Problem — Significant figures

Problem

$L = 2{,}34$ m. How many significant figures? What if it were $L = 2{,}3400$ m? What difference in uncertainty does the declared trailing zero imply?

Solution

Case $L = 2{,}34$ m. The significant figures are three (the 2, the 3 and the 4). The last declared digit is the hundredths place, so the measurement carries an implicit uncertainty on the order of $\delta L \approx \pm 0{,}01\ \text{m}.$ Writing $2{,}34$ declares that we know the length down to the centimetre, but no further.

Case $L = 2{,}3400$ m. Now the significant figures are five: the declared trailing zeros after the decimal point are significant, because they communicate that those positions were actually measured and found to be zero. The last digit is the ten-thousandths place, so $\delta L \approx \pm 0{,}0001\ \text{m}.$

Difference in uncertainty. These are not the same number “written differently”: $2{,}3400$ states a measurement a hundred times more precise than $2{,}34$, since the uncertainty goes from $0{,}01$ m to $0{,}0001$ m: $\frac{\delta L(2{,}34)}{\delta L(2{,}3400)}=\frac{0{,}01}{0{,}0001}=100.$ This is why trailing zeros must never be added or removed carelessly: they convey precise physical information about the quality of the measurement.

$\ev{2{,}34\ \text{m}=3\ \text{cifre},\ \pm0{,}01\ \text{m};\qquad 2{,}3400\ \text{m}=5\ \text{cifre},\ \pm0{,}0001\ \text{m}\ (\times 100\ \text{più precisa})}$

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Topics: Metodo sperimentale Concepts: Cifre significative e incertezza Skills: Uso delle cifre significative