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Just thought of a simpler way to explain "order of magnitude".
Here are some ranges of numbers. Each one is an order of magnitude:
0 to 9.99999.....
10 to 99.99999.....
100 to 999.99999.....
1000 to 9999.99999.....
10000 to 99999.99999.....
100000 to 999999.99999.....
1000000 to 9999999.99999.....
and so on
You can multiply these ranges by any number and have another set of
orders of magnitude.
To say that a quantity has increased by an order of magnitude means
that it has moved from one range to the next greater range. Two
orders of magnitude, two steps on the chart.
In science & engineering, to say, "You aren't even in the right order
of magnitude" means, "You are so far off, making improvements to the
details won't make any real difference. You have to change your whole
approach." Or, "You dope, you got the decimal point in the wrong
place."
To say, "We've improved by an order of magnitude" means, "Relative to
this improvement, most incremental improvements are in the realm of
insignificant digits or experimental error."
Broadly, the term means "a range of magnitude within which variations
that are considered 'big' would be considered 'small' within a higher
order of magnitude." The key idea is that you are bringing up a
*range* within which your quantity exists, and within which some
variations (usually variations of the top digit) are considered
significant and others insignificant. The variations that you mean
must not stray outside that range. Conventionally, these ranges are
between powers of ten, as illustrated above. If you intend a
different kind of range, then you'll need to specify it if you want to
be understood.
The term makes sense because in mathematics, one meaning of "order" is
just a set that fits into a certain place in a sequence of sets, and
"magnitude" means a one-dimensional quantity ("how much something
is"). "Third-order differential equations" is another such set (but
not a set of magnitudes).
Here are some real examples of the term in informed use:
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