Cayley–Dickson construction
So, imagine you have a big number that has a real part and an imaginary part. For example, the number 3+4i, where 3 is the real part and 4i is the imaginary part.
Now, the Cayley-Dickson construction is like playing with blocks to make bigger numbers. You start with two blocks, one representing the real part and one representing the imaginary part. Then you stack them on top of each other to make one bigger block.
Next, you take two of these bigger blocks and stack them on top of each other to make an even bigger block. And you keep doing this, stacking bigger and bigger blocks on top of each other, like a tower of blocks.
But here's the cool part: each time you stack two blocks, you create a new number system with its own operations. For example, when you stack two blocks to make a "quaternion", you get a new number system where you can add, subtract, multiply, and divide just like you can with real numbers, but with some extra rules that come from the imaginary part.
And if you keep stacking blocks, you can make even more number systems with even more extra rules. It's like building a tower of complexity!
So that's the Cayley-Dickson construction in a nutshell. It's a way to create new number systems by stacking blocks on top of each other, starting with real and imaginary numbers. It's kind of like playing with blocks, but instead of building a tower, you're building a tower of numbers!
Now, the Cayley-Dickson construction is like playing with blocks to make bigger numbers. You start with two blocks, one representing the real part and one representing the imaginary part. Then you stack them on top of each other to make one bigger block.
Next, you take two of these bigger blocks and stack them on top of each other to make an even bigger block. And you keep doing this, stacking bigger and bigger blocks on top of each other, like a tower of blocks.
But here's the cool part: each time you stack two blocks, you create a new number system with its own operations. For example, when you stack two blocks to make a "quaternion", you get a new number system where you can add, subtract, multiply, and divide just like you can with real numbers, but with some extra rules that come from the imaginary part.
And if you keep stacking blocks, you can make even more number systems with even more extra rules. It's like building a tower of complexity!
So that's the Cayley-Dickson construction in a nutshell. It's a way to create new number systems by stacking blocks on top of each other, starting with real and imaginary numbers. It's kind of like playing with blocks, but instead of building a tower, you're building a tower of numbers!