SIGNIFICANT FIGURES Every experient (except in soe counting situations) involves a degree of uncertainty. Thus, suppose that several people easure the length of a sheet of paper, using a ruler that is divided into tenths of a centieter, and get the following results: 7.9 c, 7.96 c, 7.90 c,.0 c. Notice that everyone agrees as to the first three digits (except the last person who is easuring fro the wrong end of a 30 c long ruler!). Clearly the fourth digit, which has been estiated by everyone, is a doubtful figure (in fact even the third figure ay be doubtful if the ruler is too short or too long due to anufacturing defect or environental factors).. DEFINITION: The digits that are considered correct and the first doubtful digit are called significant figures. The nuber of significant figures in a easureent depends upon the precision of the instruent and, to soe extent, upon the skill of the easurer. An effort should always be ade to obtain as any figures as an instruent will allow. Conversely, only significant figures should be recorded in taking data i.e., a easureent should not be written in such a way as to iply a greater precision than is actually inherent in the easuring device and/or easuring technique. Exaple : We ight use a variety of instruents to deterine the ass of a typical calculator, and obtain the following results: 0. kg ( significant figure) on a bathroo scale 0.85 kg (3 significant figures) on a good superarket scale 0.8365 kg (6 significant figures) on an analytical balance. In the above exaple, we could express the results in gras instead of kilogras clearly, the precision of the result would be unaffected by a siple change of units, so that we would still end up with the sae nuber of significant figures. Exaple : 0.8365 kg = 83.65 g = 8365 g = 0.0008365 ton All nubers have the sae nuber of significant figures. This iplies that the position of the decial point has no effect on the nuber of significant figures. Zeros which are used erely to locate a decial point are not significant figures but only spacers. On the other hand if a zero represents a value actually obtained fro an instruent, then it is a significant figure. Exaple 3: In the nuber 0.008040, the last two zeros are significant figures but the first three are not. Zeros ust be used even with whole nubers if the easureent is to convey a certain degree of precision.
Exaple 4: A easureent recorded as 64 g iplies that the balance used was one of relatively low precision and that the value lies (typically) between 63 g and 65 g. If, on the other hand, the ass has been deterined accurate to the nearest 0.0 g, then the value should be recorded as 64.00 g. This correctly iplies a precision of four significant figures, rather than two. Finally, it is iportant to recognize that soe nubers used in coputations are either exact (by definition), or have as any significant figures as we ay require. Exaple 5: (a) (b) (c) (d) for a circle, diaeter/ radius=, exactly (by definition) ft= in, exactly (by definition) conductivity resistivity =, exactly (by definition) π= 3.45965358978338466 (known to over,000,000 figures, if needed). STANDARD FORM (Scientific Notation): Scientific notation akes use of powers of ten to adjust the position of the decial point. It has two advantages: i) Convenience and copactness. Exaple 6: The ass of an electron is 9. 0-3 kg. Copare this value with 0.000000000000000000000000000000000009 kg. Try entering the latter into a calculator! ii) Eliination of abiguity regarding precision Exaple 7: Suppose we see the stateent: The average distance between Earth and Sun is 50,000,000 k. Are the zeros significant digits or are soe of the used only as spacers? By writing the value as.50 0 8 k, we are clearly iplying a precision of three significant figures, if such is actually warranted. 3. CACUATION WITH MEASURED QUANTITIES: 3. Rounding Off In soe calculations the final result will contain figures that are not significant; such a result ust be rounded off, leaving only the significant figures, in order to avoid iplying a higher degree of precision in the calculated result than justified by the data.
Exaple 8: Result of a calculation as read fro a calculator display * Fuel econoy of 0.5368 l/k Engine displaceent of Engine power output of 96.5 kw Nuber of significant figures Result correctly rounded off 3 0.54 l/k Coents A real gas-guzzler! 3.8 0 3 c 3 One reason why it is 384 c 3 a gas-guzzler! 97 kw or 96 kw =30 hp for that gas-guzzler. * The bar placed over a digit indicates the last significant figure (i.e., the first doubtful figure) 3. Addition and Subtraction Arrange the quantities to be added or subtracted in colun for and carry out the required arithetical operation. Then scan the coluns fro left to right RUE A: THE AST SIGNIFICANT FIGURE IN THE RESUT IS THE FIGURE UNDER THE AST COUMN WHICH CONTAINS ONY SIGNIFICANT FIGURES. Exaple 9: Add the following easured lengths: 800 36.5 9.3 945.8 To obtain the 8 in the su, an unknown figure was unjustifiably assued to be zero (in the first line). Thus, the result ust be rounded off to 946. 3.3 Multiplication and Division RUE B: THE NUMBER OF SIGNIFICANT FIGURES IN THE RESUT IS THE SAME AS THE NUMBER OF SIGNIFICANT FIGURES IN THE EAST PRECISE QUANTITY INVOVED IN THE CACUATIONS. Exaple 0: Calculate the volue of a cylinder whose height and diaeter were found to be 0.3 c and 4.0 c, respectively
V 4 4 = πd H = π (4.0 c) (0.3 c) = 68.76 c = 68 c 3 3 The fourth figure in 4.0 is unknown. The value of this unknown figure would affect the values of the fourth figure in the result but would not significantly change the third figure, naely 8. What about the significant figures in /4 and π? (See Exaple 5 and the coent that precedes it). Exaple : Calculate the power required to raise a 95 kg elevator 35 in 37.5 seconds at a constant speed, neglecting friction. g y (95 kg)(9.806 /s )(35 ) P = = = 309 J/s t 37.5 s =.3 kw =.3 kw The least precise quantity (35 ) has only two significant figures, which liits the precision of the result to two significant figures. 3.4 Cobined Operations Follow the usual hierarchy of operations. Whenever you carry out addition or subtraction, apply rule A. Whenever you carry out ultiplication or division, apply Rule B. Exaple : When the teperature of a solid bar is raised fro T to T, the length of the bar changes fro to according to = ( ) = α T T where α is the coefficient of linear expansion of the aterial in question (average value for this teperature range). Find using the following easured values: α =5 0-6 / C o, =58.5 c, T =4 o C, T = 99.7 o C. (Note: If T is the actual boiling point of water, it can be deterined ore precisely than T, using the known relationship between teperature and the vapor pressure of water. = + α ( T = 58.5 + 5 0 = 58.5 + 5 0 T ) = 58.5 + 0.06643 6 6 58.5 (75.7) = 58.5664 c = 58.6 c 58.5 (99.7 4) First operation (apply rule A) Second and third operations (apply rule B) Fourth operation (apply rule A)
3.5 General Notes In calculations with easured quantities it is best to work through the interediate calculations with one figure ore than the final result will contain in order to avoid the accuulated round-off error. The final result should be rounded off to the correct nuber of significant figures. The procedures for treating significant figures outlined herein do not cover all situations. Unique cases will have to be treated on their own erit. Exercises on Significant Figures (Pre-lab questions for ab #). Write the following quantities in standard for, assuing three significant figures in each: a) 5360 W b) 0,000 V c) 9500.5 A d) 345.6 e) 0.000477 kg f) 865.5 c In all the following exercises, only the significant figures have been kept in all the easured quantities.. Add the following easured lengths: 45.7 c; 03 c;.5688 0 ;.7 0 - k;.0000 0 3 3. A certain rod easures.0000 in length at 0.0 o C and.0008 at +5.0 o C. Use a forula fro Exaple to deterine which of the following gives the correct values of α. a..857 0-5 /C o b..9 0-5 /C o c..3 0-5 /C o d. 0-5 /C o 4. The tie taken by sunlight to reach the Earth is given by: 7 distance between Earth and Sun 5.0 0 k t = = 5 speed of light in vacuu (.9979 0 k/s)(60 s/in) A calculator gives 8.3397 (verify this). What is the correct answer (including units)? 5. Calculate the weight of 37 cars if each weighs.83 0 4 N. 6. Re-calculate the answer in Exaple, using a. y=36 b. y = 34
7. Calculate the surface area of a cylindrical tank whose radius and height were found to be.30 and 0.445, respectively. Using the forula A = ( π R ) + (πr) H the calculator yields 0.9478 +3.690743 = 4.638565. What is the correct answer? 8. For a cart rolling down an inclined plane, the forula s = v 0 t + at gives the displaceent s as a function of tie t. Given the following easureents: v 0 (initial velocity) = 0. /s down the plane, s=.35, t=.906 s, find the acceleration a.