complex number vs 2d vectors

complex-numberlinear-algebrathought

intuition

• equivalence of dimensions
  • in 2d vectors, the dimensions are equivalent
    • which means you can perform operations that combine the dimensions together
  • in complex number, the real and imagine components are 2 non-equivalent dimensions
    • any operation will not combine them, which means the 2 components will still be separated
    • the only way to combine them is through multiplication by $i$, which acts as a gateway to go from one to the other
• complex number is like a special case of 2d vectors where only the operation that not combine the dimensions are allowed ?ideaquestion
• parallel vs orthogonal dimensions
  • in vector space, dimensions are orthogonal (in the sense that their intersection is the lower dimension)
  • in complex number, the 2 components are like in 2 parallel dimensionsidea
    • even though they can be visualized as orthogonal
    • this is a new way of thinking about complex numbers
      • multiplication with $i$ means jump from one dimension to another
      • a combination of a real and an imagine parts is a level in between the 2 dimensions
      • the only intersection is 0, which can be thought of as the 2 parallel dimension that actually start from the same pointidea
        • this intuition might be useful in physics ?question
        • maybe that’s why it’s used in describing general relativity and quantum physics ?
    • so maybe the above question is wrong ?
    • maybe that why complex numbers make many maths problems simpler, because it unlocks access to a hidden parallel dimension
      • orthogonal dimensions are not locked, we live in a 3-d space so we used to think in orthogonal
      • but our mind is not familiar with parallelism
      • might be related to the idea of multiverse
      • we see orthogonal dimension with our eyes, but we need to “see” parallelism with our mindquote
• k-d vectors of complex numbers = k orthogonal dimensions and 2 parallel dimensions

questions

questionidea

• by the same logic, can we have 3 (and more) parallel-dimensional numbers ?
  • each dimension pair will have a special operation to “jump” between them
  • multiple orthogonal dimensions and multiple parallel dimensions
  • so can we use that to make complicated problems simpler, as it unlocks more hidden parallel dimension ?
  • or even discover more hidden problems ?
    • we understand reality and the universe better thus we have more complicated questions