So, Nyarlathotep originally mentioned it in my last question and I’m having a little bit hard time understanding term. He has helped me via forum messages but there are still some questions that remain.
The reason I want to learn more about this term is because it’s actually quite interesting from a scientific point of view. I didn’t even know there was a term to describe “wild guesses”. Anyways, here we go, the only non-complex example on the Internet is from Wikipedia:
Zeroth-order approximation is the term scientists use for a first rough answer. Many simplifying assumptions are made, and when a number is needed, an order-of-magnitude answer (or zero significant figures) is often given.
For example, you might say “the town has a few thousand residents”, when it has 3,914 people in actuality. This is also sometimes referred to as an order-of-magnitude approximation. The zero of “zeroth-order” represents the fact that even the only number given, “a few”, is itself loosely defined.
I got some pretty good examples from some people I asked the same question on a different medium. However, there’s some uncertain about their examples too:
So, here’s example #1 from a guy called Roy:
Suppose I have a metal rod with a length of say 1 metre then its length changes a bit with temperature. So a zeroth-order approximation is that its length is 1 metre, whereas a first-order approximation is that its length is 1+k*t where t is the temperature increase relative to some initial temperature (when the length was 1) and k is a constant. A second-order approximation is that the length is 1+kt+mt² where m is another constant. The speed of some vehicle at time t might be approximated by the second order approximation s=a+bt+ct². The zeroth-order approximation is s=a, the first-order is s=a+bt.
So basically, the higher the order, the more refined and accurate the calculation becomes. But was this 1 metre known to be certain from the beginning or was it also a zeroth order approximation?
Example #2 from a guy called Michael:
“Nth order” refers to a technique of estimating a value by assuming it behaves like a polynomial, and specifically a estimate called the Taylor approximation; first-order models up to linear term, second-order up to the quadratic term, etc. So zeroth order would be a degree-zero approximation, i.e., constant. If we suppose the polynomial’s variable represents information about the system we’re modeling, we can think of zeroth-order as an initial guess in the absence of any information about it.
What do I understand so far?
Zeroth order approximation is a term used to describe the first, wild approximate guess for something. In the Wikipedia’s example, the city has a “few thousand” residents, where “few thousand” is the zeroth order approximation. In other words, we don’t know the exact value, but we do know it’s somewhere in the thousands.
What’s confusing me now:
Does zero significant figures (this term is mentioned in Wikipedia’s definition) answer refer to an answer without numerical figures, hence the example’s usage of English words “few thousand”?
Michael in example #2 mentions that zeroth order approximation is the initial guess in the absence of any information about it. So, does this mean that the Wikipedia’s example, “few thousand” residents is a totally wild guess, and it co-incidentally is in the same unit of number (“thousands”, like ones, tens, hundred) as the actual figure of 3,914 residents?
If Michael is wrong, then it would mean that in both the Wikipedia’s example and Roy’s example #1, the initial figures are certain i.e. that the city’s residents are certainly in the range of thousands, and that the metal rod’s length is certainly 1 metre.
If Michael is correct, then it means that both the Wikipedia example and Roy’s example are using zeroth order approximation to make the initial guess about the possible number (few thousand and 1 metre), both of which are not certain to be correct, as they were made in the absence of any information.
All I’m trying to figure out is if the initial numbers used in all these examples were certain, or were they part of the wild guess, hence making it clear that even at the end of the calculation, we still don’t know how accurate we are.