this post was submitted on 07 Jun 2024
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One definition of the complex numbers is the set of tuples (x, y) in R^(2) with the operations of addition: (a,b) + (c,d) = (a+c, b+d) and multiplication: (a,b) * (c,d) = (ac - bd, ad + bc). Then defining i := (0,1) and identifying (x, 0) with the real number x, we can write (a,b) = a + bi.
Ok, that's actually quite interesting
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.