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6 University of South Carolina Stephen L Morgan Tutorial on the Use of Significant Figures All measurements are approximations--no measuring device can give perfect measurements without experimental uncertainty. By convention, a mass measured to 13.2 g is said to have an absolute uncertainty of 0.1 g and is said to have been measured to the nearest 0.1 g. In other words, we are somewhat uncertain about that last digit it could be a "2"; then again, it could be a "1" or a "3". A mass of g indicates an absolute uncertainty of 0.01 g. The objectives of this tutorial are: Explain the concept of significant figures. Define rules for deciding the number of significant figures in a measured quantity. Explain the concept of an exact number. Define rules for determining the number of significant figures in a number calculated as a result of a mathematical operation. Explain rules for rounding numbers. Present guidelines for using a calculator. Provide some exercises to test your skill at significant figures. What is a "significant figure"? The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number is said to have 4 significant figures Rules for deciding the number of significant figures in a measured quantity: (1) All nonzero digits are significant: g has 4 significant figures, 1.2 g has 2 significant figures. (2) Zeroes between nonzero digits are significant:

7 1002 kg has 4 significant figures, 3.07 ml has 3 significant figures. (3) Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: C has only 1 significant figure, g has 2 significant figures. (4) Trailing zeroes that are also to the right of a decimal point in a number are significant: ml has 3 significant figures, 0.20 g has 2 significant figures. (5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as: 5.06 x 10 4 calories (3 significant figures) x 10 4 calories (4 significant figures), or x 10 4 calories (5 significant figures). What is a "exact number"? Some numbers are exact because they are known with complete certainty. Most exact numbers are integers: exactly 12 inches are in a foot, there might be exactly 23 students in a class. Exact numbers are often found as conversion factors or as counts of objects. Exact numbers can be considered to have an infinite number of significant figures. Thus, the number of apparent significant figures in any exact number can be ignored as a limiting factor in determining the number of significant figures in the result of a calculation. Rules for mathematical operations In carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate measurement involved in the calculation.

8 (1) In addition and subtraction, the result is rounded off to the last common digit occurring furthest to the right in all components. Another way to state this rules, it that,in addition and subtraction, the result is rounded off so that it has the same number of decimal places as the measurement having the fewest decimal places. For example, 100 (assume 3 significant figures) (5 significant figures) = , which should be rounded to 124 (3 significant figures). (2) In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example, 3.0 (2 significant figures ) x (4 significant figures) = which should be rounded off to 38 (2 significant figures). General guidelines for using calculators When using a calculator, if you work the entirety of a long calculation without writing down any intermediate results, you may not be able to tell if an error is made. Further, even if you realize that one has occurred, you may not be able to tell where the error is. In a long calculation involving mixed operations, carry as many digits as possible through the entire set of calculations and then round the final result appropriately. For example, (5.00 / 1.235) (6.35 / 4.0)= = The first division should result in 3 significant figures. The last division should result in 2 significant figures. The three numbers added together should result in a number that is rounded off to the last common significant digit occurring furthest to the right; in this case, the final result should be rounded with 1 digit after the decimal. Thus, the correct rounded final result should be 8.6. This final result has been limited by the accuracy in the last division. Warninq: carrying all digits through to the final result before rounding is critical for many mathematical operations in statistics. Rounding intermediate results when calculating sums of squares can seriously compromise the accuracy of the result. Sample problems on significant figures Instructions: print a copy of this page and work the problems. When you are ready to check your answers, go to the next page.

9 = = = = x 2.5 = / = x 273 = 8. (5.5) 3 = x ( ) = x 3.00 = x x 10 2 = (Give the exact numerical result, then express it the correct number of significant figures). 12. What is the average of , , , , and ? Answer key to sample problems on significant figures = = = = 129 (assuming that 125 has 3 significant figures) x 2.5 = / = x 273 = (5.5) 3 = 1.7 x x ( ) = x 3.00 = 1.4 x 10 2 (assuming that 45 has two significant figures)

10 CHAPTER 01. RECITATION PROBLEMS Unit Conversion P13, For about 10 years after the French Revolution, the French government attempted to base measures of time on multiples of ten: One week consisted of 10 days, one day consisted of 10 hours, one hour consisted of 100 minutes, and one minute consisted of 100 seconds. What are the ratios of (u) the French decimal week to the standard week and (b) the French decimal second to the standard second? Express the duration of the average heart beat in seconds and in "French" seconds **If light takes about 8.4 min from sun to Earth and the speed of light is 3.0e+8m/s what is the Earth-Sun separation distance (1.5 x10^8 km) Dimensional Analysis 15. The average radius of a nucleus is R = 10.0 fin. Find the density of the nucleus which has a mass of 15u. (1 fin = in. I n = " 27 kg) (Ans: 5.94, kg/m3) 16. A drop of oil (mass = 0.90 milligram, and density = 918 kg/m 3) spreads out on a surface and forms a circular thin film of radius = 41.8 cm and thickness h. Find h in nano meter (nun). (Ans: 1.8 mu) 17. How many molecules of water are there in a cup containing 250 cm 3 of water? Molecular mass of water = 18 g'mole: density of water = 1.0 g/cm 3 : Avogadro's number = 6.02N molecules/mole. (Ans: ) A18. The speed of sound v in a fluid depends upon the fluid density p and its bulk modulus B as follows: v = tf.bm. Using dimensional analysis, find the values of constants n and in, respectively. The unit of density p is kg/m 3 and that of bulk modulus B is kgt(m.s 2). (Ans: -1/2, +1/2) A19. Consider the following physical relation: M = C.rb. where M is mass, p is density, r is distance, and a and b are exponents. C is a dimensionless constant. What are the values of a and b so that the equation is dimensionally correct? (Ans: a = 1 and b = 3) A20. The force F applied on a particle is given by the relation F = K p A B2, where K is a dimensionless constant, p is a density and A is an area. Find the dimension of B. (Ans: LIT) 021. The average density of blood is 1.06x 10 3 kg/m 3. If you donate one pint of blood, what is the mass of the blood you have donated, in grains? [1 pint = 1/2 Liter, 1 Liter = 1000 cm 3] (Ans: 530) AI. Which formula could be correct for the speed v of ocean waves in terms of the density p of sea water. the acceleration of free fall g. the depth h in the ocean. and the wave length 2? (Note: Unit for wave length X is meter (m) and unit for density p is kg/m 3) A) v,ga 13) v = 1J g C) v pgh D) v = F) Significant Digits: Express in scientific notation = 2.02 x 2.5 = x ( ) =