1.4 Significant figures  (Page 4/5)

The measured value of volume has the least “2” numbers of significant figures. In accordance with the rule, the result of multiplication is, therefore, limited to two significant digits. The value after rounding off is :

$$\Rightarrow m=17gm$$

Addition or subtraction

In the case of addition or subtraction also, a different version of guiding principle applies. Idea is to maintain least precision of the measured value in the result of mathematical operation. To understand this, let us work out the sum of three masses “23.123 gm”, “120.1 gm” and “80.2 gm”. The arithmetic sum of masses is “223.423 gm”.

Here, we shall first apply earlier rule in order to show that we need to have a different version of rule in this case. We see that third measured value of “80.4 gm” has the least “3” numbers of significant figures. In accordance with the rule for significant figures, the result of sum should be “223”. It can be seen that application of the rule results in loosing the least precision of 1 decimal point in the measured quantities.

Clearly, we need to modify this rule. The correct rule for addition and subtraction, therefore, is that result of addition or subtraction should retain as many decimal places as are there is in the measured value, having least decimal places.

Therefore, the result of addition in the example given above is “223.4 gm”.

Rounding off

The result of mathematical operation can be any rational value with different decimal places. In addition, there can be multiple steps of mathematical operations. How would we maintain the significant numbers and precision as required in such situations.

In the previous section, we learnt that result of multiplication/division operation should be limited to the significant figures to the numbers of least significant figures in the operands. Similarly, the result of addition/subtraction operation should be limited to the decimal places as in the operand, having least decimal places. On the other hand, we have seen that the arithmetic operation results in values with large numbers of decimal places. This requires that we drop digits, which are more than as required by these laws.

We, therefore, follow certain rules to uniformly apply “rounding off” wherever it is required due to application of rules pertaining to mathematical operations :

Rule 1 : The preceding digit (uncertain digit of the significant figures) is raised by 1, if the digit following it is greater than 5. For example, a value of 2.578 is rounded as “2.58” to have three significant figures or to have two decimal places.

Rule 2 : The preceding digit (uncertain digit of the significant figures) is left unchanged, if the digit following it is less than 5. For example, a value of 2.574 is rounded as “2.57” to have three significant figures or to have two decimal places.

Rule 3 : The “odd” preceding digit (uncertain digit of the significant figures) is raised by 1, if the digit following it is 5. For example, a value of 2.535 is rounded as “2.54” to have three significant figures or to have two decimal places.