Order of magnitude

Counting in tens

Written as $a \times 10^n$, the exponent $n$ is the order of magnitude. It says how big the quantity is, while the $a$ only refines it.

Quantity	Scientific notation	Order of magnitude
A classroom, in people	$3 \times 10^1$	$10^1$
A city, in people	$1 \times 10^6$	$10^6$
The Earth, in people	$8 \times 10^9$	$10^9$

A city and the Earth differ by three orders of magnitude, which means a factor of about a thousand.

Why it is the useful part

Comparing $9 \times 10^7$ with $3 \times 10^8$ takes one look at the exponents: $10^8$ beats $10^7$, so the second is larger. Written out as $90{,}000{,}000$ and $300{,}000{,}000$, the same comparison means counting digits.

Subtracting the exponents gives the ratio directly:

$$\frac{3 \times 10^8}{9 \times 10^7} = 0.33 \times 10^1 \approx 3.3$$

Below one

Negative exponents extend the same scale downward, so a quantity smaller than one still has an order of magnitude:

$$0.001 = 10^{-3} \qquad 0.0000001 = 10^{-7}$$

A bacterium at $10^{-6}$ m and a virus at $10^{-7}$ m differ by one order of magnitude — the bacterium is about ten times longer. See negative exponents for why the sign flips below one, and scientific notation for writing these quantities down.