order of magnitude

Also called: orders of magnitude

Which power of ten a number lives in: thousands (10³), millions (10⁶), billions (10⁹). Two numbers a factor of ten apart differ by one order of magnitude.

The order of magnitude of a number is the power of ten it lives in: thousands ( $10^3$ ), millions ( $10^6$ ), billions ( $10^9$ ). Two numbers within a factor of about ten of each other are of the same order; a number ten times bigger is one order of magnitude up.

Thinking in orders of magnitude is the fastest way to catch broken numbers. A result that is 20% off may be a detail; a result that is ten times off is a mistake: an extra zero, a percent typed as 2.5 instead of 0.025, a column in thousands read as units, or a month mixed up with a year.

To get the order of a product quickly, write each number with a power of ten and add the exponents: $8{,}000 \times 600 = 8 \times 10^3 \times 6 \times 10^2 = 48 \times 10^5 \approx 5 \times 10^6$ .

Example

A report says an online school sold 48.7 million UAH of lessons in March. March had 7,840 lessons at about 620 UAH.

Estimate: 8,000 × 600 = 4,800,000, about 5 million ( $5 \times 10^6$ ).
The report’s number is $\approx 5 \times 10^7$ : one order of magnitude too big.
The real figure is 4,868,640 UAH; the report had an extra digit.

Common mistakes

Trusting a number because it has many digits. Precision is not correctness; check the order first.
Ignoring factors of 12, 100 and 1,000. They point to month/year, percent and thousands mix-ups.
Losing zeros in your head. Write numbers as powers of ten and add the exponents.

Learn it in the course

Back-of-the-envelope estimates · Check any number in ten seconds: round to one significant figure, count the zeros, and give a range instead of false precision.