• MacN'Cheezus@lemmy.today
        link
        fedilink
        English
        arrow-up
        11
        ·
        5 months ago

        They did, linear algebra and vector calculus are a thing, but complex numbers have certain properties that you don’t get with vectors and that are quite useful and worth studying.

      • Kogasa@programming.dev
        link
        fedilink
        arrow-up
        4
        ·
        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
            link
            fedilink
            arrow-up
            2
            ·
            edit-2
            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.

      • Arrkk@lemmy.world
        link
        fedilink
        arrow-up
        2
        ·
        5 months ago

        For various math reasons you only get consistent systems with 2^n dimensions, so after complex you get quaternions with 4, then the next one that works is 8, then 16, etc. They become less useful because you lose various useful features, like you lose commutabiliy with quaternions (eg ab != ba), and every time you double you lose more things.