Worked example — From significant figures to the number

Problem

From significant figures to the number. A measurement has four significant figures and is about $123\;\text{mm}$, with an uncertainty such that the fourth figure is already uncertain by one unit. Write two (or more) possible ways of representing it.

Solution

Step 1 — Interpreting “four significant figures”. If four figures are significant and the value is close to $123\;\text{mm}$, the fourth figure sits in the tenths-of-a-millimetre place. Saying that “the fourth figure is already uncertain by one unit” means $\Delta = 0{,}1\;\text{mm}$.

Step 2 — Equivalent representations. The same measurement can be written in different but equivalent ways:

$123{,}4\;\text{mm}\ \text{with}\ \Delta = 0{,}1\;\text{mm}, \qquad (123{,}4\pm 0{,}1)\;\text{mm}, \qquad 1{,}234\cdot 10^2\;\text{mm}$

In scientific-notation form the uncertainty falls on the last reported significant figure.

Step 3 — What all three convey. Each way of writing it communicates the same information: the measurement is known down to a tenth of a millimetre, and beyond that figure the data is not reliable.

$\ev{(123{,}4\pm 0{,}1)\;\text{mm} \equiv 123{,}4\;\text{mm},\ \Delta=0{,}1\;\text{mm} \equiv 1{,}234\cdot 10^2\;\text{mm}}$

Linked notes

Topics: Grandezze fisiche e misura Concepts: Cifre significative e incertezza Skills: Uso delle cifre significative