I will need some context for this, I’m afraid.
same… 3² is 9, 1 mod 4 is… 1?
Modulo arithmetic is basically numbers that go in circles instead of in a straight line. 4 is equivalent to 1 in mod 3 because there’s some integer n such that 3n+1= 4.
In general: in modulo base b, x mod b= y iff there exists an integer n such that bn+y=x.
The big domino is the question.
there are exactly 4 division algebras (over the field of real numbers): the real numbers, the complex numbers, the quaternions, and the octonians. you can’t add any more complex parts because you lost associativity with the octonians. I’m not entirely sure how the first domino leads to the last, though.
Modulo arithmetic is basically numbers that go in circles instead of in a straight line. 4 is equivalent to 1 in mod 3 because there’s some integer n such that 3n+1= 4.
In general: in modulo base b, x mod b= y iff there exists an integer n such that bn+y=x.
A couple years ago I was playing with 2 different ideas at the same time: the fact that real addition is isomorphic to real positive multiplication, and the fact that the complex unit circle contains an isomorphic subgroup of each modular integer group.
This led directly to the development of the operation φ(a, b)=exp(ln(a)ln(b)), which when paired with multiplication forms a field over the positive reals that’s isomorphic to the reals. A slightly modified version: φ_β(a,b)=exp(ln(a)ln(b)×-i τ/β) (where β is a positive real (although I suspect anything other than τ or an integer is unlikely to be useful) and τ=2π) defines an infinite family of operations that forms a field over the circle group. I don’t think this qualifies as a real algebra, (and thus doesn’t contradict Zorn’s theorem) but it’s definitely a division algebra.
