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chapter 5 Measurements & Calculations

Learn about scientific notation and significant figures in measurements, including mathematical operations using scientific notation and the rules for determining significant figures. Practice solving problems using scientific notation and significant figures.

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chapter 5 Measurements & Calculations

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  1. chapter 5Measurements & Calculations

  2. measurements/observations that involve numbers are called… quantitative

  3. 5.1 scientific notation • Some numbers are just too big or too small to deal with reasonably • Scientific Notationis a method for making very large or very small numbers more compact and easier to write. • as in: 300,000,000 can be written as 3 x 108 • it’s easy! :)

  4. Description:scientific notation must be written as the product of a number between 1 and 9.99999 and the appropriate power of 10 • just count how many times you have to move the decimal point to get a number between 1 and (less than)10 • if the number is gettingsmaller the exponent willcompensate by gettingbigger and vice versa

  5. examples • 238,000 2.38 x 105 • 1,500,000 1.5 x 106 • 0.00043 4.3 x 10-4 • 0.135 1.35 x 10-1 • 357 3.57 x 102

  6. Mathematical Operations Using Scientific Notation

  7. Mathematical Operations Using Scientific Notation Addition and Subtraction: All numbers are converted to the same power of 10, and the digit terms are added or subtracted. Example: (4.215 x 10-2) + (3.2 x 10-4) = (4.215 x 10-2) + (0.032 x 10-2) = 4.247 x 10-2

  8. Mathematical Operations Using Scientific Notation Multiplication: The digit terms are multiplied in the normal way and the exponents are added. Example: (3.4 x 106)(4.2 x 103) = (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 1010

  9. Mathematical Operations Using Scientific Notation Division: The digit terms are divided in the normal way and the exponents are subtracted. Example: (6.4 x 106)/(8.9 x 102) = (6.4)/(8.9) x 10(6-2) = 0.719 x 104 = 7.2 x 103

  10. Practice 1. Write the following in Scientific Notation • 942600000 b) 0.0000000990 2. Write the following number in normal notation a) 5.38 x 103 b) 3.7 x 10-6 3. Solve: (6.0 x 102) (2.0 x 103) 4. Solve: (1.0 x 102) / (4.0 x 103) 5. Solve: (6.3 x 102) + (2.3 x 103) 6. Solve: (4.3 x 104) - (5.1 x 103)

  11. Practice (solutions) 1. Write the following in Scientific Notation a) 9.426 x 108 b) 9.90 x 10-8 2. Write the following number in normal notation. a. 5380 b. 0.0000037 3. Solve: (6.0 x 102) (2.0 x 103) = 1.2 x 106 4. Solve: (1.0 x 102) / (4.0 x 103) = 2.5 x 10-2 5. Solve: (6.3 x 102) + (2.3 x 103) = 2.93 x 103 6. Solve: (4.3 x 104) - (5.1 x 103) = 3.79 x 104

  12. 5.4 uncertainty in measurement • Write down the volume of the liquid in the graduated cylinder

  13. 31.7 31.8 31.8 31.6 31.7 31.7 31.8 31.6 • every measuring device has some degree of uncertainty • the certain numbers+the one uncertain one are called significant…

  14. Consider the uncertainty in trying to measure the length of an object. • which instrument gives us more significant digits?

  15. 5.5 significant figures • The quality of your measurements must be represented in the numbers that you record

  16. Rules: • All non-zero digitsare significant. (e.g. 123456789) 2. Leading Zeros: zeros which start a number are never significant. (e.g. 0.0000123456789)   3. Captured Zeros: zeros which are between non-zero digits are significant. (e.g. 101) 4. Trailing Zeros: zeros at the end of a number may be significant, if a decimal point is present. (e.g. 1.23400)

  17. one more thing… • Exact numbersnumbers obtained by counting ado not limit the number of significant digits

  18. Significant examples the mass of an eyelash is 0.000304 g 3 the length of the line was 1.270 x 102 m 4 A 125-g sample of chocolate chip cookie contains 10 g of chocolate 3, 1 the volume of soda remaining in a can after a spill is 0.09020 L 4 a dose of antibiotic is 4.0 x 10-1 cm3 2

  19. Practice 5.682 100 101 100. 0.00582 1.0582 0.005820 62.4050 1.6 x 10-4 4 1 3 3 3 5 4 6 2

  20. determining sigfigs in calculations • there are only two basic rules here, one to do with multiplication and division, the other addition and subtraction

  21. rounding off

  22. When multiplying or dividing numbers, count the number of significant figures. • The answer cannot contain more significant figures than the number being multiplied or divided with the least number of significant figures. • Example: 23.123123 x 1.3344 = 30.855495 • The answer with the proper number of significant figures is… 30.855

  23. When adding or subtracting numbers, count the number of decimal places to determine the number of significant figures. • The answer cannot contain more places after the decimal point than the smallest number of decimal places in the numbers being added or subtracted. • Example: 23.112233m 1.3324m + 0.25m 24.694633m The answer with the proper number of significant figures is… 24.69m

  24. Questions?

  25. Units • go into a restaurant, sit down, just tell the waitress “two,” and see what you get • Units are used to give meaning to numbers, they identify what and how something was measured

  26. 5.2 units • Which system of measurement do you prefer to use? Why? • Turn to your neighbor and discuss…

  27. The English Standard System

  28. The Metric System

  29. the English system is used in the US; the metric system is used everywhere else • scientists everywhere use a standardized metric called the International System (SI)

  30. prefixes are used to make the metric system more efficient • 1 nm is easier to use & write than 0.000000001m

  31. 5.6 problem solving and dimensional analysis

  32. 1) write down what you know (given),2) where you’re going, then3) build a bridge between them… • Change 100 mm into m. bridge where you’re going given 100mm x 1 m 0.1 = m  1000 mm

  33. Change 45mm into km. 1 km 45mm 1 m = 1000 m 1000 mm 4.5x10-5 km

  34. If my resting heart rate is 60 beats per minute, how many times will it beat in the next year?

  35. Questions?

  36. 5.3 measurements of length, volume, and mass • length is based on the meter

  37. volumeis how much 3D space something takes up • What is the formula for calculating volume? v = l x w x h for example: v = 1m x 1m x 1m = 1m3 v = 2m x 3m x 5m = 30m3 Linear to liquid volume conversion 1cm x 1cm x 1cm = 1cm3 = 1mL

  38. Often volume is measured with a graduated cylinder

  39. when you use the Graduated cylinder to read the centerof the meniscus

  40. mass is often measured in grams and is measured with a balance

  41. 5.7 temperature conversions:an approach to problem solving • here we learn both the different temp scales and how to convert between them

  42. the Big Three Temp Scales are Fahrenheit, Celsius, and Kelvin • in science we use almost exclusively C and K

  43. converting between K and C • a degree Celsius and a Kelvin are the same size; they just differ by their starting points • they only differ by 273.15 • thus, and simply • TC + 273.15 = TK

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