Imagine I’m dealing with some rough numbers – I’m in “order of magnitude” mode. Say I’m working out how many customers and orders my archaic DVD rental business needs to store in its customer database:

Customers: 1,000 (rough projected count by the end of the year)

Orders: 3,325 (based on current data which has an average 3.325 orders for each customer)

Say I’m uncomfortable dealing with such a precise number of orders, and I want to get back to a power of 10 while I’m just fooling around with rough volumes. Is 3,325 closer to 10,000 or 1,000?

Well, what’s halfway between 1,000 and 10,000? If you had to “round off” a measurement to the nearest order of magnitude, at what point would you say that the measurement’s now closer to 10,000 than 1,000?

In linear terms, it’s 1,000 + (10,000 – 1,000) / 2 = 5,500. But that’s an unsatisfying approach – intuitively, 5,500 is a lot “closer” to 10,000 than 1,000 if it’s a figure that’s been growing exponentially.

When talking in orders of magnitude, a better threshold between 1,000 and 10,000 is about 3,162.3, since:

10 ^ 3 = 1,000

10 ^ 3.5 ~= 3,162.3

10 ^ 4 = 10,000.

So, if you ever need to remove precision from a number for some reason, to get back to orders of magnitude from an overly precise estimate or measurement, the threshold is about 3.1623 times the next lower order of magnitude.

To calculate 10 ^ 3.5 using the subset of maths I (a) learned and (b) haven’t forgotten yet:

10 ^ 3.5

= 10 ^ (3 + 0.5)

= (10 ^ 3) * (10 ^ 0.5)

= (1,000) * sqrt(10)

~= 1,000 * 3.1623

~= 3,162.3

So in general, the threshold is the lower power of 10 times sqrt(10).