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Understanding Scientific Notation and Dimensional Analysis in Data Analysis

This chapter focuses on analyzing data through scientific notation and dimensional analysis. Scientific notation simplifies the representation of very large or small numbers using powers of ten, explaining the rules for coefficients and exponents. It also covers the significance of uncertainty in measurements and the importance of significant figures. The chapter outlines how to properly write numbers in both scientific and regular notation, emphasizing the role of conversion factors in dimensional analysis, essential for converting units in scientific data.

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Understanding Scientific Notation and Dimensional Analysis in Data Analysis

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  1. Chapter 2 Analyzing Data

  2. Scientific Notation & Dimensional Analysis • Scientific notation – way to write very big or very small numbers using powers of 10 3 x 108 Superscript Coefficient

  3. Superscript Rules • Numbers greater than 10 = • Ex. 257000000000000 • Numbers less than 10 = • Ex. 0.0000000000000257

  4. Rules for Scientific Notation • The coefficient must be between 1.0 and 9.99. • Your coefficient must contain all significant digits. • Move the decimal point as many places as necessary until you create a coefficient between 1.0 and 9.99. • The exponent will be the number of places you move your decimal point. • Moving the decimal to the left makes the number larger = POSITIVE EXPONENT • Numbers greater than 10 always have exponents that are positive. • Moving the decimal to the right makes the number smaller = NEGATIVE EXPONENT • Numbers less than 1.0 always have exponents that are negative

  5. Write the following in scientific notation • 1,392,000 km • 0.0000000028 • 1176.9 • 0.0123

  6. Write the following in regular notation • 3.6 x 105 • 5.4 x 10-5 • 5.060 x 103 • 8.9 x 10-7

  7. Uncertainty in Measurement

  8. Uncertainty in Measurement • A measurement always has some degree of uncertainty.

  9. Different people estimate differently. • Record all certain numbers and one estimated number.

  10. Significant Figures • Numbers recorded in a measurement. • All the certain numbers plus first estimated number

  11. Rules for Significant Digits • Every nonzero digit is significant Ex. 24.7 m • Zeros appearing between nonzero digits are significant Ex. 24.07 m

  12. 3. Zeros after significant digits are only significant if there is a decimal point Ex. 2470 Ex. 2470.0

  13. 4. Zeros in front of numbers are NOT significant, even after a decimal point Ex. 0.0000247 Ex. 0.247 5. When a number is in scientific notation, all numbers in the coefficient are significant Ex. 2.470 x 103

  14. Significant Digits in Calculations • An answer cannot be more precise than the least precise measurement from which it was calculated. • To round off an answer you must first decide how many significant digits the answer should have. • Your calculator DOES NOT keep track of significant digits, you have to do it!

  15. Addition & Subtraction • Answer can have no more decimal places than the number in the problem with the fewest decimal places. • Ex: 4.5 + 6.007 + 13.39 = 23.897 • Correct sig figs = 23.9

  16. Multiplication & Division • Answer can have no more significant digits than the number in the problem with the fewest significant digits • Ex: 3.24 x 7.689 x 12.0 = 298.94832 • Correct Sig. Figs = 299

  17. Units and Measurement • Systeme Internationale d’Unites (SI Units) – standard units of measure used by all scientists. • Why?

  18. Base Units and SI Prefixes • Base unit – measurements that can be taken with one instrument • Time • Length • Mass • Temperature • Amount

  19. Prefixes are added to base units to indicate very large or very small quantities.

  20. Second – determined by the frequency of radiation given of by cesium – 133 • Meter – distance light travels in a vacuum in 1/299,792,458 of a second • Kilogram – defined by a platinum and iridium cylinder kept in France

  21. Temperature – quantitative measurement of the average kinetic energy of the particles that make up an object

  22. Temperature Scales • Fahrenheit • Water freezes at • Water boils at • oF = 1.8(oC) + 32 • Celsius • Water freezes at • Water boils at • oC = (oF – 32)/1.8

  23. Which is warmer, 25 oF or 25 oC? • What is 98 oF in oC? • What is 20 oC in oF?

  24. Kelvin • Water freezes at 273 • Water boils at 373 • Theoretically molecule movement completely stops at 0 K (absolute zero) • K = C + 273

  25. What is 25 oC in K? • What is 300 K in oC? • What is 35 oF in K?

  26. Derived Units • Derived unit – unit that is made by combining two or more base units • m/s • g/mL • cm3

  27. Volume – space an object takes up • L x w x h • SI unit – m3 • More useful unit = L • 1 L = 1 dm3 • 1 mL = 1 cm3

  28. Volumes of irregular objects can be found by placing them into a graduated cylinder and measuring the amount of water that is displaced • What is the volume of the dinosaur?

  29. Density = amount of mass per unit volume • g/cm3 • g/mL • kg/L • Always the same for a given substance • D = M/V

  30. What is the density of a cube that has a mass of 20 g and a volume of 5 cm3?

  31. When a piece of aluminum is placed in a 25 mL graduated cylinder that contains 10.5 mL of water, the water level rises to 13.5 mL. The density of aluminum is 2.7 g/mL. What is the mass of the piece of aluminum?

  32. What is the volume of an object with a mass of 13.5 g and a density of 1.4 g/mL?

  33. Dimensional Analysis • A systematic approach to problem solving that uses conversion factors to move from one unit to another • Conversion factor is a ratio of equivalent values with different units • 1 km = 1000 m • 12 inches = 1 foot

  34. Tools for Problem Solving • Be systematic • Ask yourself these questions • Where do we want to go? • What do we know? • How do we get there? • Does it make sense?

  35. Tools for Problem Solving • Tools for Converting from One Unit to Another Step 1 Find an equivalence statement that relates the 2 units.Step 2 Choose the conversion factor by looking at the direction of the required change (cancel the unwanted units).Step 3 Multiply the original quantity by the conversion factor. Step 4 Make sure you have the correct number of significant figures.

  36. 1 step conversions Ex 1: A roll of wire is 15m long, what is the length in cm? Ex 2: convert 8.96L to milliliters

  37. Convert 100 yards to feet • Convert 5 kilometers to miles

  38. Multi step conversions • Convert 525 km to cm • Convert 10000 in to miles

  39. Convert 3,000,000 s to years

  40. Conversions with derived units • Convert 365 mm3 to m3 • Convert 15.9 cm3/s to L/h

  41. Convert 25 miles/hour to ft/second • Convert 1.004 g/cm3 to kg/mL

  42. Uncertainty in Data • All measurements contain uncertainties

  43. Accuracy vs. precision • Accuracy is how close a single measurement comes to the actual dimension or true value of what is measured • 4.555555 vs. 4.56 • More decimal places make a measurement more accurate. • Depends on quality of measuring device

  44. Precision is how close several measurements are to the same value • Depends on more than one measurement • Depends on the skill of the person making the measurement

  45. Precise Accurate

  46. Error and percent error • Experimental value – value measured during experiment • Accepted value – true or known value • Error = experimental value – accepted value

  47. Percent error: • You calculate the density of sucrose to be 1.40 g/mL. The accepted value for the density of sucrose is 1.59 g/mL. What is your % error?

  48. 2.4 Representing Data • Graphs are a visual representation of data which make it easier to see patterns and trends

  49. Circle graphs • Aka – • Show parts of a fixed whole

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