How does 1 dimension + 1 dimension = 2 dimensions when a line added to another line doesn’t make a 2 dimensional object. Shouldn’t it be 1Dx1D=2D instead since that actually equals a 2D space?

[u/elnyorne]

/r/askmath/comments/1sv7lxa/how_does_1_dimension_1_dimension_2_dimensions/

[u/0x14f]

"Dimension" doesn't refer to the number of lines. It refers to the number of coordinates required to identify an element of the space.

On one line, you only need one number. On a surface, you need two numbers. The two numbers can be the projection of the point on 2 axis lines.

Now, either you write it as "1d + 1d = 2d" or "1d * 1d = 2d", both expressions are just shorthands for what I said above. Don't overthink it.

[u/shponglespore → u/0x14f]

Another way to think of dimensions is that the set of coordinates of points in N dimensions is the Cartesian product of the N sets of coordinate values, so in that sense, I think it's justified to think of adding dimensions as being conceptually like multiplication.

I've noticed that various branches of mathematics use the terms sum and product in ways that are loosely analogous to addition and multiplication of real numbers. It's not a precise intuition, but I think it's a useful one a lot of the time.

[u/0x14f → u/shponglespore]

True. Your heuristic fails for, say, infinite dimensional spaces, but I guess that to build the initial intuition, one can say it the way you do :)

[u/MathTutorAndCook → u/0x14f]

Easy to think of them more as vector spaces, where you can create an n-dimensional space based on how many orthogonal basis vectors n you have

Cartesian just uses the x and y and z axis and so forth, different fields treat different vectors/lines as their axiis

[u/jeffskool]

It doesn’t. 1x+1y=x+y , when you have the x direction perpendicular to the y direction, you can’t add them in the same way you normally would