I’d never given any thought to the order of polar coordinates until yesterday. They’re written (r, θ). That’s just how it is. But a friend pointed out two reasons why this bothers him.
First, r is typically a function of θ, just as y is typically a function of x. But in rectangular coordinates, the independent variable is the first element of a pair while in polar coordinates it is the second element.
Second, if you’re going to walk a mile northwest, how do you proceed? You first face northwest, then you walk for a mile. You don’t walk a mile to the east and then walk 135° counterclockwise along an arc centered where you started. That is, you set your θ first, then increase your r.
The (r, θ) convention isn’t going to change. Maybe the only take-away from this discussion is to appreciate someone’s confusion who sees polar coordinates for the first time and wants to reverse their order.
Related post: Why use j for imaginary unit
(I don’t use j for imaginary unit, except when writing Python code. The i notation is too firmly engraved in my brain. But I understand why j has advantages.)