Significant Figures Bluefield High School 1 What is a significant digit? Significant digits is a set of internationally accepted rules for measurement. The rule of thumb is to record all digits that are certain, plus one that is estimated. The rules of significant digits apply to all reporting and calculating unless you are dealing with defined values (those by definition. For example 60 minutes = 1 hour). 1. Every nonzero digit is significant has three significant digits 2. Zeroes between nonzero digits are significant. 2.07m and 109m each have three significant digits. 3. Zeroes in front of all nonzero digits are not significant. They are merely place holders. The measurements of m and 0.46m each have two significant digits. 4. Zeroes at the end of the number and to the right of the decimal point are significant. The measurement 43.00m, 1.010m and 9.500m all have four significant digits. 5. Zeroes at the end of a measurement and to the left of the decimal point are unclear. You may consider all such zeroes to be not significant. To avoid confusion, express the number in scientific notation. The Rules of Significant Digits When to use Significant figures When a measurement is recorded only those digits that are dependable are written down. When to use Significant digits If you measured the width of a paper with your ruler you might record 21.7cm. To a mathematician 21.70, or is the same. But, to a scientist 21.7cm and 21.70cm is NOT the same cm to a scientist means the measurement is accurate to within one thousandth of a cm. significant figures video How many sig digits? x ,000, How many sig digits here? ,083,000, How many sig digits here? ,000,050, Counting Significant Figures Number of Significant Figures cm4 5.6 ft lb___ m___ All non-zero digits in a measured number are (significant or not significant). 10 Leading Zeros Number of Significant Figures mm oz lb____ mL ____ Leading zeros in decimal numbers are (significant or not significant). 11 Sandwiched Zeros Number of Significant Figures 50.8 mm min lb____ m ____ Zeros between nonzero numbers are (significant or not significant). 12 Trailing Zeros Number of Significant Figures 25,000 in yr1 48,600 gal3 25,005,000 g ____ Are trailing zeros, serving as place holders in numbers without decimals, significant or not significant? 13 Learning Check A. Which answer contains 3 significant figures? 1) ) ) 4760 B. All the zeros are significant in 1) ) ) x 10 3 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x Solution A. Which answers contain 3 significant figures? 2) ) 4760 B. All the zeros are significant in 2) ) x 10 3 C. 534,675 rounded to 3 significant figures is 2) 535,000 3) 5.35 x Learning Check In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and ) and 40 3) and 150,000 16 Solution 3) and 150,000 17 Learning Check State the number of significant figures in each of the following: A m B L C g D m E. 2,080,000 bees Solution A m2 B L4 C g1 D m 3 E. 2,080,000 bees3 19 Significant Numbers in Calculations A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing 20 Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places one decimal place two decimal places answer 26.5 one decimal place 21 Learning Check In each calculation, round the answer to the correct number of significant figures. A = 1) ) ) 257 B = 1) ) ) Solution A = 2) B = 3) Multiplying and Dividing Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures. 24 Learning Check A X 4.2 = 1) 9 2) 9.2 3) B 0.07 = 1) ) 62 3) 60 C X = X ) 11.32) 11 3) Solution A x 4.2 = 2) 9.2 B 0.07 = 3) 60 C x = 2) x Continuous calculator operation = 2.54 x Significant Figures 1.8 Multiplication or Division The number of significant figures in the result is set by the original number that has the smallest number of significant figures 4.51 x = = sig figsround to 3 sig figs 6.8 = sig figsround to 2 sig figs = 0.061 Accuracy how close a measurement is to the true value Precision how close a set of measurements are to each other accurate & precise but not accurate & not precise 1.8 Scientific Notation A short-hand way of writing large numbers without writing all of the zeros. The Distance From the Sun to the Earth 93,000,000 Step 1 Move decimal left Leave only one number in front of decimal Step 2 Write number without zeros Step 3 Count how many places you moved decimal Make that your power of ten The power of ten is 7 because the decimal moved 7 places. 93,000, Standard Form 9.3 x Scientific Notation Practice Problem 1) 98,500,000 = 9.85 x 10 ? 2) 64,100,000,000 = 6.41 x 10 ? 3) 279,000,000 = 2.79 x 10 ? 4) 4,200,000 = 4.2 x 10 ? Write in scientific notation. Decide the power of ten x x x x 10 6 More Practice Problems 1) 734,000,000 = ______ x ) 870,000,000,000 = ______x ) 90,000,000,000 = _____ x On these, decide where the decimal will be moved. 1)7.34 x ) 8.7 x ) 9 x 10 10 Complete Practice Problems 1) 50,000 2) 7,200,000 3) 802,000,000,000 Write in scientific notation. 1) 5 x ) 7.2 x ) 8.02 x 10 11 Scientific Notation to Standard Form Move the decimal to the right 3.4 x 10 5 in scientific notation 340,000 in standard form move the decimal Write in Standard Form 6.27 x x 10 4 tutorial 6,270,000 90,100 1.9 Dimensional Analysis Method of Solving Problems 1.Determine which unit conversion factor(s) are needed 2.Carry units through calculation 3.If all units cancel except for the desired unit(s), then the problem was solved correctly. 1 L = 1000 mL How many mL are in 1.63 L? 1L 1000 mL 1.63 L x = 1630 mL 1L 1000 mL 1.63 L x = L2L2 mL The speed of sound in air is about 343 m/s. What is this speed in miles per hour? 1 mi = 1609 m1 min = 60 s1 hour = 60 min 343 m s x 1 mi 1609 m 60 s 1 min x 60 min 1 hour x = 767 mi hour meters to miles seconds to hours 1.9 Please do the following questions Page 349 in your text #s 2,3,4,5,6,8 and 9 43