Significant Figures

Significant Figures:

 

In  the  previous  sections,  we  have  studied  various  types  of  errors,  their  origins  and  the  ways to minimize them. Our accuracy is limited to the least count of the instrument used during the  measurement.  Least  count  is  the  smallest  measurement that can be made using the given instrument.  For  example  with  the  usual  metre  scale, one can measure 0.1 cm as the least value. Hence its least count is 0.1cm.

Suppose we measure the length of a metal rod  using  a  metre  scale  of  least  count  0.1cm.  The  measurement  is  done  three  times  and  the  readings are 15.4, 15.4, and 15.5 cm. The most probable length which is the arithmetic mean as per  our  earlier  discussion  is  15.43.  Out  of  this  we are certain about the digits 1 and 5 but are not certain about the last 2 digits because of the least count limitation.

The  number  of  digits  in  a  measurement  about which we are certain, plus one additional digit, the first one about which we are not certain is  known  as  significant  figures  or  significant  digits.

Thus   in   above   example,   we   have   3   significant  digits 1, 5 and 4.

The larger the number of significant figures obtained  in  a  measurement,  the  greater  is  the  accuracy  of  the  measurement.  If  one  uses  the  instrument of smaller least count, the number of significant digits increases.

Rules for determining significant figures

  1)  All   the   nonzero   digits   are   significant,   for  example  if  the  volume  of  an  object  is  178.43 cm³, there are five significant digits which are 1,7,8,4 and 3.

2) All  the  zeros  between  two  nonzero  digits  are  significant,  eg.,  m  =  165.02  g  has  5  significant digits.

3)  If the number is less than 1, the zero/zeroes on  the  right  of  the  decimal  point  and  to  the  left  of  the  first  nonzero  digit  are  not  significant e.g. in 0.001405, the underlined zeros  are  not  significant.  Thus  the  above  number has four significant digits.

 4) The zeros on the right hand side of the last nonzero  number  are  significant  (but  for  this,  the  number  must  be  written  with  a  decimal point), e.g. 1.500 or 0.01500  have both 4 significant figures each.

On  the  contrary,  if  a  measurement  yields  length L given as

L  =  125  m  =  12500  cm  =  125000  mm,  it  has only three significant digits.

To avoid the ambiguities in determining the number of significant figures, it is necessary to report every measurement in scientific notation (i.e., in powers of 10) i.e., by using the concept of order of magnitude.

The magnitude of any physical quantity can be  expressed  as  A×10ⁿ  where  ‘A’  is  a  number  such that 0.5 d<A<5 and ‘n’ is an integer called the order of magnitude.

(i) radius of  Earth  = 6400 km 

                                      = 0.64×10⁷m

 The order of magnitude is 7 and the number of significant figures are 2.

(ii) Magnitude of the charge on electron e= 1.6×10^-19 C

Here the order of magnitude is -19 and the number of significant digits are 2.