Guide
Orders of magnitude
Why the exponent matters more than the digits in front of it, how scientific notation makes unlike quantities comparable, and what an order of magnitude actually buys you.
The exponent carries the information
When two quantities are wildly different in size, the interesting part is almost never the leading digits. The Earth is about 12,700 kilometres across and the Sun about 1,392,700; what matters is that one is roughly a hundred times the other, not that the digits happen to be 127 in both.
An order of magnitude is a factor of ten. Two quantities are of the same order when their ratio is under about ten, and separated by an order for every factor of ten beyond that. It is a deliberately coarse measure, and the coarseness is the point: it lets you compare things that no precise figure would let you hold in mind at once.
This is why scientific notation writes a number as a coefficient and a power of ten. Written 1.4 × 10⁹ and 8.6 × 10¹⁰, two quantities announce their relationship in the exponents before you have read the digits.
Why this catalogue stores a magnitude
A quantity here records its order of magnitude alongside its value, and the reason is comparison rather than tidiness. Avogadro's constant is 6.022 × 10²³ and there is no English word for 10²³ that anyone would recognise, so a scale word would make it harder to read rather than easier.
Recording the magnitude also makes contradictions findable. Two figures for the same thing that differ in their exponent are disagreeing about something real; two that differ in the third digit are usually disagreeing about precision.
A ladder of familiar quantities
| Quantity | Roughly | Order |
|---|---|---|
| Everest, in metres | 8.8 × 10³ | The tallest thing anyone climbs, and only four orders above a person's height. |
| The equator, in kilometres | 4.0 × 10⁴ | Four and a half orders above Everest once both are in the same unit — which is the point of putting them in one. |
| The Moon's distance, in kilometres | 3.8 × 10⁵ | About ten times the equator's length, so one order up. |
| Light, in metres per second | 3.0 × 10⁸ | A different dimension entirely, and comparable only because the notation makes the scales explicit. |
Each rung is roughly a thousand times the one below. Nothing about the words million, billion and trillion conveys that, which is why the exponent is the part worth reading.
When the coarseness misleads
An order of magnitude is a blunt instrument and occasionally the bluntness matters. Two figures within the same order can still differ by a factor of nine, which is the difference between a comfortable margin and a failure.
It is also asymmetric in a way people forget: rounding 9.4 × 10⁵ to 'about 10⁶' overstates by six per cent, while rounding 1.1 × 10⁶ down to 10⁶ understates by ten. The notation invites treating everything in a decade as equivalent, and it is not.