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Chapter Two

Chapter Two. Standards for Measurement Scientific Notation Measurement and Uncertainty Significant Figures Significant Figures in Calculations Metric System Problem-Solving Measuring Mass and Volume Measurement of Temperature Density. Observations in Science.

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Chapter Two

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  1. Chapter Two • Standards for Measurement • Scientific Notation • Measurement and Uncertainty • Significant Figures • Significant Figures in Calculations • Metric System • Problem-Solving • Measuring Mass and Volume • Measurement of Temperature • Density

  2. Observations in Science • Qualitative Observation – a less specific observation describing the quality of something. • Quantitative Observations – a measurement of a specific quantity of something. • Question: What are some qualitative and quantitative observations of our classroom?

  3. Scientific Notation • Science often must use very large and very small numbers. • The average distance from the Sun to the Earth is 150,000,000 kilometers. • The diameter of an atom is about 0.0000000001m. • Writing these out is not always easy.

  4. Scientific Notation • Rules for converting a large or small number into scientific notation: • Move the decimal point in the original number until it is located after the first non-zero digit. • Multiply this by the number ten raised to an exponent (called the “power”) equal to the number of decimal places that was required in step one. • The sign on the exponent is positive if you moved the decimal point to the left and negative if you moved it to the right.

  5. Learning Check • Write each of the numbers into scientific notation. • Ex) 150,000,000 km • Ex) 0.0000000001 m

  6. Scientific Notation • To convert a number from scientific notation back to a decimal number: • Move the decimal point back to the right for a positive exponent – will need to add zero’s at the end of the number. • Move the decimal point back to the left for a negative exponent – will need to add zero’s before the number.

  7. Learning Check • Write the following scientific notation numbers as a decimal equivalent. • Ex) 4.5 x 105 • Ex) 6.3 x 10-6

  8. Scientific Notation • On your calculator – use either the EE or EXP key for entering scientific notation. • Ex) 4.5 x 105 is entered as: 4 5 EE 5  Each colored box represents a the key on the calculator to push.

  9. Learning Check • Multiply or dividing two (or more) scientific notation numbers. • Ex) (4.5 x 105) * (3.6 x 107) = • Ex) (8.1 x 1012)  (4.9 x 10-5) =

  10. Types of Numbers • Numbers can be either Exact or Measured. • An Exact number is one that is either a counted number or a number that is part of an established relationship. • Ex) 21 chairs in a room • Ex) 12 inches in a foot • Ex) 100 centimeters in a meter • A Measured number is one that is obtained using any measuring instrument like a ruler, graduated cylinder, or thermometer.

  11. Learning Check • Describe as either an Exact or Measured number. • Ex) 12 apples in a bag • Ex) 25.0mL of water in a 25-mL graduated cylinder • Ex) 16 ounces in a pound • Ex) The mass of a U.S. quarter is 5.67 grams • Ex) 1000 grams in one kilogram • Ex) The indoor temperature is 74.8oF

  12. Uncertainty in Measurements • All measured numbers are made up of certain and one uncertain digit. • Certain digits – all will agree. That is: we all agree that is at least 45 but not 46 degrees Fahrenheit. • Uncertain digit – not all will agree! It is each person’s best “guess” between two markings. • Can only have ONE uncertain digit!

  13. Uncertainty in Measurements • Each marking is 0.2mL. • We would (hopefully) agree that it is between 6.6 and 6.8mL. • Must provide a best “guess” to nearest 0.05mL.

  14. Uncertainty in Measurements • What if it seems to be “exactly” on a mark???

  15. Uncertainty in Measurements • Many of the measuring devices used today are electronic like a scale or a digital thermometer. • The last digit is always the uncertain digit and should always be included even when it is a zero.

  16. Significant Figures • All of the certain digits plus the one uncertain digit are called significant digits. • Thus, our volume from earlier – 6.65mL would have three significant figures. • Our temperature, 36.60oC, would have four significant figures. • And our scale reading, 45.22g, would also have four significant figures. • Note: Exact numbers have an infinite significance because they are exact.

  17. Rules for Significant Figures • Rules for counting significant figures are: • All non-zero digits are significant. • Zeros are significant when: • Between non-zero digits like in 205. • At the end of a number WITH a decimal point like in 0.250 or 12.00 or 850. • Zeros are not significant when: • Before the first non-zero digit like in 0.0025. • After non-zero digits without a decimal point at the end like 95,000 or 45,000,000

  18. Learning Check • How many significant digits does each measured number have? • 0.000305 • 125.0 • 1.0 • 0.023012 • 0.0001 • 100 • 45.050

  19. Rounding • A calculation involving measured numbers will almost always need to be rounded to a certain place. • Rules: • When the first digit to be rounded after those that will be retained is a 4 or less, then drop all of those extra digits. • When the first digit to be rounded is 5+, the drop all of those digits and adjust the last digit retained up by one.

  20. Rounding • If it is EXACTLY 5, then round the final retained digit to an even number. • Ex) 45.62 rounded to two digits is ______ • Ex) 89.145 rounded to three digits is ______ • Ex) 9.15 rounded to two digits is _____ • Ex) 84.5 rounded to two digits is _____

  21. Significant Figures & Calculations • Rules for Multiplication and Division • The answer must be rounded to the same number of significant figures as the number with the least amount of significant figures. • Reason? The result of a calculation cannot be more precise than the least precise measurement. • Ex) 45.6 x 0.023 = 10.488 (calculator answer) • Ex) 8.02  45.23 = 0.1773159… (calculator answer)

  22. Learning Check • Round each to the correct number of significant figures. Answers are “calculator” answers. • 89.00 x 3.25 = 289.25 • 6.516  482 = 0.01351867… • 1.25 x 80. = 100 • 35.7 x 89.52  0.028 = 114,138

  23. Significant Figures & Calculations • Rules for Addition and Subtraction • The answer cannot contain any more decimal places as the value with the fewest decimal places. • Can get more or less number of significant figures than either of the two numbers you are adding or subtracting. • Example: Let’s say you had a briefcase containing roughly $1.1 million dollars. You open the briefcase and pull out a $5 bill. How much is in the briefcase?

  24. Learning Check • Round each to the correct number of significant figures. Answers are “calculator” answers. • 45.2 + 3.5 = 48.7 • 73.21 – 71.58 = 1.63 • 151 + 3.28 = 154.28 • 173.18 – 167.98 = 5.2

  25. Mixed Calculations • For a mixed calculation, you must follow each of two rules individually and follow order of operations. • Ex) (45.23 – 3.15)  4825 = 0.87212435… • You must do the subtraction first! • Ex) (11.35 + 2.55) x 0.12 = 1.668 • You must do the addition first! • Ex) (23.33 – 22.93) x 4.58 = 1.832

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