Significant Figures

 

In any measurement, the accurately known digits and first doubtful digit are known as significant figures.

we must know the following rules in deciding how many significant figures are to be retained in the final result. 

     1. All digits 1,2,3,4,5,6,7,8,9 are significant. 

However, zeros may or may not be significant. ln case of zeros, the following rules may be adopted.

• A zero between two significant figures is itself significant.
• Zeros to the left of significant figures are not significant. For example, none of the zeros in 0.00467 or 02.59 is significant.
• Zeros to the right of a significant figure, may or may not be significant. In decimal fraction, zeros to the right of a significant figure are significant. For example, all the zeros in 3.570 or 7.4000 are significant.

However, in integers such as 8,000 kg, the number of significant zeros is determined by the accuracy of the measuring instrument. If the measuring scale has a least count of 1 kg then there are four significant figures written in scientific notation as 8.000 x 10^3 kg. 

If the least count of the scale is 10 kg, then the number of significant figures will be 3 written in scientific notation as 8.00 x 10^3 kg and so on.

• When a measurement is recorded in scientific notation or standard form, the figures other than the powers of ten are significant figures. For example, a measurement recorded as 8.70 x 10^4 kg has three significant figures.

       2. In multiplying or dividing numbers, keep a number of significant figures in the product or quotient not more than that contained in the least accurate factor i.e., the factor containing the least number of significant figures. 

                    For example, the computation of the following using a calculator gives 

    

    5.348×10^-2×3.64×10^4  = 1.45768982×10^2

                   1.336  

As the factor 3.64 x 10^4 the least accurate in the above calculation has three significant figures, the answer should be written to three significant figures only. The other figures are insignificant and should be deleted. 

While deleting the figures, the last significant figure to be retained is rounded off for which the following rules are followed.

a) If the first digit dropped is less than 5, the last digit retained should remain unchanged.

b) If the first digit dropped is more than 5, the digit to be retained is increased by one.

c) If the digit to be dropped is 5, the previous digit which is to be retained, is increased by one if it is odd and retained as such if it is even. For example, the following numbers are rounded off to three significant figures as follows. The digits are deleted one by one.

43.75 is rounded off as 43.8

56.8546 is rounded off as 56.8

73.650 is rounded-off as 73.6

64.350 is rounded off as 64.4

               3. In adding or subtracting numbers, the number of decimal places retained in the answer should equal the smallest number of decimal places in any of the quantities being added or subtracted. 

In this case, the number of significant figures is not important. It is the position of decimal that matters. For example, suppose we wish to add the following quantities expressed in metres. 

                i) 72.1                      ii) 2.7543

               +   3.42                    +        4.10

               +  0.003                 +       1.2737

                  5.523                           8.1273

Correct answer:     75.5 m        8.13 m

In case (i) the number 72.1 has the smallest number of decimal places, thus the answer is rounded off to the same position which is then 75.5 m. In case (ii) the number 4.10 has the smallestnumberof decimal places and hence the answer is rounded off to the same decimal positions which is then 8.13m.