Help me understand the zeroth order approximation

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:

  1. 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”?

  2. 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?

  3. 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.

  4. 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.

A meter is easy to measure. No reason for it to be an approximation. It is often common in the physical sciences to set values equal to 1 whenever possible (distances, speed of light, Plank’s constant, gravitational constant, etc) to make the math easier.


I think Michael was exaggerating when he said absence of any information about it; and you are taking that too literally.

If you have absence of any information about a system; you probably shouldn’t be offering estimates about it; IMO.

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Technically, the orders of approximation are centered around the idea of Taylor expansion of functions. For example, assume a function (any function) has the expansion
f(x) = a0 + a1x + a2x2 + a3x3 + … + anxn + … = Σk=0,1,2,…akxk
where ak are constants. For technical reasons, to make sure this will not be too long-winded (this is a post in forum, not a textbook), we assume that x is limited to values from -1 to 1.

The order of approximation here is determined by how many terms we include. So the zeroth order approximation stops at k=0, yielding
f0(x) = a0
The first order approximation gives an extra linear term as correction:
f1(x) = a0 + a1x,
the second order approximation gives an extra quadratic term as correction:
f2(x) = a0 + a1x + a2x2
and so on. For each term we include, the correction gets smaller and smaller, as we close in on the true value.

As for the practical use for this: in physics, when the equations get too complex, we often solve the equations by first solving to the zeroth order (f0(x) = a0), e.g. getting a rough approximation when everything is in equilibrium and nothing much happens. Then we use the zeroth order approximation as a starting point to get the first order approximation, i.e. to find the linear term, to get f1(x) = a0 + a1x, also called linearization. And so on. For each step, the solution becomes more and more accurate. This is called perturbation theory.

In other contexts, we can use the term order of approximation more loosely. The zeroth order approximation of, say, the population of London would be to say “a few millions”. For the next order of approximation, when you have more information, could be to say “roughly 10 million”. Or we could use census data from a few years back, and then extrapolate to today using the population growth. For the next order of approximation, we could use census data, plus a second-order polynomial to correct for the population growth since the census. And so on. For each step, we get closer and closer to the true value.

Using the term even more loosely, we can say that the zeroth-order approximation is a rough guess. The first order approximation is when we have some more information, and so on, with each step including smaller and smaller corrections.

I hope this wasn’t too technical.

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Okay, that seems to clear that you can’t make the rough guess with absolute absence of information. We need some information, otherwise, it’s not an approximation, it’s an imaginative number.

Just one clarification left. I’m aware of what the significant figures are but I’ve never heard of this term “zero significant figures”.

Are we talking about English words here like one hundred, two thousand etc. instead of 100, 2000, because the English words don’t have significant figures?

Maybe I’m taking it too literally. I mean, zero significant figures seems like absence of any number to me.

Like @Get_off_my_lawn uses “a few millions” to describe the estimated population of London.

Sorry if I’m sound completely dumb here. I’m just trying to learn as much as I can.

Yeah, I guess you can say that. Or, to put it another way, with no significant digits, what you are specifying is the order of magnitude, i.e. for “thousands”, you say that it is a four-digit number, with “hundreds of thousands” that it is a six-digit number, and with “millions” that it is a seven-digit number (or even eight or nine). Or, more technically, you can specify the order of magnitude directly, using scientific notation:
Y=a×10n
Here a is a real number where the absolute value is in the range 1 ≤ |a| < 10 and n is an integer, and specifies the number of digits after the first. So for n=3 we have “thousands” and for n=6 we have “millions”. For the order of magnitude estimate you specify n, and say that Y is “on the order of ten to the n”.

The magnitude n can also be negative, then for n=-3 we use the ISO-prefix “milli-” (like in millimeters/-seconds), for n=-6 we use “micro-” (like in micrometers/-seconds"), for n=-9 we use “nano-” (like in nanometers/-seconds). For n=-10, we have the length unit Ångstrøm, which is a unit used in atomic physics to conveniently specify the size of atoms. So the zeroth-order estimate for the size of an atom, when you don’t know which one it is, is that it’s “on the order of a few Ångstrøms” or “Ångstrøm sized” (actually ranging from approximately 0.25Å to 3Å).

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Incredibly grateful mate, much thanks!