I'm the number one fan 0:00:56.760 --> 0:00:58.840 of numbers. I think I know a number of those 0:00:59.320 --> 0:01:17.600 in physics and math. Too many to number. Hi. I'm 0:01:17.640 --> 0:01:21.000 Jorge Mack, cartoonists and the author of Oliver's Great Big Universe. 0:01:21.400 --> 0:01:24.040 Hi, I'm Daniel. I'm a particle physicist, and I've been 0:01:24.040 --> 0:01:27.319 a professor at UC Irvine for a large number of years. 0:01:27.760 --> 0:01:30.240 Oh, too many to count? Or do you go into 0:01:30.240 --> 0:01:33.200 some sort of like weird subspace when you do physics. 0:01:34.200 --> 0:01:36.080 No, I think I'm going to enter a puba phase 0:01:36.120 --> 0:01:38.760 eventually and then emerge as an emeritus professor. 0:01:39.680 --> 0:01:44.040 That's a butterfly. That's a beautiful tenured emeritus. 0:01:43.520 --> 0:01:45.320 Butterfly, as a white haired moth. 0:01:45.440 --> 0:01:49.040 I think instead of wings, do you just have a 0:01:49.080 --> 0:01:51.200 bunch of research papers taped together. 0:01:53.200 --> 0:01:55.200 I'm going to fly too close to the universe on 0:01:55.240 --> 0:01:56.960 wings of research papers. 0:01:56.720 --> 0:01:59.600 Right, But you know, butterflies don't have mouths, right, so 0:01:59.640 --> 0:02:00.600 you won't be to talk. 0:02:00.840 --> 0:02:03.000 Yeah, but they have those long newses, and so I'm 0:02:03.000 --> 0:02:03.880 already said. 0:02:04.560 --> 0:02:09.480 There you go. You can sniff out science from there. 0:02:09.720 --> 0:02:10.000 M hmm. 0:02:10.160 --> 0:02:12.640 I can snort up all the information in the universe. 0:02:12.960 --> 0:02:15.280 Oh boy, that sounds a little illegal. 0:02:15.440 --> 0:02:17.560 Are you going to arrest a butterfly? Is that what 0:02:17.720 --> 0:02:18.200 has come to? 0:02:18.880 --> 0:02:22.600 Maybe a cocaine sniffing butterfly. Maybe maybe it will be 0:02:22.639 --> 0:02:25.520 our informant for, you know, figuring out how the universe works. 0:02:26.680 --> 0:02:30.240 Maybe altering your mental state is important for understanding the universe. 0:02:30.320 --> 0:02:32.480 M I hear that's called the butterfly effect. 0:02:33.760 --> 0:02:35.440 I think that's something else. 0:02:35.280 --> 0:02:38.760 Meant something else. Did I get that wrong. 0:02:39.040 --> 0:02:40.360 I think it's called microducing. 0:02:40.639 --> 0:02:43.920 Yeah, I think this is called micro punning, which it 0:02:43.919 --> 0:02:46.120 doesn't work as well. But anyways, welcome to our podcast 0:02:46.200 --> 0:02:50.240 Daniel and Jorge Explain the Universe, a production of iHeartRadio. 0:02:49.680 --> 0:02:52.000 In which we give you a drip, drip, drip, little 0:02:52.080 --> 0:02:55.440 doses of the incredible wonder we discovered about the universe, 0:02:55.880 --> 0:02:58.480 everything that's out there that makes sense, and everything out 0:02:58.480 --> 0:03:00.880 there that still puzzles us as we try to fit 0:03:00.960 --> 0:03:04.320 the entire universe into a pattern that makes sense to 0:03:04.440 --> 0:03:06.000 our little human minds. 0:03:06.400 --> 0:03:08.200 That's right, We try to get you high on the 0:03:08.240 --> 0:03:11.560 little dopamin head of wonder and amazement at how our 0:03:11.639 --> 0:03:14.840 universe works and the surprising ways in which it still 0:03:15.000 --> 0:03:16.639 puzzles scientists even today. 0:03:16.800 --> 0:03:20.120 And one thing that still puzzles scientists and philosophers is 0:03:20.160 --> 0:03:23.760 at any of it makes sense that these mathematical models 0:03:23.760 --> 0:03:26.440 we build on our head can actually describe and even 0:03:26.480 --> 0:03:29.840 predict what's going to happen out there in the universe. 0:03:30.400 --> 0:03:33.720 That somehow mathematics seems to be not just the currency 0:03:33.760 --> 0:03:36.760 of our physics, but the currency of the universe itself. 0:03:36.920 --> 0:03:39.720 Wait, wait, are you saying math is more important than physics. 0:03:41.760 --> 0:03:44.360 I mean, is English more important than Shakespeare. You can't 0:03:44.400 --> 0:03:45.760 really compare the two things, you know. 0:03:47.280 --> 0:03:51.160 Wait with this analogy, Shakespeare's math in English is physics. 0:03:51.280 --> 0:03:54.640 No, Shakespeare's physics and English is math. Physics is speaking 0:03:54.640 --> 0:03:57.640 in the language of mathematics. We are writing poetry in 0:03:57.680 --> 0:03:58.600 the language of math. 0:03:58.840 --> 0:04:02.040 I see. I see. So physics a comedy or a tragedy. 0:04:02.840 --> 0:04:04.320 It's a tragic comedy, for sure. 0:04:04.400 --> 0:04:05.720 It's a tragic comedy. 0:04:06.720 --> 0:04:09.120 We're all waiting for the final twist. Nobody knows how 0:04:09.120 --> 0:04:09.480 it ends. 0:04:09.520 --> 0:04:12.279 That's right, it never ends well for Shakespeare. 0:04:12.920 --> 0:04:14.320 But I hope we laugh along the way. 0:04:14.520 --> 0:04:17.000 And you also have to wear tights when you do physics. 0:04:17.040 --> 0:04:17.400 Mmmm. 0:04:17.680 --> 0:04:20.360 And as always, the jester is the wisest one. 0:04:21.240 --> 0:04:24.680 I see. That's the engineer and the team. 0:04:24.800 --> 0:04:27.400 Did you just call engineers jokers? I mean, I'm just 0:04:27.400 --> 0:04:28.720 gonna step slowly away from that. 0:04:29.160 --> 0:04:30.680 No. I know. We have a great sense of humor, 0:04:30.800 --> 0:04:35.280 is what I'm saying. We're the smaltest person in the room. 0:04:35.000 --> 0:04:36.880 Often laughed at and unwisely ignored. 0:04:37.040 --> 0:04:38.600 But yeah, it seems like there are a lot of 0:04:38.680 --> 0:04:41.560 numbers in physics. There's an infinite number of numbers in math, 0:04:41.640 --> 0:04:43.400 let's face it. But in physics it seems like there 0:04:43.400 --> 0:04:45.599 are special numbers out there that have kind of a 0:04:45.680 --> 0:04:48.599 special status because they're sort of significant in how the 0:04:48.680 --> 0:04:49.360 universe works. 0:04:49.440 --> 0:04:52.280 Yeah, there are certain constants which seem to be important 0:04:52.360 --> 0:04:55.320 to tell us something about the universe, the speed of light, 0:04:55.760 --> 0:04:59.440 planks constant, the gravitational constant. There are even numbers that 0:04:59.480 --> 0:05:02.279 don't have units, you know, like the number of particles 0:05:02.320 --> 0:05:05.200 we've discovered or ratios of masses that we think reveal 0:05:05.320 --> 0:05:08.040 something deep about the universe. But even more than that, 0:05:08.080 --> 0:05:12.200 there are systems of numbers. There are patterns of numbers 0:05:12.440 --> 0:05:17.080 that reflect patterns we see in the universe, mathematical constructs 0:05:17.120 --> 0:05:20.400 and all sorts of fancy mathematics that really are crucial 0:05:20.440 --> 0:05:22.880 to understanding how the universe operates. 0:05:23.120 --> 0:05:26.039 And this is kind of especially true at the microscopic level, right, 0:05:26.040 --> 0:05:27.600 That's where a lot of these numbers and a lot 0:05:27.600 --> 0:05:29.440 of this math comes from, and it comes from our 0:05:30.560 --> 0:05:33.280 understanding how things work at the particle level. 0:05:33.520 --> 0:05:36.279 Math an interesting point. I would say that math describes 0:05:36.320 --> 0:05:38.680 the universe at every level. You know, we have mathematics 0:05:38.720 --> 0:05:41.839 for fluid dynamics and for planetary evolution and for the 0:05:41.880 --> 0:05:45.919 expansion of the whole universe. It's mathematics up and down. 0:05:46.000 --> 0:05:48.240 Okay, so it's at every level, but it seems like 0:05:48.360 --> 0:05:51.040 in the Standard model there are especially some interesting numbers. 0:05:51.200 --> 0:05:55.520 Yes, absolutely, the theories of particle physics are very mathematical, 0:05:55.680 --> 0:05:58.520 and not just in the sense that they're predicting numbers 0:05:58.560 --> 0:06:01.960 and places in time. But the patterns we see in 0:06:02.000 --> 0:06:06.040 the Standard Model use complex mathematical theories like group theory 0:06:06.080 --> 0:06:08.200 and field theory and all sorts of like heavy hitting 0:06:08.279 --> 0:06:10.320 mathematical apparatuses. 0:06:09.960 --> 0:06:11.919 Right, which sort of eraises a question of whether the 0:06:12.000 --> 0:06:15.479 universe itself is mathematical. Like maybe if you dig down 0:06:15.520 --> 0:06:17.760 deep enough into the stuff we're all made out of 0:06:17.800 --> 0:06:21.720 at the end, maybe we're just mathematical equations or formulas. 0:06:21.839 --> 0:06:23.880 Yeah, and it makes you wonder, like what does that mean? 0:06:24.000 --> 0:06:27.440 And our numbers real the way like stuff is real? 0:06:28.000 --> 0:06:30.360 You know, like where is the number two? If two 0:06:30.440 --> 0:06:33.839 is a physical thing, does it exist somewhere in the universe. 0:06:34.120 --> 0:06:37.280 There's a whole fascinating branch of philosophy. But like how 0:06:37.320 --> 0:06:39.880 we learn things about numbers because you can't like do 0:06:40.000 --> 0:06:42.920 experiments with the number two and the number seven. It's 0:06:42.960 --> 0:06:44.839 all sort of mental games. 0:06:44.760 --> 0:06:47.440 Right, right, Like maybe the universe is not really physical, 0:06:47.480 --> 0:06:50.039 maybe it's just sort of like conceptual theoretical. 0:06:50.200 --> 0:06:52.719 Yeah, but then you wonder, like what breeds fire into 0:06:52.720 --> 0:06:56.320 all those concepts it makes us experience it. The other 0:06:56.360 --> 0:06:59.440 side of that argument is that numbers are not universal, 0:06:59.440 --> 0:07:02.080 they're not physics, they're not natural. They're just sort of 0:07:02.120 --> 0:07:05.040 the way that our human mind works. That you can 0:07:05.200 --> 0:07:07.880 use numbers as a way to describe the universe, but 0:07:07.880 --> 0:07:10.600 it doesn't mean they're part of the universe. The way 0:07:10.600 --> 0:07:13.640 you can like describe the color orange using a bunch 0:07:13.680 --> 0:07:16.280 of words, but none of those words are orange or 0:07:16.280 --> 0:07:18.720 fully capture the oranginess of an orange. 0:07:19.000 --> 0:07:23.040 Mm and is one really the loneliest number? Is another 0:07:23.040 --> 0:07:24.280 big physics question, right. 0:07:25.680 --> 0:07:28.760 Yes, yes, a very deep physics question. But we have 0:07:28.840 --> 0:07:31.000 to see a lot of value in math. There's lots 0:07:31.040 --> 0:07:34.960 of times when mathematicians have developed some cool little technique 0:07:35.120 --> 0:07:37.320 just for fun, because they see cool patterns and they 0:07:37.400 --> 0:07:39.840 like playing with them, and then later on physicists will 0:07:39.840 --> 0:07:41.920 come along and be like, hey, that looks useful, and 0:07:42.000 --> 0:07:44.240 just like pluck it out of their hands and go 0:07:44.280 --> 0:07:47.600 insert it into our physics equations and get great insights. 0:07:47.680 --> 0:07:49.760 There's a lot of hints there that the laws we 0:07:49.880 --> 0:07:52.360 use to describe the universe are deeply mathematical. 0:07:52.560 --> 0:07:54.000 Yeah, it seems like there are a lot of new 0:07:54.000 --> 0:07:56.800 ideas coming up all the time about how to explain 0:07:56.840 --> 0:07:58.800 the patterns that we see in the universe, maybe with 0:07:58.920 --> 0:08:01.720 new kinds of math and so to be. On the podcast, 0:08:01.720 --> 0:08:11.000 we'll be asking the question, what are octonions? Did I 0:08:11.000 --> 0:08:12.880 say that right? Or is it actonions? 0:08:13.240 --> 0:08:16.280 I think it's oct onions, Like, give me eight onions 0:08:16.280 --> 0:08:17.239 for that recipe, please. 0:08:17.560 --> 0:08:20.120 Isn't it like a blooming onion? Isn't that a dish 0:08:20.120 --> 0:08:23.520 in a fried dish in one of these famous restaurants? 0:08:24.240 --> 0:08:27.320 Is this like a fried onion that looks like an octopus? 0:08:28.560 --> 0:08:30.480 Are you shilling for Outback Steakhouse? Now? 0:08:30.600 --> 0:08:31.520 Oh? Is it out back? 0:08:31.640 --> 0:08:32.480 I don't know, never been. 0:08:32.520 --> 0:08:33.800 Maybe Applebee's is interested. 0:08:36.320 --> 0:08:40.040 I think it's maybe octonians or octanians, depending on where 0:08:40.080 --> 0:08:40.840 you are in the world. 0:08:41.040 --> 0:08:43.280 So this is an interesting word. It sort of sounds 0:08:43.320 --> 0:08:47.359 like octo, which is eight. Then you're ending it with onions, 0:08:47.679 --> 0:08:52.440 which is a vegetable or a tuber. But it's also 0:08:52.480 --> 0:08:55.040 sort of how some particle names ends, right. 0:08:55.040 --> 0:08:57.439 Mm hmm, yeah, ions exactly. 0:08:57.200 --> 0:09:00.720 Oh yeah, right right yeah ions. So is it denions? 0:09:00.840 --> 0:09:04.920 Then I think we should have eight different pronunciations of 0:09:04.960 --> 0:09:05.320 this word. 0:09:05.400 --> 0:09:07.800 Now a trace you go, and then we'll spend eighty 0:09:07.800 --> 0:09:11.600 eight minutes talking about it here on the podcast as. 0:09:11.480 --> 0:09:14.560 Part of our Cult of eight. There you go, send 0:09:14.600 --> 0:09:15.520 eight dollars to join. 0:09:15.720 --> 0:09:17.920 There you go. We'll have eight dollars in our pockets. 0:09:18.240 --> 0:09:22.000 In our eight pockets because I wear cargo pants exactly. 0:09:22.040 --> 0:09:23.760 Maybe these are the spiders of the universe. 0:09:23.840 --> 0:09:27.120 It's all an intricate web of eight. Well, anyways, as usually, 0:09:27.120 --> 0:09:29.120 we were wondering how many people out there had thought 0:09:29.120 --> 0:09:32.080 about or even heard of the word octonions and what 0:09:32.120 --> 0:09:33.560 they think it might mean. 0:09:33.800 --> 0:09:37.000 Thanks very much to everybody who participates in this audience 0:09:37.080 --> 0:09:41.000 contribution segment of the podcast. We'd love if you joined 0:09:41.040 --> 0:09:44.400 the crew. Please write to me two questions at Danielandjorge 0:09:44.600 --> 0:09:45.320 dot com. 0:09:45.360 --> 0:09:46.920 So think about it for a second. What do you 0:09:46.960 --> 0:09:51.000 think are octonions? And are they sold at Applebee's or 0:09:51.080 --> 0:09:52.079 out back to steakhouse? 0:09:52.240 --> 0:09:54.000 And would they make you cry if you chop them? 0:09:54.040 --> 0:09:55.319 Here's what people had to say. 0:09:55.160 --> 0:09:59.280 What are onions, so octors, the prefix for eight onions 0:09:59.320 --> 0:10:05.640 being something players, probably so eight layers stars. 0:10:05.920 --> 0:10:09.520 I don't know, Honestly, I have no idea, but if 0:10:09.520 --> 0:10:14.320 I were to have to guess, I would deconstruct the word. 0:10:14.679 --> 0:10:19.160 So I think the prefix oct means it has to 0:10:19.200 --> 0:10:21.440 do with eight and so maybe it has to do 0:10:21.600 --> 0:10:27.560 with glue ons or any type of similar particle or 0:10:27.760 --> 0:10:36.520 force carrier that is dependent on matrices for transformation. 0:10:37.080 --> 0:10:40.040 All right, well, pretty much the same as what I've 0:10:40.080 --> 0:10:43.160 been guessing. Something to do with onions and eight and 0:10:43.320 --> 0:10:44.000 or particles. 0:10:44.760 --> 0:10:47.280 Yeah, reasonable guess is absolutely. 0:10:46.960 --> 0:10:49.800 Is it like what happens when you smash together eight onions? 0:10:50.960 --> 0:10:51.960 You get an ooked onion. 0:10:52.480 --> 0:10:54.560 We're working on the onion collider right now. We just 0:10:54.559 --> 0:10:56.880 need a little bit more funding, eight dollars more and 0:10:56.920 --> 0:10:57.440 we'll be there. 0:10:58.280 --> 0:11:01.080 You need a little bit more garlic, You're gonna say 0:11:01.280 --> 0:11:04.440 a bit more seasoning. You need eighty eight billion dollars. 0:11:05.760 --> 0:11:07.880 I won't say no to eighty eight billion dollars, that's 0:11:07.880 --> 0:11:08.240 for sure. 0:11:08.280 --> 0:11:11.599 Yeah, there you go. And then what happens at the 0:11:11.720 --> 0:11:13.440 end when you collide the onions? Would you have to 0:11:14.040 --> 0:11:14.600 eat them? 0:11:14.679 --> 0:11:18.440 I disappeared to my private island with all the cash. 0:11:18.640 --> 0:11:20.120 I think they can give you eighty years in prison 0:11:20.160 --> 0:11:24.040 for that, Daniel, only if they catch me, only they 0:11:24.040 --> 0:11:25.240 smell of the crime. 0:11:26.920 --> 0:11:30.200 I'll have an army of cocaine infused butterflies to protect me. 0:11:30.360 --> 0:11:34.600 Oh my gosh, you are a super villain there. I mean, 0:11:34.640 --> 0:11:38.760 I've heard of sharks with lasers, but m drug butterflies 0:11:38.800 --> 0:11:40.560 is a new level of villainy and they will never 0:11:40.600 --> 0:11:44.160 see me coming. All right, Well, stick into it, Daniel. 0:11:44.200 --> 0:11:46.160 What is an octonian and what do you think is 0:11:46.160 --> 0:11:47.360 the right way to pronounce it. 0:11:47.559 --> 0:11:50.600 I pronounced it octonians, but it does make it sound 0:11:50.679 --> 0:11:53.440 like creatures from the planet Octo or something. 0:11:53.840 --> 0:11:58.120 M I see you're going for the more tony pronunciation Octonians. 0:11:58.360 --> 0:11:59.880 Octonians exactly. 0:12:00.720 --> 0:12:00.800 So. 0:12:01.000 --> 0:12:04.520 Octonians are a kind of number, and they're sort of 0:12:04.520 --> 0:12:08.600 in the category of complex numbers, but they're an extension 0:12:08.600 --> 0:12:11.640 of complex numbers the way complex numbers have like two 0:12:11.720 --> 0:12:14.920 components to them, the real and the imaginary part, which 0:12:14.920 --> 0:12:17.440 are an extension of real numbers. They just have one component. 0:12:17.720 --> 0:12:23.920 Octonians have eight components, one real and seven imaginary components. 0:12:24.320 --> 0:12:26.280 Well, okay, wait, hold on, I think maybe some of 0:12:26.360 --> 0:12:29.920 us might not be super familiar with our high school math. 0:12:30.080 --> 0:12:32.240 What is an imaginary number in the first place. 0:12:32.320 --> 0:12:34.800 Imaginary numbers were invented in the sixteen hundreds, and they're 0:12:34.800 --> 0:12:38.280 called imaginary because you don't see them in reality. They're 0:12:38.320 --> 0:12:42.400 very useful in math. And there's basically one imaginary number, 0:12:42.480 --> 0:12:46.040 which is I, which is the square root of minus one. 0:12:46.240 --> 0:12:49.240 There's no real number which, if you multiplied by itself, 0:12:49.600 --> 0:12:52.520 gives you minus one. So they invented a number. They 0:12:52.559 --> 0:12:55.240 just call it I, and they say, if you multiply 0:12:55.360 --> 0:12:57.840 I by itself, you get minus one. So it's a 0:12:57.880 --> 0:13:00.440 new kind of number. It's different from any of the 0:13:00.520 --> 0:13:02.920 numbers on the normal number line. 0:13:02.600 --> 0:13:05.600 Right, because like we had the number negative one, and 0:13:05.640 --> 0:13:08.400 we have the function to take a square root of something, 0:13:08.480 --> 0:13:10.400 but when you mix the two, it's like you get 0:13:10.400 --> 0:13:13.680 something that's almost impossible, or that it's not on the 0:13:13.720 --> 0:13:16.920 regular number line that we all use in our everyday. 0:13:16.559 --> 0:13:19.920 Lives exactly, because any number on the regular number line 0:13:19.960 --> 0:13:22.000 we call the real numbers, if you square it, you 0:13:22.040 --> 0:13:25.520 get a positive number. Three squared is nine. Negative three 0:13:25.559 --> 0:13:28.720 squared is also nine because the two negatives give you 0:13:28.760 --> 0:13:31.880 a positive. So there's no number on the real number line, 0:13:31.880 --> 0:13:34.480 which if you square it will give you a negative answer. 0:13:34.920 --> 0:13:37.320 So there's no number that's an answer to the question 0:13:37.679 --> 0:13:39.680 what is the square root of negative one? So they 0:13:39.679 --> 0:13:42.400 had to invent a new kind of number, sort of 0:13:42.480 --> 0:13:45.880 imagine the real number line is like the x axis. 0:13:45.960 --> 0:13:48.600 They invented a new axis, like the y axis, which 0:13:48.679 --> 0:13:51.439 is like the imaginary direction. So instead of having a 0:13:51.559 --> 0:13:54.280 number line, and they have a number plane where every 0:13:54.360 --> 0:13:58.200 number is two components, a real component and this imaginary component. 0:13:58.760 --> 0:14:01.360 Right, because you came up with us or mathematicians came 0:14:01.440 --> 0:14:03.200 up with I, and then you can have a whole 0:14:03.280 --> 0:14:05.360 number line based on I. You can have like two 0:14:05.400 --> 0:14:07.960 I and three I or four point seven I or 0:14:08.000 --> 0:14:08.439 eight I. 0:14:08.720 --> 0:14:11.880 Or negative two I. Right, the imaginary number line goes 0:14:11.960 --> 0:14:14.240 both ways, right, right? 0:14:14.280 --> 0:14:15.480 And can you also have not I? 0:14:15.880 --> 0:14:17.520 You can have not it and like I'm not doing the. 0:14:17.440 --> 0:14:21.240 Dishes, Yes, that's what I mean, Like zero I, would 0:14:21.280 --> 0:14:22.080 you call it not I? 0:14:23.040 --> 0:14:24.840 Oh? You can have zero? Yeah, the zero in the 0:14:24.920 --> 0:14:28.240 number line is just zero Comma zero, right, zero real 0:14:28.320 --> 0:14:29.800 numbers and zero imaginary. 0:14:30.000 --> 0:14:33.480 Right. So in physics, when you talk about an imaginary number, 0:14:34.000 --> 0:14:35.960 you talk about like a number that has both a 0:14:36.000 --> 0:14:39.000 real component and an imaginary component, so you write it 0:14:39.040 --> 0:14:43.360 as two numbers like seven plus eight I exactly. 0:14:43.400 --> 0:14:46.040 That's what we call a complex number, something with a 0:14:46.120 --> 0:14:49.560 real and an imaginary component, sort of like a coordinate 0:14:49.640 --> 0:14:53.440 on the two dimensional complex plane. If you imagine real 0:14:53.520 --> 0:14:56.000 numbers or x and imaginary numbers are y. 0:14:56.400 --> 0:14:58.680 Right, And I think like all of most of quantum 0:14:58.680 --> 0:15:01.400 particle physics is based on imaginary numbers, right, Like it's 0:15:01.400 --> 0:15:05.320 a convenient mathematical space to do all of the math in. 0:15:05.720 --> 0:15:07.920 Yeah, exactly. So there's a few important things to know 0:15:07.960 --> 0:15:10.600 about complex numbers. Number one, they turned out to be 0:15:10.680 --> 0:15:13.640 really useful. Back in the sixteen hundreds when they were invented, 0:15:13.680 --> 0:15:17.200 before we had quantum mechanics, they were already useful. Italian 0:15:17.200 --> 0:15:20.239 mathematicians came up with them as a way to solve problems, 0:15:20.360 --> 0:15:22.960 and they found that they could get to the solutions 0:15:22.960 --> 0:15:25.680 of tricky math problems much more rapidly if they used 0:15:25.720 --> 0:15:28.400 these imaginary numbers, even though they didn't believe in them, 0:15:28.480 --> 0:15:31.400 they didn't believe that they represented anything in the universe. 0:15:31.880 --> 0:15:34.720 But now all of our quantum mechanics relies on these 0:15:34.800 --> 0:15:37.720 complex numbers. Like the wave function that we talk about 0:15:37.720 --> 0:15:41.160 all the time. In quantum mechanics, that's actually a complex 0:15:41.360 --> 0:15:44.840 valued object. That's why you can't observe it directly. You 0:15:44.880 --> 0:15:47.640 can only observe it when it's squared. That gives you 0:15:47.680 --> 0:15:51.160 the probability. The wave function itself is a complex number, 0:15:51.240 --> 0:15:54.760 has a real component and an imaginary component. We couldn't 0:15:54.760 --> 0:15:58.400 do quantum mechanics without complex numbers. They represent something real 0:15:58.520 --> 0:15:59.440 about the universe. 0:16:00.120 --> 0:16:01.640 Wait, wait, what are you saying. Are you saying like 0:16:01.720 --> 0:16:06.320 somehow the universe or particles are themselves imaginary, or that 0:16:06.400 --> 0:16:09.400 they sort of operate in a two dimensional space. 0:16:09.640 --> 0:16:11.400 We don't know if the wave function is real, if 0:16:11.440 --> 0:16:14.600 it actually exists in the universe. What we do know 0:16:14.800 --> 0:16:18.680 is that the mathematics needed to describe what particles do 0:16:18.760 --> 0:16:22.080 in which direction they go has to have complex numbers 0:16:22.080 --> 0:16:24.680 in it. Like that. Math just does not work without 0:16:24.680 --> 0:16:28.680 complex numbers. So in our models, in our calculations, the 0:16:28.720 --> 0:16:32.000 wave function that describes all these particles have complex values 0:16:32.000 --> 0:16:34.640 in them. Now, when you predict the outcome of an experiment, 0:16:34.680 --> 0:16:37.120 it always just gives you actual numbers or real numbers 0:16:37.120 --> 0:16:38.920 that you can measure the particle will be at this 0:16:39.040 --> 0:16:42.800 location at that time. But the intermediate states. The calculations 0:16:42.800 --> 0:16:46.800 we do between the observations require complex numbers. Does that 0:16:46.840 --> 0:16:49.480 mean that the universe is complex, that there are imaginary 0:16:49.480 --> 0:16:53.360 components to it? We don't know. Deep question for philosophy. 0:16:53.880 --> 0:16:57.760 Well, the universe is definitely complicated, whether it's complex or 0:16:57.800 --> 0:17:02.160 real or imaginary. Maybe it is the topic for another podcast. 0:17:02.160 --> 0:17:04.880 But here today we're talking about maybe a different kind 0:17:04.880 --> 0:17:08.320 of imaginary number, one that is maybe eight times more 0:17:08.320 --> 0:17:11.600 imaginary than an imaginary number. So to dig into that 0:17:11.680 --> 0:17:14.000 and see whether or not it can help explain more 0:17:14.000 --> 0:17:15.919 of the universe that we see around this and what 0:17:15.960 --> 0:17:18.879 would that mean. But first, let's take a quick break. 0:17:31.400 --> 0:17:35.160 All right, we're talking about Oktonians or Kanyans, which are 0:17:35.359 --> 0:17:39.800 maybe extra imaginary numbers that might describe how the universe 0:17:39.840 --> 0:17:43.960 works at the particle level. We're digging into that here today. Now, Daniel, 0:17:44.000 --> 0:17:47.119 you're saying that imaginary numbers play a big part in 0:17:47.320 --> 0:17:50.399 describing how things work at the quantum level. 0:17:50.440 --> 0:17:54.200 Exactly, we need complex numbers in order to do quantum mechanics. 0:17:54.520 --> 0:17:57.880 Like if they hadn't been invented already early this century 0:17:57.920 --> 0:18:00.680 when we were developing quantum mechanics, we might have invented 0:18:00.680 --> 0:18:03.240 them then, As it is, they already existed in the 0:18:03.280 --> 0:18:05.520 sort of mathematical toolbox, and so we could just pluck 0:18:05.560 --> 0:18:08.040 them out and say, oh, this is helpful, let's use it. 0:18:08.240 --> 0:18:10.919 Well, was that like a big surprise, Like, wait a minute, 0:18:10.920 --> 0:18:12.440 what does that mean about the universe that you need 0:18:12.480 --> 0:18:14.280 imaginary numbers to describe it? 0:18:14.359 --> 0:18:18.399 Yeah? It has deep consequences for mathematical philosophy. You know, 0:18:18.440 --> 0:18:20.840 you can wonder if the wave function is real, does 0:18:20.880 --> 0:18:24.680 that mean that imaginary numbers are real in some philosophical sense, 0:18:24.720 --> 0:18:26.840 even if we don't call them real numbers in a 0:18:26.920 --> 0:18:30.200 mathematical sense. We don't know what that means. But imaginary 0:18:30.280 --> 0:18:33.879 numbers are also important in mathematics itself. It tells us 0:18:33.880 --> 0:18:37.520 something about how numbers work even before they were applied 0:18:37.520 --> 0:18:38.520 to quantum mechanics. 0:18:39.160 --> 0:18:42.320 Interesting, so there may be something point to something fundamental 0:18:42.320 --> 0:18:43.640 about the nature of the universe. 0:18:43.840 --> 0:18:47.120 Yeah. Perhaps, because complex numbers are not just like something 0:18:47.160 --> 0:18:49.800 mathematicians invented. You can't just say, well, I'm going to 0:18:49.840 --> 0:18:51.560 take two numbers and stick them together, and now I've 0:18:51.560 --> 0:18:53.359 got a new kind of number. When you do that, 0:18:53.400 --> 0:18:55.440 you also have to decide, like, what are the rules 0:18:55.440 --> 0:18:57.240 of those numbers, how do you add them? How do 0:18:57.280 --> 0:18:59.840 you subtract them? What happens when you multiply the two numbers. 0:19:00.320 --> 0:19:02.160 So you have to come up with what mathematicians call 0:19:02.240 --> 0:19:04.879 a division algebra, which is basically just like all the 0:19:05.000 --> 0:19:07.800 rules of how the math works for these numbers. And 0:19:07.880 --> 0:19:10.520 it's not always easy to come up with that system. 0:19:10.640 --> 0:19:12.760 For example, you can do it for complex numbers, but 0:19:12.800 --> 0:19:15.520 you can't do it for triplets of numbers. So it 0:19:15.520 --> 0:19:18.240 works for single numbers, it works for pairs of numbers, 0:19:18.480 --> 0:19:20.680 it doesn't work for triplets of numbers. 0:19:21.080 --> 0:19:22.880 Okay, wait, hold on, and I think you're talking about 0:19:22.920 --> 0:19:26.600 now increasing the imaginariness of a number. So like a 0:19:26.640 --> 0:19:29.879 regular imaginary number has a real component and then an 0:19:29.920 --> 0:19:34.080 imaginary component which you get by multiplying another number by 0:19:34.080 --> 0:19:37.040 the number I. So like an imaginary number is seven 0:19:37.160 --> 0:19:40.919 plus eight I. Now you're talking about like adding a 0:19:40.960 --> 0:19:41.760 third component. 0:19:42.040 --> 0:19:45.520 Yeah, call it J. Right, J is like another square 0:19:45.600 --> 0:19:48.800 root of negative one, a different a unique square root 0:19:48.840 --> 0:19:51.560 of negative one. Now thing says you can't have more, right, 0:19:51.600 --> 0:19:55.399 Maybe there's multiple square roots of negative one. And after 0:19:55.440 --> 0:19:58.560