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Significant Figures in a Pure Number

Measurement of a physical quantity is nothing but the determination of the number which on multiplication with its unit value gives the value of the physical quantity being measured.  Two types of measurements are possible.  These involve (1) discrete variation, and (2) continuous variation.  Normal counting of positive integers is an example of discrete variation.  For example, we can count the number of peas in a bottle, the number of eggs in a basket, the number of bananas, etc.  In each case, we get an exact number.

The mass of a substance can be determined in the laboratory by using an analytical balance.

Analytical Balance

To measure temperature thermometers are used. There are three common scales to measure temperature — °C (degree celsius), °F (degree fahrenheit) and K (kelvin).
In the laboratory, volume of liquids or solutions can be measured by graduated cylinder, burette, pipette etc. A volumetric flask is used to prepare a known volume of a solution.
The measurement involving continuous variation, such as the height of a man, volume of water in a cup, length of an iron rod, etc., is always associated with uncertainty as the value obtained depends on the least count of the apparatus used for this purpose. For example, the volume of water measured with a measuring cylinder may come out to 150 cm3, whereas the same volume, if measured with the help of a burette or a caliberated pipette (precision 0.1), may come out to be 150.4 cm3.

The exact value of volume may be 150.40 cm3 or a slightly less or more (say, 150.38 cm3 or 150.41 cm3), we cannot get this value unless we employ an apparatus of precision 0.01. Thus, we conclude that the measurement of a continuous variable cannot be more precise than the precision of the apparatus used.

The precision of the apparatus used should be reflected from the way we write the measurement. The number expressing measurement should include all those digits which are certain and a last digit which is uncertain. For example, earlier we wrote the volume of water as 150 cm3 if measured with a measuring cylinder since the precision of the cylinder is 1. This means that the volume of water may lie anywhere between 149 cm3 and 151 cm3. Similarly, when we measured the volume with a burette, we reported it as 150.4 cm3. This means that the volume may lie anywhere between 150.3 cm3 and 150.5 cm3. In the results reported above, the last figure represents the uncertainty of the measurement or the precision of the apparatus used.

The total number of digits in a number is called the number of significant figures. While reporting a number, care must be taken to include as many significant figures as permitted by the precision of the apparatus used for the measurement. To report the data in a larger number of significant figures is misleading. On the other hand, if data is reported in a lesser number, it is equivalent to suppressing some information that may be useful.

The following rules are observed in counting the number of significant figures in a given measured quantity.
  1. All digits are significant except zeros appearing in the beginning of a number.  For example, 152 cm, 0.152 cm and 0.0152 cm all have three significant figures.
  2. The zeros appearing on the right of a digit, including those appearing after a decimal point, are significant. For example, 150 cm3, 152.0 cm3 and 152.00 cm3 carry three, four and five significant figures respectively.
In reporting the results according to the above two rules, the last digit may be rounded off if required.  Suppose the final result of 55.673 is to be reported in three significant numbers, then the result would be 55.7.  Here the last number 6 has been rounded to 7 as the next digit is more than 5.  The rule of rounding off an integer may be stated as follows.

"If the figure following the last number to be retained is less than or equal to 5, the last number is left unchanged.  However, if the figure is greater than 5, the last number to be retained is increased by one".

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