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.
NeatNit@discuss.tchncs.de 1 year ago
I will need some context for this, I’m afraid.
PotatoesFall@discuss.tchncs.de 1 year ago
same… 3² is 9, 1 mod 4 is… 1?
mathemachristian@hexbear.net 1 year ago
3^2^ (mod 4) = 9 (mod 4) = 1 (mod 4)
is whats meant I presume
NeatNit@discuss.tchncs.de 1 year ago
The big domino is the question.
Leate_Wonceslace@lemmy.dbzer0.com 1 year ago
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.
fossilesque@mander.xyz 1 year ago
oop
Leate_Wonceslace@lemmy.dbzer0.com 1 year ago
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.