Significant places

Make a number are significant figures : (also applicable / valid points / numbers called, if they are significant). For this, possible deviations of this number must lie within the limits of the deviation of the last digit. Leading zeros are not meaningful. Whether ending zeros are significant has to be questioned on a case-by-case basis - a suitable notation can provide clarity here.

In science and technology, many numerical values ​​have their origin as a measured value that is subject to a measurement uncertainty . This makes the numerical value uncertain at a decimal point; all lower-valued places are then meaningless. Conversely, the number of significant digits is the minimum number of digits that is required to specify a given numerical value in scientific information without any loss of accuracy . There is a natural tendency to "play it safe" and perform a calculation to a greater number of decimal places than experimental accuracy warrants. In such a case, the calculation result incorrectly represents the quantity to be determined. The temptation to drag along too many decimal places is great when using pocket calculators. A person familiar with the recognized rules of technology (DIN, GUM ) indicates how “good” a numerical value is by only specifying the points that are known with certainty, plus one more that is uncertain.

Spelling of numbers in the decimal system

Significant digits of a number with decimal places

As a decimal in the are decimal representation of a number used digits to the right of commas referred. The number of decimal places must be distinguished from the number of significant places.

Examples of digits in a number:

number	Significant places	Decimal places
98.76	4th	2
0.009 876	4th	6th

Significant digits of a number without decimal places

It is more difficult to make a statement about the significant digits - for example, whether a “60” contains one, two or even more significant digits. Depending on the context, a number is to be evaluated exactly if it is e.g. B. is used as a natural number ; or it is to be interpreted as a rounded number if it is used as a numerical value for a physical quantity.

Scientific notation with a power of ten factor helps to avoid ambiguity with a value of 60 for a quantity determined using measurement technology . This allows a ending zero to be shifted to a decimal place. A non-significant zero is omitted; by writing the zero it is marked as significant:

• a significant digit: 6 · 10 ¹
• two significant digits: 6.0 · 10 ¹
• three significant digits: 60.0 or 6.00 · 10 ¹

Further examples

number	Significant places	Decimal places
9 876 000.00 · 10 −2	9	2
9 876 000	unexplained: 4 to 7	0
98 760 · 10 2	unexplained: 4 or 5	0
987.6 · 10 4	4th	1
9.876 · 10 6	4th	3

Exactly known values

Some numerical values ​​in science and technology are known exactly, i.e. without measurement uncertainty. This can be

• Whole numbers . Example: the number of protons in an atomic nucleus
• Numbers with a finite number of decimal places. Example: Planck's quantum of action h was set to exactly 6.62607015 · 10 ⁻³⁴  Js for the definition of the units of measurement .

In all of these cases, the concept of significant digits is not applicable because the number of digits indicated does not correspond to the measurement accuracy. In the case of an infinite number of decimal places , it is usual to write ellipses after the last specified place to indicate that any number of further places could be specified.

Definition and point rule

DIN 1333 defines the significant digits as the first digit different from zero up to the rounding point . This is the last digit that can still be specified after rounding; see notation of numbers .

The digits to be omitted by rounding should not be padded with zeros. By shifting the decimal point and the power of ten factor, the rounding point can be shifted to the ones place or a decimal place, see also measured value .

In measurement technology, the position of the decimal point can not only be adjusted using the power of ten factor, but also by selecting the unit (e.g. for length mm → cm → m → km).

Example: Anyone who paraphrases a specification 20 km in 20,000 m has filled in with ending zeros, which are not significant. If the length can be specified to within one meter, 20,000 km would have to be written beforehand (all positions up to the rounding point). When a number is given without further information, it is generally interpreted to mean that the last digit is rounded. The number 20,000 is assumed to represent a value between 19,999.5 and 20,000.5.

Result of a calculation

Here are two rules of thumb first; a more reliable procedure follows in the next chapter.

• The result of an addition / subtraction has as many decimal places as the number with the fewest decimal places.
• The result of a multiplication / division gets as many significant digits as the number with the fewest significant digits:

numbers	Smallest number of decimal places	Smallest number of significant digits	Result
20.567 + 0.0007	3		20,568
12 + 1.234	0		13
12.00 + 1.234	2		13.23
12,000 + 1,234	3		13.234
1.234 x 3.33		3	4.11
1.234 x 0.0015		2	0.0019
28 · ${\displaystyle \pi }$		2	88

The result also depends on whether one of the numbers is exact and whether the number of digits before or after the calculation is fixed:

1. In the following table in the first example, 3 is a parameter to be assessed as exact ; the significant digits result from the value 1.234 in the sense of a measured value.
2. In the second example, the number 1.234 is a parameter; the significant digits result from the value 3, so that there is only one significant digit in the result.

numbers	Significant places	Result
3 x 1.234	4th	3.702
	1	3 (before the calculation: 1.234 ≈ 1)
		4 (According to the calculation: 3.702 ≈ 4)

Hints:

• Rounding should only be carried out as late as possible in the invoice. Otherwise, several rounding deviations can add up to a larger total deviation. In order to avoid this enlargement, known quantities with at least one place more should be used in intermediate calculations than can be stated in the result.
• If the diameter of a circle is measured to the nearest millimeter, and the circumference is calculated with the closest possible approximation to pi , the circumference can again be given with millimeter precision at best, despite the calculation with a perhaps ten-digit factor.
• If a drawing is enlarged to a scale of 10: 1 and the coordinates are drawn with an accuracy of ½ millimeter, the enlargement is accurate to 5 millimeters. The number of significant digits of the coordinates does not change due to the scale factor 10, which is assumed to be exact.

Significant places in measurement technology

For measurement technology it is always the safest method to observe the error limits of the input data and to determine their effects on the result of a calculation, see error propagation . Exact numbers have a margin of error of zero. The error limit of the result provides information on which digit is still significant as the lowest digit.

Example: A circle radius is measured as 17.5 cm. The scope is sought . In contrast to the above, it should not be specified with a large number of decimal places, but only with a number of digits that match the number of digits . ${\displaystyle U=2\pi r}$ ${\displaystyle \pi }$ ${\displaystyle r}$

Exactly: ${\displaystyle 2=2{,}000\cdot (1\pm 0\,\%)}$

Rounded: ${\displaystyle \pi =3{,}14\pm 0{,}005=3{,}14\cdot (1\pm 0{,}2\,\%)}$

A number rounded according to commercial or mathematical rules can deviate between −5 and +5 on the first cut-away digit.

Measured: ${\displaystyle r=\mathrm {17{,}5\,cm\pm 0{,}1\,cm=17{,}5\,cm\cdot (1\pm 0{,}6\,\%)} }$

It is assumed from the measured value that the least significant digit can be specified incorrectly by ± 1.

Invoice: ${\displaystyle U=109{,}90\,\mathrm {cm\cdot (1\pm (0+0{,}2+0{,}6)\,\%)=109{,}90\,cm\pm 0{,}9\,cm} }$

Result :, nothing more precise cannot be given, because in this case the first decimal place with ± 9 is already maximally uncertain. It is therefore better to state that the next higher digit can be wrong by a maximum of ± 1: The result is accurate to a maximum of a centimeter. ${\displaystyle U=(110\pm 1)\,\mathrm {cm} }$

In other words: the ending zero in the ones place of 110 is significant in this case. To make this clear without writing down the error limits, it is better to write because the explicit specification of the decimal point shows that it was determined to be significant in this calculation. would have the same purpose. It is not allowed to write or , because the ending zero or ending nine is not a significant digit due to the error limit. ${\displaystyle U=1{,}10\,\mathrm {m} }$ ${\displaystyle U=11{,}0\,\mathrm {dm} }$ ${\displaystyle U=1100\,\mathrm {mm} }$ ${\displaystyle U=1099\,\mathrm {mm} }$

Even a more exact one would only have produced a result of , the number of digits of the result would be the same. ${\displaystyle \pi }$ ${\displaystyle U=109{,}96\,\mathrm {cm} \cdot (1\pm 0{,}6\,\%)=109{,}96\,\mathrm {cm} \pm 0{,}7\,\mathrm {cm} }$

The fact that in this example the result is only centimeter-accurate, although the original measurement was carried out with millimeter precision, shows the importance of the number of digits for metrological problems: Because the result is roughly a power of ten greater than the specification and the case here is unfavorable, the one also shifts Accuracy to the power of ten from millimeters to centimeters. During the multiplicative calculation, the magnitude of the accuracy only remains constant relative to the respective value; the millimeter-precise measurement does not guarantee a result that is accurate to the millimeter. In more complicated calculations, the accuracy can no longer be estimated from the number of significant digits, but only a correct error propagation calculation guarantees the reliability of a result. The subsequently determined number of digits then represents the result of the error analysis.

Indication of significant places in measurement technology according to GUM

According to the internationally recognized "Guide to the Expression of Uncertainty in Measurement (GUM)", the number of meaningful significant digits to be indicated (based on a determination of the measurement uncertainty ) is given by the following procedure:

• Perform measurement and determine uncertainty,
• Round the result of the expanded uncertainty to a maximum of two significant digits,
• Round the result of the measurement to the same places.

Individual evidence

1. ↑ ^{a b c d} DIN EN ISO 80000-1: 2013-08 Sizes and units - Part 1: General. Cape. 7.3.4.
2. ^ Daniel C. Harris: Textbook of Quantitative Analysis. Springer, 8th edition 2014, p. 64.
3. ↑ Wilbert Hutton, quoted in Richard E. Dickerson: Principles of Chemistry. Walter de Gruyter, 2nd ed. 1988, p. 997.
4. ↑ Klaus Eden, Hermann Gebhard: Documentation in measurement and testing technology: measuring - evaluating - displaying - protocols - reports - presentations. Springer Vieweg, 2nd edition 2014, p. 27.
5. ^ Ulrich Müller: Chemistry: The basic knowledge of chemistry. Georg Thieme, 12th edition 2015, p. 29.
6. ↑ DIN 1333: 1992-02 figures. Cape. 10.2.2.
7. ↑ Josef Draxler, Matthäus Siebenhofer: Process engineering in examples: problems, approaches, calculation methods. Springer Vieweg, 2014, p. 3.
8. ↑ Douglas C. Giancoli: Physics: Upper Secondary School. Pearson School, 2011, p. 5.
9. ↑ Evaluation of measurement data - Guide to the Expression of Uncertainty in Measurement (GUM). JCGM 100: 2008.