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UNITS AND MEASUREMENT 5 decimal place. The final result should, therefore, or 2.0000 as required. Similarly, in T t , n is be rounded off to 663.8 g. n an exact number. Similarly, the difference in length can be expressed as : 1.3.1 Rules for Arithmetic Operations with –3 0.307 m – 0.304 m = 0.003 m = 3 ×10 m. Significant Figures The result of a calculation involving approximate Notethatweshouldnotusethe rule(1)applicable for multiplication and division and write 664 g as measured values of quantities (i.e. values with the result in the example of addition and limited number of significant figures) must 3.00×10 m in the example ofsubtraction. They reflect the uncertainties in the original measured values. It cannot be more accurate do not convey the precision of measurement properly. For addition and subtraction, the rule than the original measured values themselves is in terms of decimal places. on which the result is based. In general, the final result should not have more significant 1.3.2 Rounding off the Uncertain Digits figures than the original data from which it was The result of computation with approximate obtained. Thus, if mass of an object is measured numbers, which contain more than one to be, say, 4.237 g (four significant figures) and uncertain digit, should be rounded off. Therules its volume is measured to be 2.51 cm , then its for rounding off numbers to the appropriate density, by mere arithmetic division, is 1.68804780876 g/cm upto 11 decimal places. significant figures are obvious in most cases. A number 2.746 rounded off to three significant It would be clearly absurd and irrelevant to figures is 2.75, while the number 1.743 would record the calculated value of density to such a be 1.74. The rule by convention is that the precision when the measurements on which the value is based, have much less precision. The preceding digit is raised by 1 if the insignificant digit to be dropped (the following rules for arithmetic operations with underlined digit in this case) is more than significant figures ensure that the final result of a calculation is shown with the precision that 5, and is left unchanged if the latter is less is consistent with the precision of the input than 5. But what if the number is 2.74 5 in which the insignificant digit is 5. Here, the measured values : convention is that if the preceding digit is (1) In multiplication or division, the final result should retain as many significant even, the insignificant digit is simply figures as are there in the original number dropped and, if it is odd, the preceding digit is raised by 1. Then, the number 2.745 rounded with the least significant figures. Thus, in the example above, density should off to three significant figures becomes 2.74. be reported to three significant figures. On the other hand, the number 2.73 5 rounded off to three significant figures becomes 2.74 4.237g -3 Density  3  1.69 g cm since the preceding digit is odd. 2.51 cm In any involved or complex multi-step Similarly, if the speed of light is given as calculation, you should retain, in intermediate 3.00 × 10 m s -1(three significant figure) and steps, one digit more than the significant digits 7 one year (1y = 365.25 d) has 3.1557 × 10 s (five and round off to proper significant figures at the significant figures), the light year is 9.47 × 10 mend of the calculation. Similarly, a number (three significant figures). known to be within many significant figures, 8 (2) In addition or subtraction, the final result such as in 2.99792458 10 m/s for the speed should retain as many decimal places as are of light in vacuum, is rounded off to an approximate value 3 10 m/s , which is often there in the number with the least decimal places. employed in computations. Finally, remember For example, the sum of the numbers that exact numbers that appear in formulae like 436.32g,227.2 g and 0.301 g by mere arithmetic L addition, is 663.821 g. But the least precise 2 in T  2 , have a large (infinite) number measurement (227.2 g) is correct to only one g Reprint 2026-27