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This question originates from a quandary about the meaning of the statement that two values are within the same order of magnitude. I wonder whether there is an established usage, of (rather more generally) whether an intuitive meaning follows naturally from established usage of closely related concepts.

The problem:

Are $4.6 \ast 10^{-5}$ and $1.22 \ast 10^{-4}$ within the same order of magnitude?

No, of course not (was my precipitous reaction). However, I did look up the relevant Wikipedia entry, as well as a few random rigorous and popular texts using this and similar constructions, and had to admit that I simply am not sure what the phrase really conveys.

The Wikipedia entry (section:Uses) appears to distinguish three notions:

  • the order of magnitude of a number (the exponent of the power of $10$ in the scientific notation for the number);

  • the order of magnitude estimate (the exponent of the nearest power of $10$);

  • the order of magnitude difference (left unexplained; apparently the order of magnitude of the factor of two numbers).

This seems somewhat convoluted, especially combined with the John Baez quote at the beginning of the entry (implying that the above values have the same order of magnitude, their ratio being $2.65$).

Stating that a value $a'$ is within the same order of magnitude as a value $a$, do I commit myself to anything beyond their ratio being less that $10$?

anemone
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    The common usage in most (all?) of science and engineering is that if $$ \frac 1{10} < \frac ab < 10$$ then "$a$ and $b$ are within an order of magnitude". – Simon S Jan 14 '16 at 22:58

3 Answers3

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The phrase "order of magnitude" is frequently used in engineering when you aren't trying to be very precise. However, the key word in your problem would appear to be "same". This implies that you first calculate an order of magnitude for the numbers and then compare. So based on that, I would in fact say "no", that $$4.6∗10^{−5}, o.m. = -5$$ and $$1.22∗10^{−4}, o.m. = -4,$$ are not within the same order of magnitude.

However, in response to the question "are they within an order of magnitude", I would say yes.

gariepy
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    Indeed the first interpretation would lead to unwanted results: As $10^{100}$ and $10^{1000}$ are certainly not of the same order of magnitude, there exists $a\ge 10^{100}$ such that the nearly equal numbers $a$ and $a+1$ are mot of the same order of magnitude. That is of course not desired, so "of same order of magnitude" is not intended to be an equivalence relation. – Hagen von Eitzen Jan 15 '16 at 07:09
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In case someone randomly ends up on this old thread and is coming from more of an engineering perspective:

As Simon mentioned in the comment to the original post, I understand this phrase to mean the factor between two values is [0.1, 10].

However, from a more applied point of view: Oftentimes we like to simplify our life and drop anything from calculations that only has a minor impact on the result.

Example

Calculate battery life of a circuit. Circuit draws current I_c and the battery has a self-discharge I_s.

If I_c >> I_s, we simply drop I_s cause our theoretical calculations are off by a few percent anyways due to production, material and temperature differences. Any value that only impacts the third, fourth, ... digit of our result has no meaningful impact.

If different values with high impact on our results are "within the same order of magnitude" it means that neither of the values can be neglected for estimations and calculations.

L. Heinrichs
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No, but your question is tricky, because each value is part of several orders of magnitude.

If measuring a continuous value, the order(s) of magnitude depend(s) on the units. $4.6 \ast 10^{-5}$ m and $1.22 \ast 10^{-4}$ m may be considered in orders -5 and -4, but $1.51 \ast 10^{-4}$ feet and $4 \ast 10^{-4}$ feet may both be considered in order -4, yet the values in these sets are equal.

But even a discrete value is part of several orders of magnitude, because there is no standard definition of where each order starts and ends. What the English Wikipedia's article conveys (in an indeed convoluted―and broken―way) is that there are several scales of orders. Your precipitous reaction suggests you were only aware of what I may call the "number-of-digits-minus-one" (at least for quantities above 1) convention, which I illustrate first in the following diagram:

Comparison of 2 common order of magnitude conventions

In the problem you quote, $4.6 \ast 10^{-5}$ clearly falls in different orders (-5 vs -4) depending on the convention, but even $1.22 \ast 10^{-4}$ is technically in several orders, i.e. both in "number-of-digits order -5" and in "engineering notation order -5". And there are even more conventions than these 2 (as the OomKind argument of the ord_of_mag function of the big_num_math library shows).

Therefore, if taken literally, the problem/question is based on a mistaken assumption and technically incorrect. But in fact, "within the same order of magnitude" is a locution. Its meaning is explained by Vocabulary.com:

two quantities are of the same order of magnitude if one is less than 10 times as large as the other

A much less misleading phrasing for "within the same order of magnitude" would be "within an order of magnitude" (but that variant is unfortunately falling out of favor).

Philippe Cloutier
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