I don't care if the dataset in memory is 553MB or 632MB - what I really need to know is whether it's "a few tens of MB", "a few hundreds of MB", or a "a few thousand MB".
I don't care if the API server can service 7321 simultaneous requests or 6578 - I just need to know if its "a few hundred", "a few thousand", or "a few tens of thousands".
You can solve an enormous number of engineering and architecture problems with a reliable order-of-magnitude estimate - at the very least you can quickly exclude solutions that are vastly under (or over) provisioned for the problem you're trying to solve.
A good order-of-magnitude estimate is also a great error check for a more detailed calculation, if my quick estimate said "5000-ish plus or minus 50%", and your calculation says "24,152", one of us has got something wrong.
And then the teacher just took the range from the integration, and the formula, multiplied the two and put a ~= sign between them. I believe I actually stood up and said you can't do that and we had the first of many discussions about exactness.
That was scary.
That was my first run-in with what I considered the central article of my then faith : that you can derive the structure of the physical world from first principles. Throwing away terms in an equation in order to arrive at correct physics laws, I don't know, I considered it sacrilege or something. Of course I've since learned that deriving all of physics from it's own basic laws doesn't work, and the way we fix that is that we delete "inconvenient" terms in the equations when required. Deriving physics from a few mathematical laws is completely impossible. You can't even correctly derive the (mathematical) fields used in physics, so the very numbers that one uses to do physics aren't actually valid mathematical numbers.
So the relation between physics and mathematics is not that one is based on the other, because that was tried and didn't work out, and people have almost completely given up. So it was replaced by a marriage of convenience (this works ! Sure it won't validate mathematically but the numbers look really similar), ignoring at least a dozen elephants that stood in the way, and we just act like they don't exist.
When you deal with the real world there are always a lot of errors and uncertainty in measurement. Simply being within 10% of the right answer is generally sufficient and quickly getting that answer over getting the 99.99% accurate answer is better if it takes you one-tenth the time.