The Picard Maneuver@lemmy.worldM to memes@lemmy.world · 5 months agoThey keep digging that hole deeperlemmy.worldimagemessage-square88fedilinkarrow-up1844arrow-down136
arrow-up1808arrow-down1imageThey keep digging that hole deeperlemmy.worldThe Picard Maneuver@lemmy.worldM to memes@lemmy.world · 5 months agomessage-square88fedilink
minus-squareKogasa@programming.devlinkfedilinkarrow-up2·edit-25 months agoYup, you’ll notice the only thing distinguishing C from R^(2) is that multiplication. That one definition has extremely broad implications. For fun, another definition is in terms of 2x2 matrices with real entries. The identity matrix 1 0 0 1 is identified with the real number 1, and the matrix 0 1 -1 0 is identified with i. Given this setup, the normal definitions of matrix addition and multiplication define the complex numbers.
Yup, you’ll notice the only thing distinguishing C from R^(2) is that multiplication. That one definition has extremely broad implications.
For fun, another definition is in terms of 2x2 matrices with real entries. The identity matrix
is identified with the real number 1, and the matrix
is identified with i. Given this setup, the normal definitions of matrix addition and multiplication define the complex numbers.