Orders of magnitude

Humans can easily distinguish things 1/10th or 10 times their average size. Beyond that? It becomes fuzzier. To get a better grasp on the very small or the very large, we invented orders of magnitude.

The Wikipedia page for “Order of Magnitude” is very detailed but not very useful in the context of this article. To simplify, the order of magnitude is the number of digits used to write a number after the most significant one. This gives this handy table:

Number Order of magnitude
1-9 0
10-99 1
1,000,000-9,999,999 6
0.01-0.09 -2

Orders of magnitude give an easier way to describe a reality whose scale is hard to grasp with conventional numbers. A famous use of orders of magnitude is Richter’s scale used to measure the energy released by earthquakes. Between low intensity earthquakes and catastrophic ones, there is a difference of several orders of magnitude in released energy, which is a shorter way to express a massive difference by a factor of 10,000,000 or more.

However, orders of magnitude can also help us when no earthquake is going on. I’ve identified two uses in every day life for orders of magnitude: inaccurate estimations and appraising large numbers.

Wild guesses

Whether it is size, time or cost, we often have to estimate numbers with relatively low accuracy. However, it is possible to bound the estimate by finding its definite order of magnitude. For example, without looking at a catalog, a computer price order of magnitude is most probably 3, meaning that most computers will cost between $1,000 and $9,999. It still is a wide range but it still is a better estimation than no range at all.

Especially with time or general cost, orders of magnitude can help prioritize or decide whether to engage in a new activity. For a software project starting from scratch, roughly how many files will have to be written by hand? 1, 10, 100 or 1,000? The time investment won’t look the same at all in each general case.

Get a feel for large numbers

Paradoxically, the greater a number is, the least we are able to appreciate its actual importance. For example, Malcolm Gladwell’s Outliers book establish the “10,000-Hour Rule” as the amount of correct practice time it takes to achieve world-class mastery of any given discipline. I do not think it was meant as an exact number, rather as an order of magnitude, as disciplines greatly vary in time demands, but I believe it is the general ballpark for world-class mastery. However, many people have been subjugated by the large number itself, clocking themselves to reach the magical number, although I don’t believe it was ever the point of the “10,000-Hour Rule”.

In a different category, I witnessed people being confused about the fundamental difference between millionaires and billionaires. Millionaires are generally a couple orders of magnitude richer than most people. It’s already a lot, but billionaires are 3 orders of magnitude richer than millionaires, which is a mind-bogglingly massive difference! And this means that most people are closer to millionaires than millionaires are to billionaires.

Tangent on billionaires

This also means that billionaires are 5 orders of magnitude richer than most people and as such any correlation between their wealth and their work capacity is moot. Can you imagine working 10 times as hard (=long) as you do know? How about 100 times? How about the 100,000 times it would require to earn billions from your work alone?

Of course not. But millions and billions have a tendency to blend together in a vague infinity-like concept, even though they have nothing in common with each other. And orders and magnitude can help mentally separate them.

(Banner is CC-BY-SA 4.0 and is adapted from Pablo Carlos Budassi’s work)

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