Significant figures

Digits of a number are significant digits (also: the current locations ) called when this number is within the limits of the deviation of the last of these points. These include the meaningful digits without leading zeros. Whether ending zeros are significant, must be occasionally questioned; should be made for clarity here by appropriate notation.

The significance of a number is the number of significant digits. It is one of the accuracy.

Spelling of numbers in the decimal system

Significant digits in a number with decimal places

As the number of decimal places used in the decimal representation of a number are referred to the right of the decimal point.

The number of decimal places to be different from the number of significant digits. But any responsible appended to the number ( not lightly assumed by the calculator) decimal place is one significant digit.

Examples of digits of a number:

Significant digits in an integer

Integers have no decimal places.

More difficult is the testimony to the significant digits: Does " 20" one, two or even more significant digits? Depending on context, a number of exact values ​​, when it is used for example as a natural number; or they can be seen as round number when it is used as a numerical value to a physical quantity.

For an exact number, the question of the significance is not, because it can be extended with any number of decimal zeros.

To avoid a determined Messtechnik by size on numerical value " 20", an ambiguity to choose scientific notation with exponent factor. Thus, a zero ending will be moved to one decimal place. A non-significant zero is omitted; by writing the zero it is marked as significant:

  • One significant digit: 2 · 101
  • Two significant figures: 2.0 · 101
  • Three significant figures: 20.0 or 2.00 · 101

Definition and commas rule

DIN 1333 defines the significant digits than the first nonzero digit to the rounding position. This is the last point that can be entered after rounding.

The to be omitted due to rounding figures are not to be filled by zeros. Comma- shift and power-of- factor the rounding position is to move to the units digit or a decimal point, see also measured value.

In metrology, the decimal position can be adjusted not only by the power-of- factor, but also by the choice of the unit (eg, length mm cm → → → m km).

Example: Who circumscribes an indication of 20 km in 20 000 m, has filled with ending zeros are not significant. If the length is exactly but can be specified to within a few meters, would previously write 20.000 km ( all digits to the rounding position ).

Result of a calculation

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

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

The result is also dependent on whether one of the numbers is accurate, and whether the number of digits is fixed before or after the invoice:

Notes:

  • Rounding should only be carried out as late as possible within the accounting transition. Otherwise, multiple rounding differences can be assembled into a larger overall deviation. To avoid this increase, known quantities to be used with at least one point more than in the result can be specified in interim bills.
  • If a diameter of a circle measured in millimeter, and if we include the scope of this with a close approximation of Pi, so the scope can be given only a millimeter again, despite the bill with a maybe ten digit factor.
  • If a drawing at a scale of 10:1 increased, and the coordinates on ½ millimeters are drawn exactly the magnification to 5 millimeters is accurate. The number of significant digits of the coordinates does not change by the assumed as exact scale factor of 10

Significant digits in measurement technology

For the measurement technique, it is always the safest way to observe the limits of error of the input data and to determine their impact on the result of a calculation, see error propagation. Exact numbers have the margin of error is zero. The accuracy of the result gives an indication of how far lower order digits are significant.

Example: A circle radius is measured at 17.5 cm. Wanted is the scope. In contrast to above should not be specified here with a lot of decimal places, but only with a significance to match the significance of.

The fact that in this example the result is only the centimeter, although the original measurement was performed with millimeter accuracy, showing the importance of places of significance for measurement problems: Because the result is to roughly an order of magnitude greater than the specification, and the case here is unfavorable, also shifts the accuracy of an order of magnitude of millimeters to centimeters. The magnitude of accuracy remains constant during the multiplicative account only relative to the respective value constant, accurate to the millimeter measurement does not guarantee a millimeter accurate result. In more complex calculations the accuracy of the number of significant digits can not be estimated, but only one correct error propagation calculation guarantees the reliability of a result. The subsequently determined locations of significance then represents the result of the error analysis.

Perimeter
285981
de