Scientific notation and orders of magnitude

When numbers are very large or very small, writing them out in full becomes cumbersome and error-prone (one slip in counting the zeros is enough). It is then convenient to use scientific notation: every number is written in the form

$a\cdot 10^n \qquad \text{with}\quad 1 \leq a < 10,$

where $a$ is the mantissa (a number with a single digit before the decimal point) and $n$ is an integer, positive for large numbers and negative for small ones. A few examples from the scales of nature:

• Electron mass: $9{,}1\cdot 10^{-31}\;\text{kg}$
• Atomic nucleus radius: $\sim 10^{-15}\;\text{m}$
• Earth’s radius: $6{,}4\cdot 10^6\;\text{m}$
• Earth-Sun distance: $1{,}5\cdot 10^{11}\;\text{m}$
• Sun’s mass: $2\cdot 10^{30}\;\text{kg}$

The order of magnitude of a number is the nearest power of 10. It is a quick way of saying “how large” a quantity is, ignoring the details. For instance, the Earth’s radius is of order $10^7\;\text{m}$: more precisely it equals $6{,}4\cdot 10^6\;\text{m}$, but since $6{,}4 > \sqrt{10} \approx 3{,}2$ the mantissa “rounds” upward, and the nearest power of 10 is $10^7$ rather than $10^6$. The criterion, in general, is exactly this: the mantissa is compared with $\sqrt{10}\approx 3{,}2$ to decide which power to round towards.

Sensitivity to orders of magnitude

In physics it is essential to develop an “intuition” for orders of magnitude. If a calculation gives the mass of the Moon as $5\;\text{kg}$, it is immediately clear there is an error, without even redoing the sums: the result should be of order $10^{22}\;\text{kg}$. Comparing the order of magnitude of a result with what is expected is the first, fastest check of any calculation.

Collegamenti

Argomenti: Grandezze fisiche e misura Concetti: Notazione scientifica e ordini di grandezza Competenze: Stime di Fermi

Esercizi collegati: Problema — Volume di un parallelepipedo · Problema — Capelli in testa · Problema — Mar Adriatico