A zeroth-order approximation

[freebeard]

I thought I'd found a solution to the problem of 'rules of thumb' rubbing thin-skinned individuals the wrong way. But the upshot was that the thread in Aerodynamics was locked. I choose to share here instead to spare the feelings of the last poster.

en.wikipedia.org/wiki/Order_of_approximation

Usage in science and engineering
In formal expressions, the ordinal number used before the word order refers to the highest order of derivative in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero order approximation is also common. Cardinal numerals are occasionally used in expressions like an order zero approximation, an order one approximation, etc.

The omission of the word order leads to phrases that have less formal meaning. Phrases like first approximation or to a first approximation may refer to a roughly approximate value of a quantity.[1][2] The phrase to a zeroth approximation indicates a wild guess.[3] The expression order of approximation is sometimes informally used to mean the number of significant figures, in increasing order of accuracy, or to the order of magnitude. However, this may be confusing as these formal expressions do not directly refer to the order of derivatives.

So don't say rule of thumb when there is a nerdier expression available.

[oil pan 4]

We could start a hurt feelings sub forum to aerodynamics.
It's really a good thing I didn't bother to notice that one till after it was locked.

[freebeard]

I took me a while to figure out why there was no Quick Reply field. My first reaction? Sorry, not sorry; let's take it to The Lounge, where you-know-who doesn't go.

I will point out that zeroth order would be a Wild *ss Guess. First order is more appropriate for Rule of Thumb. What it comes down to is the significant digits. 4000 vs 3900 vs 3914 or 0.1 vs 0.01 vs 0.001, etc. So zeroth order has one significant digit that could be off by an order of magnitude.

[Vman455]

To use the topic that provoked this as an example, I would equate "rule of thumb" in this context with "zeroth-order approximation." If I measure a 5% fuel economy improvement on my already low-drag, CVT, electric-assist (all of which are differences from the cars which generated the RoT) car from a change and I assume that means I improved drag 10%, I have no basis on which to guess that. Maybe my car actually needs a 20% improvement in drag to get a 5% fuel economy improvement at 55mph, in which case increased resolution of the 10% guess doesn't improve its accuracy--it's still just a guess. I have absolutely no way of knowing on my car specifically without measuring something else to corroborate or disprove, and I could be way off as a result.

Put another way: If the RoT was a first-order approximation, we would have to know that drag improvement between 5%-14% on any car results in 5% fuel economy improvement, and we simply don't know that. The friction wasn't with the terminology; it's with the concept.

[freebeard]

I'd say the concept is meta to the content.

Like the programmer at the airport that kept counting his luggage and coming up one short. (Off-by-one error joke)

en.wikipedia.org/wiki/Order_of_approximation#Colloquial_usage

These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (e.g. "Of course the rotation of the Earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect. The terminology, in this context, represents a high level of precision required to account for an effect which is inferred to be very small when compared to the overall subject matter.

[ME_Andy]

Originally Posted by freebeard

let's take it to The Lounge, where you-know-who doesn't go.

Voldemort?



(Harry Potter reference)

[Piotrsko]

Systematic wild a##ed guess?

I ve seem him here, with a zero looked at couple of posts, BTW.

So much for a working ignore function.

[Ecky]

I appreciate this. I'm going to start using this in common, everyday speech.

[freebeard]

I just like to say 'zeroth'.

[Piotrsko]

Well it does cause a certian academic aura to your conversation.