I’m pretty certain that 'on the order of means within a unit of 10 up or down. (5 and 500 would be ‘on the order of’ 50) but I’m unclear on the phrase ‘on the close order of’.
Can someone enlighten me?
Heh. Been reading The Moon is a Harsh Mistress, have we?
I don’t think the terms are defined specifically; “on the order of…” to a man like Heinlein, means “being within one log of…” “On the close order of…” would then mean “being within some fraction of a log of…” Being within a tenth of a log of 50, for example, would be between 40 and 63.; that strikes me as a reasonable definition, but not necessarily a common one. I mainly regard the phrase to be a pre-emptive strike against those who would claim that “on the order of…” is too vague when taken literally.
You are correct about “on the order of,” which is more formally “on the same order of magnitude as”, and is basically useless for real estimates. It’s used for giving a naive person an idea of what scale they can expect: “Measurements from this instrument are typically on the order of tens of meters.” That implies that a measurement of one meter would be abnormally low, or that one hundred meters would be abnormally high.
“On the close order of” implies more precision, although how much more is probably subjective. I always think of it this way: if I were graphing results on a logarithmic scale, numbers “on the close order” of each other would be within at most half a major division. That is, “on the close order of 10” would definitely be between 3 and 33, but probably closer. You might just as easily say that “on the close order of 10” is between 5 and 20, though.
Scientists use the phrases “zeroth order -” “first order -” and “second order estimates” to refer to this kind of approximation. For an actual value of 3,543, you might have a zeroth order estimate of “in the thousands,” a first order estimate of “three thousand” or “four thousand”, and a second-order estimate of “thirty-five hundred” or “thirty-six hundred.” Mapping the two sets of terms, I would say that if you give a second significant digit (a second-order approximation) then the listener may assume that you mean “on the close order of”; if you only give one significant digit, the listener might assume “on the order of” unless you specified otherwise.
One last example: suppose the real number is 3,012. If I said it’s “on the order of 3,000”, you might assume it to be anywhere between 1,000 and 9,000. If I specify “on the close order of” then you might assume it to be between 2,000 and 4,000. If I specify “on the very close order of” then you might assume that it’s between 2,900 and 3,100. This is easier than me saying “It’s about three-point-zero-zero times ten to the third,” but the latter really conveys my precision most … precisely.
Actually, you kind of have it backwards. 50 would be on the order of 5 - 500. But it’s really more common to talk about powers of 10 when talking about “on the order of”. If you always write numbers in scientific notation, you’ll see how this comes about: 536 becomes 5.36 x 10^2. You ignore the first part and look at the second part to see what “order” it is.
Which is a very condescending and unnecessarily disparaging definition of a phrase that many people find perfectly useful and applicable to real situations.
I often deal with quantities that vary over many, many orders of magnitude, such as radiance, time constants, scattering ratios, dopant concentrations, and other things that have exponential or power relationships to independant variables such as temperature, angle, or wavelength.
Thus, an estimate of radiance, for instance, within an order of magnitude can prove very helpful in determining what kind of exposure time I will need to achieve a decent signal level in an image, and to estimate the amount of storage and bandwith I will need to handle the image data.
When one deals with quantities that have logarithmic and exponential relationships with each other, “on the order of” is a frequently used and perfectly valid qualifier.
I used “naive” to mean “unfamiliar with the subject matter” above, and didn’t mean to sound condescending. Nonetheless, the subject matter you listed (is that computer vision or overhead imagery?) seems pretty esoteric even to me, so consider that most people don’t have much use for the phrase “on the order of,” as useful as it might be in your daily work.
As an engineer, I work with quantities that tend to vary linearly, or with an inverse-square at most, so I’ll rephrase “basically useless for real estimates” to “too broad for most people’s purposes.” Feel better?
…on the order of two times better?
…on the close order of… okay, I’ll stop pestering you.
Not quite. “nth order approximation” actually has a very precise definition, and it’s not applicable to all estimates. It’s only useful where you’re expressing some quantity using a series, and in particular, using the first few terms of a series. For instance, suppose I want the cosine of a small number, x. I might say that to a zeroth order approximation, cos(x) ~= 1. To a second-order approximation, though, cos(x) ~= 1 - x[sup]2[/sup]/2. How good an approximation this is depends on how large x is. If, for instance, x = 0.9, then cos(x) = 0.6216…, my zeroth order approximation is 1, and my second order approximation is 0.595. So my zeorth order approximation is off by 61%, but my second-order approximation is only off by 4.3%. In this case, the second-order approximation is a big improvement over the zeroth-order. On the other hand, suppose that x = 0.1. Now, cos(x) = 0.995004…, my zeroth-order approximation is off by about 0.5%, and my second-order approximation of 0.995 is only off by 0.00042%. So here, I nearly nailed it with my zeroth-order approximation, which was much better than my second-order approximation for x = 0.9. So sometimes a low-order approximation can be better than a high-order approximation, and you can’t say what order of approximation a number is unless you know how the number was obtained.
Um, not really. Say you’ve got a process you can’t solve exactly, but there’s a very similar process that you can solve exactly and you can describe deviations from the idealized process and the real one in terms of some process whose effect depends on some parameter A. The solution to the real process is often solved by “perturbation”, which means writing things out in a power series.
<Ideal Solution> + <one correction>A + <two corrections>A[sup]2[/sup] + …
The first term is the “zeroth order” estimate, the second is the “first order”, and so on.