I am interested in understanding a visual representation of results of operations between numbers, on a 2D plane.
I wonder if a visual representation on cartesian graph could be eventually "mapped" to operations that can be performed on matrices, using vectors.
So to coherently represent a visual representation of operations, as 1 + 1 , 1 x 1, 1 x 1 x 1 (1^3).
Consider the following images:
summation
multiplication
power
I interpreted that:
- a summation a + b could be represented by a vector of length |a + b| , in the same direction : in the case of planar graph, 1+1 might be represented by a vector of length 2 along axes x.
This could be interpreted with a Cartesian graph as well.
(see FIG. 1)
- a multiplication a x b would be represented by a vector normal to a, b - thus, a vector along axes z
Though, on a Cartesian graph this would be represented by a cube - a power, or with a bit of imagination, "a plane" times another "plane", and I figure axes are represented as normal to each other because they may "capture" the concept of vector multiplication.
- on a Cartesian graph, a x b could be represented as the surface between axes x, y - as a square. But a plane may represent a cosine similarity concept between two vectors.
I figure a conceptual glitch is in one case one we use vectors and in the other planes and volumes. Although, in both we recur to augmenting a dimension when using multiplication.
I would like to know which visual representations may better represent the operations of summation, multiplication and power, on a 2D plane. Visual representation of operations like : 1 + 1 + 1 + 1 and 2 x 2 should be equivalent.
I raise this question inspired by https://youtu.be/lRZ4aMaXPBI?t=542 : having a formal visual language to represent what a resulting "number" is, it may be of help to understand concepts of numerability and concepts like factorisation in primes, and matrices decompositions.