2.7 Significant figures - Units and Measurements - Class 11 Physics

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2.7 SIGNIFICANT FIGURES

NCERT Class 11 Physics text book made accessible for blind and visually impaired students.

As discussed above, every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures. If we say the period of oscillation of a simple pendulum is 1.62 s, the digits 1 and 6 are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has three significant figures. The length of an object reported after measurement to be 287.5 cm has four significant figures, the digits 2, 8, 7 are certain while the digit 5 is uncertain. Clearly, reporting the result of measurement that includes more digits than the significant digits is superfluous and also misleading since it would give a wrong idea about the precision of measurement.

The rules for determining the number of significant figures can be understood from the following examples. Significant figures indicate, as already mentioned, the precision of measurement which depends on the least count of the measuring instrument. A choice of change of different units does not change the number of significant digits or figures in a measurement. This important remark makes most of the following observations clear: (1) For example, the length 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 mm or 23080 µm.

All these numbers have the same number of significant figures (digits 2, 3, 0, 8), namely four. This shows that the location of decimal point is of no consequence in determining the number of significant figures.


The example gives the following rules :


• All the non-zero digits are significant.


• All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.


• If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. [In 0.00 2308, the underlined zeroes are not significant].


• The terminal or trailing zero(s) in a number without a decimal point are not significant.

[Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.] However, you can also see the next observation.

• The trailing zero(s) in a number with a decimal point are significant. [The numbers 3.500 or 0.06900 have four significant figures each.]

(2) There can be some confusion regarding the trailing zero(s). Suppose a length is reported to be 4.700 m. It is evident that the zeroes here are meant to convey the precision of measurement and are, therefore, significant. [If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4.7 m]. Now suppose we change units, then


4.700 m = 470.0 cm = 4700 mm = 0.004700 km

Since the last number has trailing zero(s) in a number with no decimal, we would conclude erroneously from observation (1) above that the number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures.

(3) To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10). In this notation, every number is expressed as a × 10^b, where a is a number between 1 and 10, and b is any positive or negative exponent (or power) of 10. In order to get an approximate idea of the number, we may round off the number a to 1 (for a ≤ 5) and to 10 (for 5

It is often customary to write the decimal after the first digit. Now the confusion mentioned in (a) above disappears :


4.700 m = 4.700 × 10^2 cm


= 4.700 × 10^3 mm = 4.700 × 10^−3 km

The power of 10 is irrelevant to the determination of significant figures. However, all zeroes appearing in the base number in the scientific notation are significant. Each number in this case has four significant figures.

Thus, in the scientific notation, no confusion arises about the trailing zero(s) in the base number a. They are always significant.

(4) The scientific notation is ideal for reporting measurement. But if this is not adopted, we use the rules adopted in the preceding example :


• For a number greater than 1, without any decimal, the trailing zero(s) are not significant.


• For a number with a decimal, the trailing zero(s) are significant.

(5) The digit 0 conventionally put on the left of a decimal for a number less than 1 (like 0.1250) is never significant. However, the zeroes at the end of such number are significant in a measurement.

(6) The multiplying or dividing factors which are neither rounded numbers nor numbers representing measured values are exact and have infinite number of significant digits. For example in r = d/2 or s = 2π r, the factor 2 is an 2 exact number and it can be written as 2.0, 2.00 or 2.0000 as required. Similarly, in T = t/n , n is an exact number.

2.7.1 Rules for Arithmetic Operations with Significant Figures

The result of a calculation involving approximate measured values of quantities (i.e. values with limited number of significant figures) must reflect the uncertainties in the original measured values. It cannot be more accurate than the original measured values themselves on which the result is based. In general, the final result should not have more significant figures than the original data from which it was obtained. Thus, if mass of an object is measured to be, say, 4.237 g (four significant figures) and its volume is measured to be 2.51 cm^3, then its density, by mere arithmetic division, is 1.68804780876 g/cm^3 up to 11 decimal places. It would be clearly absurd and irrelevant to record the calculated value of density to such a precision when the measurements on which the value is based, have much less precision. The following rules for arithmetic operations with significant figures ensure that the final result of a calculation is shown with the precision that is consistent with the precision of the input measured values :

(1) In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.


Thus, in the example above, density should be reported to three significant figures.


Density = 4.237g/ 2.51 cm^3 = 1.69 g cm^−3


Similarly, if the speed of light is given as 3.00 × 10^8 m s^−1 (three significant figures) and one year (1y = 365.25 d) has 3.1557 × 10^7 s (five significant figures), the light year is 9.47 × 10^15 m (three significant figures).

(2) In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

For example, the sum of the numbers 436.32 g, 227.2 g and 0.301 g by mere arithmetic addition, is 663.821 g. But the least precise measurement (227.2 g) is correct to only one decimal place. The final result should, therefore, be rounded off to 663.8 g.

Similarly, the difference in length can be expressed as :


0.307 m − 0.304 m = 0.003 m = 3 × 10^−3 m.

Note that we should not use the rule (1) applicable for multiplication and division and write 664 g as the result in the example of addition and 3.00 × 10^−3 m in the example of subtraction. They do not convey the precision of measurement properly. For addition and subtraction, the rule is in terms of decimal places.

2.7.2 Rounding off the Uncertain Digits

The result of computation with approximate numbers, which contain more than one uncertain digit, should be rounded off. The rules for rounding off numbers to the appropriate significant figures are obvious in most cases. A number 2.746 rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74. The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined digit in this case) is more than 5, and is left unchanged if the latter is less than 5. But what if the number is 2.745 in which the insignificant digit is 5. Here, the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. Then, the number 2.745 rounded off to three significant figures becomes 2.74. On the other hand, the number 2.735 rounded off to three significant figures becomes 2.74 since the preceding digit is odd.

In any involved or complex multi-step calculation, you should retain, in intermediate steps, one digit more than the significant digits and round off to proper significant figures at the end of the calculation. Similarly, a number known to be within many significant figures, such as in 2.99792458 × 10^8 m/s for the speed of light in vacuum, is rounded off to an approximate value 3 × 10^8 m/s , which is often employed in computations. Finally, remember that exact numbers that appear in formulae like

2 π in T = 2 π √ (L/g) , have a large (infinite) number of insignificant figures. The value of π = 3.1415926.... is known to a large number of significant figures. You may take the value as 3.142 or 3.14 for π, with limited number of significant figures as required in specific cases.

Example 2.13

Each side of a cube is measured to be 7.203 m. What are the total surface area and the volume of the cube to appropriate significant figures?

Answer:
The number of significant figures in the measured length is 4. The calculated area and the volume should therefore be rounded off to 4 significant figures.


Surface area of the cube = 6(7.203)^2 m^2


= 311.299254 m^2 = 311.3 m^2


Volume of the cube = (7.203)^3 m^3


= 373.714754 m^3


= 373.7 m^3

Example 2.14

5.74 g of a substance occupies 1.2 cm^3. Express its density by keeping the significant figures in view.

Answer:


There are 3 significant figures in the measured mass whereas there are only 2 significant figures in the measured volume. Hence the density should be expressed to only 2 significant figures.


Density = 5.74/1.2 g cm^ −3


= 4.8 g cm^ −3.

2.7.3 Rules for Determining the Uncertainty in the Results of Arithmatic Calculations

The rules for determining the uncertainty or error in the number/measured quantity in arithmetic operations can be understood from the following examples.

(1) If the length and breadth of a thin rectangular sheet are measured, using a metre scale as 16.2 cm and, 10.1 cm respectively, there are three significant figures in each measurement. It means that the length l may be written as


l = 16.2 ± 0.1 cm


= 16.2 cm ± 0.6 %.

Similarly, the breadth b may be written as


b = 10.1 ± 0.1 cm


= 10.1 cm ± 1 %

Then, the error of the product of two (or more) experimental values, using the combination of errors rule, will be


l b = 163.62 cm^2 ± 1.6%


= 163.62 ± 2.6 cm^2


This leads us to quote the final result as


l b = 164 + 3 cm^2


Here 3 cm^2 is the uncertainty or error in the estimation of area of rectangular sheet.

(2) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures.


However, if data are subtracted, the number of significant figures can be reduced.


For example, 12.9 g − 7.06 g, both specified to three significant figures, cannot properly be evaluated as 5.84 g but only as 5.8 g, as uncertainties in subtraction or addition combine in a different fashion (smallest number of decimal places rather than the number of significant figures in any of the number added or subtracted).

(3) The relative error of a value of number specified to significant figures depends not only on n but also on the number itself.

For example, the accuracy in measurement of mass 1.02 g is ± 0.01 g whereas another measurement 9.89 g is also accurate to ± 0.01 g. The relative error in 1.02 g is


= (± 0.01/1.02) × 100 %


= ± 1%


Similarly, the relative error in 9.89 g is


= (± 0.01/9.89) × 100 %


= ± 0.1 %


Finally, remember that intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement. These should be justified by the data and then the arithmetic operations may be carried out; otherwise rounding errors can build up. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (three) is 0.104, but the reciprocal of 0.104 calculated to three significant figures is 9.62. However, if we had written 1/9.58 = 0.1044 and then taken the reciprocal to three significant figures, we would have retrieved the original value of 9.58.

This example justifies the idea to retain one more extra digit (than the number of digits in the least precise measurement) in intermediate steps of the complex multi-step calculations in order to avoid additional errors in the process of rounding off the numbers.