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  • If real, irrational, rational, whole and complex numbers are the only numbers you're familiar

  • with...boy have I got news for you.

  • Some special kinds of numbers, like quaternions and octonions, describe more than our three

  • dimensions.

  • Just let that sink in a little bit.

  • Can you even picture the kind of math involved for a reality that exists in more than 3D?

  • How would we even think about those shapes?

  • Let's take it back to like, middle school for a sec.

  • If you want to travel in one dimension, you add and subtract real numbers.

  • Here you are, the dot, on your little line, and you want to move 7 units to the right?

  • Great, that's plus seven.

  • Let's take it into two dimensions.

  • To make moves like this you can use what's called an imaginary number These can be combined

  • with real numbers to make what are called complex numbers.

  • Real numbers and complex numbers are the first two of four number systems in pure mathematics

  • that we call the normed division algebras over the real numbers.

  • We can think of complex numbers as two-dimensional numbers...because they help you move things

  • in 2D.

  • But something kinda funny happens when you use these division algebras to manipulate

  • an object in 3-dimensions.Say you've got a cube, and you need to be able to rotate

  • it around to see all its sides.

  • What kind of math are we looking at?

  • After the 1D real numbers and the 2D complex numberscomes a division algebra with 4-dimensions,

  • not 3, like you'd expect!

  • And this 4-dimensional number system is known as the quaternions!

  • And they are awesome and SUPER crazy, so we got in touch with Cohl Furey, a postdoctoral

  • researcher at Cambridge who lives and breathes this stuff.

  • She helped us get a lot of this straight, and if you're a math PhD, you can go watch

  • her series on quaternions and get real nitty-gritty.

  • But for the rest of us--quaternions are the number system we use to describe 3D shapes

  • in 4D space.

  • That sounds a little trippy, but that fourth dimension doesn't necessarily represent

  • a spatial dimension, like up and down.

  • Sometimes we use extra dimensions to represent a different concept, like time.

  • So in your quaternion, you have a real number, nested inside your complex number, and two

  • new terms, j and k, which represent your two other dimensions--four components, hence the

  • 'quat' in quaternion.

  • What makes quaternions extra hard to work with is that they're noncommutative.

  • Stick with me here--that means the order of the calculations matters in a way it doesn't

  • with real and complex numbers.

  • Think about it: If we go back to our 2D axis, it doesn't matter if you rotate in the plane

  • by 10 degrees first, and then by 20 degrees, or 20 degrees first and then 10 degrees--either

  • way you end up in the same place.

  • But in 3D space, if we have our cube, it can matter which move you do first, partly because

  • you have more options.

  • And while they may sound insane, quaternions are super useful--they come in handy when

  • calculating the launch trajectory of an aerospace vehicle.

  • Your phone can use them to calculate your device's position and orientation in space.

  • We use them all the time in generating computer graphics and for programming robots, which

  • move around in 3D space.

  • They are also especially important because a version of quaternions underpin Einstein's

  • theory of special relativity--allowing us to calculate the movement of an object in

  • 3 physical dimensions, plus the extra dimension, which can be defined as time.

  • But strap in--'cause here come the octonions.

  • That's the next step up.

  • Now, because of some complex mathematical rules, these kinds of numbers come only in

  • dimensions 1, 2, 4, and 8, so after quaternions, we make the jump to octonions--numbers that

  • help us navigate eight-dimensions.

  • That's the quaternion, which contains complex numbers, which contain real numbers, plus

  • 4 new units.

  • E1, e2, and e4 are one way that i, j, and k from the quaternions are represented in

  • the octonion system.

  • These eight 'dimensions' again, don't have to be spatial--they can represent characteristics

  • of particle behavior, like how certain particles behave under the strong force, making octonions

  • really useful for quantum calculations.

  • And there we have our nice neat set of the four normed division algebras [over the reals]

  • [over the real numbers] The thing is, with each next level, the math

  • of these division algebras gets even more tangled.

  • Quaternions are noncommutative, and octonions are too.

  • But they're also nonassociative--which means not only does order matter, but it also matters

  • how you group the multiplication of octonions.

  • But the very structure that makes the math so difficult is one reason that the octonions

  • can be so useful in describing fundamental physics.

  • One recent paper revealed that octonion mathematics allows for the existence of sterile neutrinos--a

  • particle that scientists are actively looking for experimentally because...it might be dark

  • matter.

  • That's just one of a whole host of applications--many of our models of particle physics are still

  • incomplete, but in the words of Dr. Furey “...there have been a number of very tantalizing

  • hints that the octonions might somehow underlie certain aspects of the structure of elementary

  • particle physics.”

  • So they're famously difficult to work with.

  • But incredibly useful and underpin the forefront of mathematical exploration happening today!

  • They're so crazy physicists even disagree on how to approach them, but researchers hope

  • octonions will help them peel back the layers of reality to explain why the universe behaves

  • the way we observe it to.

  • Wow.

  • We made it through that, pretty exciting.

  • Now we know more about extra-dimensional math!

  • For more mind-blowing close-up looks at the universe, check out this video here.

  • A huge thanks again to Dr. Cohl Furey for helping us make sure this was all correct,

  • and let us know in the comments if you have any questions or any other in-depth mathematics

  • that you'd like us to explore.

  • Thanks for watching!

If real, irrational, rational, whole and complex numbers are the only numbers you're familiar

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この数字は宇宙の新次元を解き明かすことができるのか? (Could These Numbers Unravel New Dimensions in Space?)

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    林宜悉 に公開 2021 年 01 月 14 日
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