1.4 Significant figures  (Page 2/5)

In the figure, elements of measurement of a vernier scale are shown. The reading on the main scale (upper scale in the figure) is taken for the zero of vernier scale. The same is shown with an arrow on the left. This reading is "5.3". We can see that zero is between "5.3" and "5.4". In order to read the value between this interval, we look for the division of vernier scale, which exactly matches with the division mark on the main scale. Since 10 divisions of vernier scale is equal to 9 divisions on main scale, it ensures that one pair of marks will match. In this case, the seventh (7) reading on the vernier scale is the best match. Hence, the final reading is "5.37".

In this example, the last measurement constitutes the suspect reading. On repeated attempts, we may measure different values like "5.35" or "5.38".

Rules to identify significant figures

There are certain rules to identify significant figures in the reported value :

Rule 1 : In order to formulate this rule, we consider the value of measured length as "5.02 cm". Can we drop any of the non-zero digits? No. This will change the magnitude of length. The rule number 1 : All non-zero digits are significant figures.

Rule 2 : Now, can we drop “0” lying in between non- zeros “5” and “2” in the value considered above? Dropping “0” will change the value as measured. Hence, we can not drop "0". Does the decimal matter? No. Here, “0” and “decimal" both fall between non-zeros. It does not change the fact that "0" is part of the reported magnitude of the quantity. The rule number 2 : All zeros between any two non-zeros are significant, irrespective of the placement of decimal point.

Rule 3 : Let us, now, express the given value in micrometer. The value would be 0.000502 micrometer. Should expressing a value in different unit change significant figures. Changing significant figures will amount to changing precision and changing list count of the measuring instrument. We can not change least count of an instrument – a physical reality - by mathematical manipulation. Therefore, rule number 3 : if the value is less than 1, then zeros between decimal point and first non-zero digit are not significant.

Rule 4 : We shall change the example value again to illustrate other rule for identifying significant figures. Let the length measured be 12.3 m. It is equal to 123 decimeter or 1230 cm or 12300 millimeter. Look closely. We have introduced one zero, while expressing the value in centimeter and two zeros, while expressing the value in millimeter. If we consider the trailing zero as significant, then it will again amount to changing precision, which is not possible. The value of 1230 cm, therefore has only three significant figures as originally measured. Therefore, Rule number 4 : The trailing zeros in a non-decimal number are not treated as significant numbers.

Rule 5 : We shall again change the example value to illustrate yet another characteristic of significant number. Let the measurement be exactly 50 cm. We need to distinguish this trailing “0”, which is the result of measurement - from the “0” in earlier case, which was introduced as a result of unit conversion. We need to have a mechanism to distinguish between two types of trailing zeros. Therefore, this rule and the one earlier i.e. 4 are rather a convention - not rules. Trailing zeros appearing due to measurement are reported with decimal point and treated as significant numbers. The rule number 5 is : The trailing zeros in a decimal number are significant.