Please don’t use “an order of magnitude” to mean a factor of ten. Orders are logarithmic. Factors of about 3 to 32 round to one power of 10.

# Please don’t use “an order of magnitud

*Posted by Andy Skelton on January 7, 2010*

http://andy.wordpress.com/2010/01/07/please-dont-use-an-order-of-magnitud/

## Otto

/ January 7, 2010Errr… An “order of magnitude” is a power of whatever number makes the most sense in the context, and that context is usually such that “10” makes the most sense.

For example, Saturn is two orders of magnitude more massive than the Earth.

Stars, on the other hand, are measured by a doubling in brightness. So a 5 order of magnitude difference is 32 times brighter. :)

## Matt

/ January 8, 2010Could you expand on this a bit more?

## Andy

/ January 8, 2010I thought of this while reading a technical article on another blog. The author wrote “an order of magnitude” on the same page with mentions of more precise factors, as in “a factor of 5.” The former is longer, harder to understand, and not nearly as precise as the intended “factor of ten”.

The math can be seen a number of ways. The literal transposition to “a factor of ten” probably means between 9.5×-10.5x. However, there is no decimal point in the word “ten” so it could as well be 5x-15x.

I chose to use logarithmic rounding because “order of magnitude” is based on the counting of powers. If one order of magnitude is 10^1, anything between 10^0.5 and 3. 10^1.5 can be rounded to that. Those values are approximately 3 and 32. So if you increased a value by any factor between 3 and 32, the precise “orders of magnitude” increase will be between 0.5 and 1.5, which rounds to one order of magnitude.

A more extreme view is that a difference of one order of magnitude can be achieved by a factor of 1.01 or 99.99. It all depends on the original value. In the case above, the original value was never mentioned.

Finally, a point touched by Otto that I couldn’t fit into 140, the base of the factor has to be assumed or gleaned from knowledge of the subject. It’s usually 10 but he points out a good exception.