2.2. Numeric expression of quantities

2.2.1. The accuracy of numbers, significant figures

Except the numbers we obtain by counting (the number of items), all numbers are inexact because their values are determined with a certain degree of uncertainty. The source of uncertainty is the limited capacity of either the measuring device (due to the flaws in its construction or improper calibration) or the measuring person (due to improper skills). We know the degree of uncertainty only in the case of our own measurements. Without a detailed discussion of the problem, the following is useful to know. In order to estimate and give the uncertainty of data, we need two (related) pieces of information: the order of magnitude and the significant figures.

The significant figures are determined by the achievable precision (exactness) of the measurement. Usually, the last figure of a measured value bears uncertainty, i.e. this figure should be considered as “estimated” (significant figures = all certain digits + one estimated digit). The value—and thus the significance—of zero digits in numbers is dependent on their position. Zeros are not significant if they are at the beginning of a number (leading zeros), or if they are at the end (trailing zeros) without a decimal point in the number (even though such zeros carry information about the magnitude). In contrast, zero digits are significant if they are at the end of a number containing a decimal point (because they show the exactness of measurement), as well as if they are inside the number (confined zeros, located between nonzero digits).

We must consider these aspects and the achievable precision of measurements when we simplify our numbers by rounding them up or down. We usually do this when we recalculate (mathematically transform) measured data or when we obtain them by calculation. (For example, a given amount of substance should be dissolved in a calculated solvent volume of 5.4786 mL. If our measuring device is calibrated with 0.1 mL division, then the figures “6” and “8” (corresponding to 0.0006 and 0.008 mL, respectively) are immeasurable, and figure “7” (corresponding to 0.07 mL) is uncertain. In this case, the desired volume can be approximated by pipetting 5.5 mL, considering the above uncertainties.) Carefully performed rounding is always recommended because series of digits of meaningless length make calculations unnecessarily difficult and bear the danger of calculation errors.

2.2.2. Expression of large and small quantities: exponential and prefix forms

In most cases, measured quantities differ from the unit of the given quantity by several orders of magnitude. In these cases, the length of the number cannot be substantially reduced by rounding. Two procedures are in use to avoid the writing of many zeros in the case of very large or small numbers, which would be uncomfortable and may be a source of error. During one of these procedures, large or small numbers are converted into their exponential form while keeping the unit of quantity (i.e. the scale). The other procedure replaces zeros with a prefix, i.e. it changes the unit of quantity (the scale). For example, we can write a quantity of 0.0000043 litre (L) as 4.3x10-6 L (exponential form) or 4.3 μL (microlitre, prefix form) according to the first and the second procedure, respectively. In the latter expression, the “μ” sign (read as “micro”) is called a prefix, which reflects the degree of change in the scale relative to the basic unit of quantity (in this example, six orders of magnitude downward)—this way keeping the magnitude information that was originally conveyed by the “eliminated” (replaced) zeros.

The definition of prefixes specifies the relationship between the two procedures (Figure 2.1). Official SI prefixes are defined at steps of three orders of magnitude above or below the unit of the given quantity, i.e. at 103, 106, 109 and 10-3, 10-6, 10-9 etc. Other (non-SI) prefixes including the deci (10-1), the centi (10-2) and the hecto (102) are not official, although widely used in civil life outside the laboratory. A mixture of prefix and exponential forms within an expression should be avoided (e.g. 5.2x10-4 μg is expressed correctly as 0.52 ng), similarly to the way in which the normal and exponential forms of numbers are used. (In exponential notation we do not write values larger than 10 or smaller than 1 such as 3100x104 or 0.12x10-2, because these numbers are correctly expressed as 3.1x107 and 1.2x10-3, respectively). The use of prefix or exponential forms of numbers is often insufficient to reduce their length. Therefore, rounding (see above) is also a necessary simplification.

Magnitude scales, prefixes and their relationship

Figure 2.1. Magnitude scales, prefixes and their relationship