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Physics is a natural science, and accurate measurement is crucial. However, measurements are rarely 100% precise, leading to what are often called measurement errors or uncertainties. These are not real errors, but rather limitations of the measuring instruments and human perception.

Using a ruler, a measurement of 10.2 cm actually implies 10.2 +/- 0.05 cm, meaning the actual length could be anywhere between 10.15 cm and 10.25 cm. This +/- value represents the error range. Different scales and instruments (like personal scales, kitchen scales, and analytical balances) have varying levels of precision, each with its own error range.

Instead of cumbersome error ranges, significant figures (or 'gültige Ziffern') are used to express the precision of a measurement. You count significant figures from the first non-zero digit. For example, 1.2 and 2.0 both have two significant figures, while 0.003537 has four. The number of significant figures indicates the precision of the measurement.

If a personal scale measures 4 kilograms of potatoes, it should be recorded as 4.0 kg to reflect its precision of 0.1 kg, thereby showing two significant figures. Using different scales to measure the same amount of milk illustrates how the number of significant figures directly corresponds to the instrument's precision, eliminating the need for an explicit error range.

When performing calculations (multiplication, division) with measurements of varying precision, the result should have no more significant figures than the measurement with the fewest significant figures. For example, multiplying 1.25 m (3 sig figs) by 3 m (1 sig fig) results in 3.75 m², which should be rounded to 4 m² to match the single significant figure of the least precise measurement.

Sometimes, simply rounding isn't enough to get the correct number of significant figures. For instance, an area of 19.5 m² (3 sig figs) calculated from 3.9 m (2 sig figs) and 5 m (1 sig fig) would need to be 20 m². To ensure it has only one significant figure, it can be expressed in scientific notation as 2 x 10^1 m².

The rule of thumb for significant figures is a simplification and can sometimes lead to an underestimated error range. By calculating the minimum and maximum possible values based on the actual error ranges of each measurement, we can see that the true range of the calculated area is much wider than what the rule of thumb suggests. The exact calculation, considering the actual error margins, reveals a more accurate and larger error range than the simplified rule of thumb provides.