We can’t physically see or understand how complex numbers exist or work in our world in a nice way, but we know they do exist. Because we’ve made massive advancements in science and technology off the assumption that they exist and work, and our understanding of many things in the world including stuff as basic as the solutions to quadratic equations would fall apart. By the same token, there are many problems for which vectors and problem spaces of nth degree are used, where n>3, and there’s that whole adage where time is considered a 4th dimension. In that way, we often solve many problems, even rudimentary linear algebra ones, using sets in R⁴, R⁵, etc, and there are many, many invisible forces at work in our world such as gravity. We know how easily our brain can trick us, we still are easily fooled by optical illusions even when we know they’re there and what they are/how they work, despite our visual cortex being the one of the most powerful and most used part of our brain. So the idea of forces and things which we don’t have the capacity to perceive existing in the world is not anything new or foreign. There are frequencies we can’t hear, colors we can’t see, etc which other animals can and do. So why is the concept of n dimensions in the world so widely rejected? There must be a simple reason, I have heard that it has to do with the volume of a gas in a container being proportionate to its dimensionality or something
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We fully understand complex numbers, just as much as we understand rational numbers. In both cases, they are just pairs of numbers from some simpler set with some definition about how to add them and multiply them and so forth. To wit:
For rational numbers, where we write a/b as (a,b), to show that it’s just a pair:
* (a,b) + (c,d) = (ad + bc, bd)
* (a,b)*(c,d) = (ac,bd)
* etc.
For complex numbers, where we write a+**i**b as (a,b)
* (a,b) + (c,d) = (a+c,b+b)
* (a,b)*(c,d) = (ac-bd,ad+bc)
* etc.
What’s not nice about that? They are just vectors on **R^2** and you can convince yourself (with some basic trigonometry — you said ELI5, but you also brought up linear algebra, so deal with it) that adding is just normal vector addition and multiplication is just normal multiplication of the lengths and addition of the angles.
How many dimensions there are is a very different kind of question. It’s a matter of observation, and we don’t really know that there are only three dimensions, though the people arguing for maybe more (string theorists) have not been very successful. We know there are only three *useful* spatial dimensions because that’s what dimension means: it’s the minimum number of directions that you can define such that going along each in some combination you can get anywhere — or at least anywhere nearby. There’s no other direction other than east/west, north/south, and up/down that we have ever found anything, but using just two of those, there are lots of places that are missing.
Equivalently, we only need three angles of rotation to achieve any attitude – two to point anywhere and one more to determine which way is “up” when we are facing somewhere. Airplanes have three angles that they control: pitch, roll, and yaw. Same for space ships. That’s all we have ever needed and it works. Those can be described by rotation matrices, in a group called O(3). O(3) successfully describes the space we live in, so it’s three dimensional. (Specifically, the fundamental representation of O(3), but that’s more ELI5th-year-math-student-in-college-or-grad-school.)
In the Standard Model of physics, there are other “dimensions” in a sense — called “internal degrees of freedom”, and they have pretty high dimensionality. Note this is *not* string theory. This is about understanding the world as being made of, essentially, an infinite lattice of little tiny springs all coupled to each other, and the extra “dimensions” are how/where those “springs” are deflected.
In this case, the local geometry of these internal degrees of freedom is *not* like **R^n**, described locally by the O(3) matrix group, but by something much more complicated called SU(3)xSU(2)xU(1). And that is thought to be just an approximation.
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