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Dimension in this sense means degree of freedom, not a spatial dimension. That is, a number on the "imaginary dimension" cannot be represented on the "real dime
by rhdunn 5y ago
Dimension in this sense means degree of freedom, not a spatial dimension. That is, a number on the "imaginary dimension" cannot be represented on the "real dimension". This is similar to how a length does not represent a width or height.
When visualizing the real and imaginary dimensions, it is convenient to represent them using x/y axes. However, some visualizations of complex functions use colour/hue as a way of representing the real or imaginary part of the result.
Dimensional analysis is used in physics, etc. when manipulating fundamental units (time, length, luminosity, etc.). That is, m/s^2 has dimensions length=1 and time=-2.
- ralusek 5y agoRight, my point is that length and time don't exist perpendicular to each other simply because there might be some utility in visualizing them that way. The complex plane seems to be generally taught as if imaginary numbers in some capacity exist on this perpendicular axis. Even Veritasium, who is supposed to be teaching deconstruction/first principles approaches to these concepts, is saying this.
- rhdunn 5y agoThe spatial unit vectors for a 3 spatial dimension [x,y,z] vector are i^=[1,0,0], j^=[0,1,0], and k^=[0,0,1]. These are all said to be orthogonal to each other, which is a generalization of perpendicularity [1] to non-spatial vector spaces. As a lay person who is not familiar with the mathematics would recognize the term perpendicular, using that in the video is fine. [1] https://en.wikipedia.org/wiki/Orthogonality https://en.wikipedia.org/wiki/Orthogonality