A View from the Back of the Envelope top

What is "order of magnitude"?
How to simplify a number
Rounding to an order of magnitude
Exponential notation
Rough Draft
a number's nearest power of ten
(extreme rounding)


Rounding to an order of magnitude
Exponential notation
How to write and speak the exponential notation
Scientific Notation

Order of Magnitude at Paradise Valley.

What exactly does "order of magnitude" mean?

Regretably, it is used to mean various things.
On this site, it means a number's nearest power of ten.

Note that the phrase "order of magnitude" is also used in other ways.
  A number's oom, we have discussed.
  An oom by itself (eg, "it went up by three orders of magnitude"),
    is a factor of ten (so, "it went up by 10^3").
  And there is a somewhat unrelated, "big-O" "little-o" notation for
    describing how fast a function's output grows as its input grows,
    but I won't go into that.
  And sometimes it is just used to mean rough/approximate/etc,
    especially with "within a factor of ten"-ish kinds of things.
And instead of 10x, powers of ten, decimal oom ("doom"), it can sometimes mean 2x, powers of two, binary oom ("boom"). Also ex, powers of the natural log. What a zoo. But 10x is the most common.

How do I find a number's order of magnitude?

See Rounding to an order of magnitude.

Why round up from 3, rather than from 5 as usual?


If a number is in scientific notation, is the 10x the oom?

Only sometimes.

99 is written in scinote as 9.9 × 101.
So is it's order of magnitude 101? Just 10 ?
100, 102, is much closer.

So, in scientific notation A × 10x,
the 10x is only the nearest power of ten if the A is < sqrt(10). That's 3.162... .
Otherwise, it is only the 2nd nearest. :)

Thus just taking the 10x has a worst case error of ×10, rather than only ×sqrt(10).
(And introduces a systematic error - numbers are always made smaller.)

This is analogous to using truncation ("floor") rather than rounding when converting Real numbers (floats) to Integers.

But sometimes this is good enough.
And it is sometimes called "order of magnitude" - oh well.

You can use exponential notation without it being scientific notation.

"Scientific notation" is a particular convetion for writing a number in exponential notation. While it can be somewhat useful for indicating precision ("significant digits"), it obscures order of magnitude.

For example, 99 and 101, both obviously order 100, get written as
9.9 × 101 and
1.01 × 102,
with different 10x's.

Writing these as
0.99 × 102 and
1.01 × 102
makes the order of magnitude clearer.

So that is what I usually to do on this site.

Excerpt from an email

Oom is a slightly "offset" version of scientific notation.

In "scientific notation", the numbers between
   1.00 x 10^3 and
   9.99 x 10^3
are all written as "x 10^3".

In oom, it is the numbers between
   0.316 x 10^3 and
   3.16 x 10^3
which are called "order 10^3".

The range in both cases are a factor of 10 big - the top number is 10 times bigger than the bottom number.

But where one scheme has the middle of its range, the other scheme puts its edge.

In scientific notation, the "middle" of the range, multiplying-wize, is 3.16 (the square root of 10). As in 3.16 times 1.00 (bottom) is 3.16 (middle), and 3.16 times 3.16 (middle) is 10 (top).

In oom, the middle is just a power of ten (ten raised to the whatever). For example, order 10^0 (ie, order 1) goes from 0.316 to 3.16. As before, 3.16 times 0.316 (bottom) is 1 (middle), and 3.16 times 1 is 3.16 (top).

              scinot says            oom calls it
  0.316                                order 10^blah's bottom
  1.         x 10^blah's bottom        order 10^blah's middle
  3.16       x 10^blah's middle        order 10^blah's top
 10.         x 10^blah's top

Now, our objective is to make multiplication easier.
By rounding off to a power of ten (10^blah).
Because it is easy to multiply tens.
And, of course, because we often dont care about the details ("What?!? my library fine is 10^4 dollars!?! It can't be! No, I don't care if it is 2 x 10^4 or 3 x 10^4 ! Just that it is 10^4 !").

But we want to round off to the _nearest_ power of ten.
Consider 9.99 x 10^3. It is written in scientific notation with 10^3, but 10^4 is sooo much closer. Using 10^3 would round the whole range down to its lower edge. So instead we call it oom 10^4.

With a x10 big range, the maximum distance to an edge is, well, x10 (the distance from the other edge). But the maximum distance to the middle is just x3.16. So using the middle as our rounded approximation introduces a smaller maximum error, and avoids us systematically making all our numbers smaller.

I usually just treat the 3.162[...] of sqrt(10) as simply "3". So order 1 goes from 1/3 to 3, order 10 from 3 to 30, order 100 from 30 to 300, etc. A range, centered on a power of ten, and extending up and down by a factor of 3.

So, a number's order of magnitude is just whatever power of ten is closest. Numbers smaller than 3.162 are closer to 1, and bigger than that, to 10.

A View from the Back of the Envelope
Comments encouraged. - Mitchell N Charity <mcharity@lcs.mit.edu>

  Calculator  number/scinote/oomnote/oom
  2001-Apr-17  Created - a quick and crufty late-night draft.
  2001-Jan-31  A reader asked about scinote vs oom.