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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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