Homework Answers p.148 #6 and #7 Express as decimal: 6a) 4.83 x 10 2 = 483 b) 7.221 x 10-4 = 0.0007221 c) 6.1x 10 0 = 6.1 Put in standard scien?fic nota?on: 7a) 142.3 x 10 3 = 1.423 x 10 5 b) 0.0007741 x 10-9 = 7.741 x 10-13 c) 22.7 x 10 3 = 2.27 x 10 4 Scientific notation Complete the chart below Decimal notation Scientific notation 127 1.27 x 10 2 0.0907 9.07 x 10 2 0.000506 5.06 x 10 4 2 300 000 000 000 2.3 x 10 12 Reminder: bring a calculator to class Significant Digits How big? How small? How accurate? What time is it? Someone might say 1:30 or 1:28 or 1:27:55 Each is appropriate for a different situation In science we describe a value as having a certain number of significant digits The # of significant digits in a value includes all digits that are certain and one that is uncertain 1:30 likely has 2, 1:28 has 3, 1:27:55 has 5 Here is a one sentence rule for coun/ng sig figs: All digits ARE significant except 1. Zeros preceding a decimal frac/on Example:.0025m contains 2 sig figs.0008g contains 1 sig fig 2. Zeros at the end of a number containing NO decimal point Example: 5000 contains 1 sig fig So. 45.50 has 4 signifcant figures while 45.5000 has 6 sig figs and.0005 has only 1 sig fig Look at the difference adding a 1 makes!.0045 has 2 significant figures but 1.0045 has 5 significant figures Numbers with no decimal are ambiguous... Does 5000 ml mean exactly 5000? Maybe... Maybe Not! So 5000, 500, 50, and 5 are all assumed to have 1 significant figure If a writer means exactly 5000, he/she must write 5000. or 5.000 x 10 3 1
A Significant Figures Rules Summary To Determine the Significant Figures in a Number: Non-zero digits are ALWAYS significant Ex: 1234.56 à 6 sig figs Counted numbers or conversion factors are considered to have infinite significance and will NEVER be considered when determining the number of digits to report in an answer. ZEROS have to be examined carefully. ZEROs may be: A Significant Figures Rules Summary Leading (placeholders) and are NEVER significant Ex: 0.00031 à 2 sig figs Captive (between nonzero numbers) and are ALWAYS significant Ex: 1013 à 4 sig figs Trailing (at the end of a number) IF there is a decimal and a non zero number before the zero à ALWAYS significant 131.400 à 6 sig figs IF no decimal à NEVER significant 131,400 à 4 sig figs Now you try: 1. How many significant figures in 4.50? 1. How many significant figures in 4.50? More Prac?ce: 2. How many significant figures in 0.005? 2. How many significant figures in 0.005? And More Prac?ce: 3. How many significant figures in 1.005? 3. How many significant figures in 1.005? Now Try A 2
Answers to question A) 1. 3 2.83 2. 4 36.77 3. 3 14.0 4. 2 0.0033 5. 1 0.02 6. 4 0.2410 7. 4 2.350 x 10 2 8. 6 1.00009 9. infinite 3 10. 5 0.0056040 Significant Digits It is better to represent 100 as 1.00 x 10 2 Alternatively you can underline the position of the last significant digit. E.g. 100. This is especially useful when doing a long calculation or for recording experimental results Don t round your answer until the last step in a calculation. Adding with Significant Digits Note accuracy of measurements (nearest.1?.01?.001?) Answer can be no more accurate than the LEAST accurate number that was used to calculate it (line up the decimals). E.g. a) 13.64 + 0.075 + 67 b) 267.8 9.36 13.64 + 0.075 + 67. 80.715 81 267.8 9.36 258.44 Math and Significant Figures Rounding The measured values are used to determine the number of digits in your final answer. If the number cannot be rounded to the correct sig figs in expanded form, use scientific notation. Ex: 400 to 2 sig fig à 4.0 x 10 2 Addition and subtraction The number with the least significant place value determines the sig figs for the answer. Sometimes your answer will have more counted sig figs than one of the numbers used in the calculation. Ex: 200 + 4.02 = 200 (200 is sig to 100s place) Ex: 200.02 + 4.021 = 204.04 (sig is limited to 1/100s place, giving 5 sig figs for final answer even though 4.021 has only 4 sig figs) Adding Prac?ce: to: 4.50 + 0.5? to: 4.50 + 0.5? (5.0) Subtrac?ng Prac?ce: to 1.0-0.005? to 1.0-0.005? (1.0) Now Try B 3
i) 83.25 0.1075 83.14 B) Answers ii) 4.02 + 0.001 4.02 iii) 0.2983 + 1.52 1.82 Multiplication and Division Determining sig. digits for questions involving multiplication and division is slightly different You must COUNT significant figures The answer can have only AS MANY significant figures as the LEAST of the numbers used to get it E.g. a) 608.3 x 3.45 b) 4.8 392 a) 3.45 has 3 sig. digits, so the answer will as well 608.3 x 3.45 = 2.10 x 10 3 b) 4.8 has 2 sig. digits, so the answer will as well 4.8 392 = 0.012 or 1.2 x 10 2 Math and Significant Figures Multiplication and Division Count the sig figs in all numbers; report answer to least number of sig figs counted. No answer will ever have more sig figs than a number used in the calculation. Ex: 200 x 4.02 = 800 (1 sig fig in 200, 1 sig fig in 800) Ex: 200.02 x 4.021 = 804.3 (4 sig fig in 4.021, 4 sig fig in 804.3) Mul?plying Prac?ce: to: 4.50 x 0.5? to: 4.50 x 0.5? (2.25 rounds to 2) a. 2 b. 3 c. 4 Dividing Prac?ce: to 1.00 / 0.0050? to 1.00 / 0.0050? (2.0 x 10 2 ) Here s a tougher one... 3.00 m/s x 60 s/min x 60 min/hr = Note: standard conversion factors (such as 60 s/min) never limit significant figures- - instruments and equipment do. Did you get it? 3.00 m/s x 60 s/min x 60 min/hr = 10800 m/hr The measurement of 3.00 m/s has 3 sigfigs. The other numbers are conversion factors and are considered to be infinitely significant (they never limit your answer). 4
Unit conversions Sometimes it is more convenient to express a value in different units. When units change, basically the number of significant digits does not. E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km Notice that these all have 3 significant digits This should make sense mathematically since you are multiplying or dividing by a term that has an infinite number of significant digits E.g. 123 cm x 10 mm / cm = 1230 mm Try question E on the handout E) Answers i) 1.0 cm = 0.010 m ii) 0.0390 kg = 39.0 g iii) 1.7 m = 1.7 x 10 3 mm THAT S ALL THERE IS TO IT! Use least accurate measurement (line up the decimals) when adding and subtrac/ng Count sig figs when mul/plying and dividing Conversion factors and counted numbers have infinite significance and will not limit your answer. 5