As stated earlier physics is based on measurements. But unfortunately whenever a physical quantity is measured, there is inevitably some uncertainty about its determined value. This uncertainty may be due to a number of reasons. One reason is the type of instrument, being used. We know that every measuring instrument is calibrated to a certain smallest division and this fact put a limit to the degree of accuracy which may be achieved while measuring with it.
Suppose that we want to measure the length of a straight line with the help of a meter rod calibrated in millimeters. Let the end point of the line lies between 10.3 and 10.4 cm marks. By convention, if the end of the line does not touch or cross the midpoint of the smallest division, the reading is confined to the previous division. In case the end of the line seems to be touching or have crossed the midpoint, the reading is extended to the next division.
By applying the above rule the position of the edge of a line recorded as 12.7 cm with the help of a metre rod calibrated in millimeters may lie between 12.65 cm and 12.75 cm. thus in this example the maximum uncertainty is ± 0.05 cm. it is , in fact, equivalent to an uncertainty of 0.1 cm equal to the least count of the instrument divided into two parts, half above and half below the recorded reading.
The uncertainty can be indicated conveniently by using significant figures. The recorded value of the length of the straight line i.e. 12.7 cm contains three digits (1, 2, 7) out of which two digits (a and 2) are accurately known while the third digit i.e. 7 is a doubtful one. As a rule:
In any measurement, the accurately known digits and the first doubtful digit are called significant figures.
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In other words, a significant figure is the one which is known to be reasonably reliable. If the above mentioned which is exact up to a hundredth of a centimeter, it would have been recorded as 12.70 cm. in this case, the number of significant figures is four, thus, we can say that as we improve the quality of our measuring instrument and techniques, we extend the measured result to more and more significant figures and we must know he following rules in deciding how many significant figures are to be retained in the final result.
(i) All digits 1,2,3,4,5,6,7,8,9, are significant. However, zeros may or may not be significant. In case of zeros, the following rules may be adopted.
a) A zero between two significant figures is itself significant.
b) Zeros to the left of significant figure may or may not be significant. In decimal fraction, zeros to the right of a significant figure are not significant. For example, none of the zeros in 0.00467 or 02.59 is significant.
c) Zeros to the right of a significant figure may or 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 
. if the least count of the scale is 10 kg, then the number of significant figures will be 3 written in scientific notation as 
kg and so on.
d) 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 
kg has three significant figures.
(ii)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
Do you Know?
Mass can be thought of as a form of energy. In fact the mass is highly concentrated form of energy Einstein’s famous equation,
means
Energy = mass x speed of 
according to this equation 1 kg mass is actually 
energy.
As the factor 
, 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 of as 56.8
73.650 is rounded off as 73.6
64.350 is rounded off as 64.4
Following this rule, the correct answer of the computation given is section (ii) is 
.
(iii) 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 meters
i) 72.1
ii) 2.7543
3.42 4.10
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 75.5m. in case (ii), the number 4.10 has the smallest number of decimal places and hence, the answer is rounded off to the same decimal positions which is then 8.13m.