• Kogasa@programming.dev
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    5 months ago

    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.

      • Kogasa@programming.dev
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        5 months ago

        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

        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.