1. around (As a preposition, so in phrasings like "going around the [x]".)
2. "over, down" (As a separable prefix - the core idea is that you're somehow turning an entity.)
3. "around, avoiding" (As a non-separable prefix - the core idea is that you place or move an entity around another one.)
4. o'clock (for the full hour in "street speak")
5. roughly (Pretty much only used in the phrasing "um die + [Plural number]".)
6. by (Specifically for giving the amount of a rotation, but also used for a few other changes of quantity.)
My english brain thinks ‘at 10 degrees…’ in never occured to me to think of the cirular degrees so now um make sense in that way.
So from what you say um never means: the precise position, as the English ‘at‘ does. EG at work, at the table, at one o’clock…?
I had to roll back my database just now because I messed something up while working on a new feature. So the original comments are lost.
But no, I was not saying that “um” never means the precise position. “um den Tisch”/”around the table” is just as precise as “at the table”.
And “um 9” means “at 9 o’clock”.
The notion of approximation is ONE aspect of “um”, borne out of the core notion of “around”. But not everything the word does is an approximation. Hope that helps :)
Thanks, that helps me squeeze my little language brain a bit closer to understanding!! :)