using the expression "order of magnitude"

If something is ten times larger would you say “one order of magnitude larger” or “one the order of one manitude larger?” What if it is 1,000 times larger? “Three magnitudes?” or “Three orders of magnitude?” See what I mean?

(Oh, and why or when would you say it that way anyway?)

In my experience something is on the same “order of magnitude” if it is within the same power of ten (i.e. has the same number of digits). If a number is ten times greater than another, it is one order of magnitude larger. More info here: Order of Magnitude

It is useful if you only care about precision to that level. You hear something similar when someone claims to earn “six figures” - it doesn’t matter if its $100,000 or $900,000 they are the same order of magnitude.


If you earn $900,000 and I earn $1,000,000, am I still one order of magnitude higher? (Ok, I’ll go read the link now.)

I would say that ‘number of digits’ is only a rough approximation.

By the logarithmic scale, going from 100,000 to 900,000 is 95% of an order of magnitude, and from 900,000 to one million is 5% of an order of magnitude. But that’s probably being excessively geeky.

And I would use the phrasing of ‘one order of magnitude larger’, ‘three orders of magnitude’ as oppose to the other constructions.

For this one, I would definitely use “three orders of magnitude”. I’m in an astronomy-related field, and “three magnitudes” would be interpreted as a reference to the stellar brightness scale (which is also logarithmic, but with a different base).

It depends on what log scale you are using. I like the term “binary ballpark” (as in “it’s in the same binary ballpark”) for things 1/2X to 2X of each other.

I’d use three orders of magnitude also. We sometimes use “binary orders of magnitude” for doubling, quadrupling, etc. Three binary orders of magnitude higher actually makes sense when talking about memory size, for instance, rather than 8X. I’ve never hard anyone use octal or hex orders of magnitude, though.

This is the meaning my physics prof in college taught us. I acknowledge that terms like this are often used more loosely than their strict mathematical definitions, though.

I would say if you’re starting at $900,000 an order of magnitude would be $9M. But your usage may vary.

You could say that something that is 100 times larger is two orders of magnitude larger, or two decades larger (at least in some engineering circles this is reminiscent of a “decade box” with switches for setting resistance or capacitance or inductance by ones, tens, hundreds, and so forth, all added).

You can also use nepers, which are like decades except with a factor of e instead of 10. Nepers are named after, well, Napier. The other one.

For that matter, you sometimes also see “octaves” used for powers of two, imported from the usage in music (a note one octave above another has twice the frequency).

Yes, 1,000,000 is not an order of magnitude larger than 900,000.