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Introduction to Scientific Notation in Expressing Numbers

This chapter covers the use of scientific notation to express extremely large or small numbers. It also introduces the concept of units and prefixes in the metric system.

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Introduction to Scientific Notation in Expressing Numbers

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  1. Chapter 5

  2. ObjectiveThe student will be able to: express numbers in scientific and decimal notation. SOL: A.10, A.11 Designed by Dr. Shah, Blue Valley High School

  3. How wide is our universe? 210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

  4. Scientific Notation A number is expressed in scientific notation when it is in the form a x 10n where a is between 1 and 10 and n is an integer

  5. Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

  6. 2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 1023

  7. 1) Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10-8

  8. Write 28750.9 in scientific notation. • 2.87509 x 10-5 • 2.87509 x 10-4 • 2.87509 x 104 • 2.87509 x 105

  9. 2) Express 1.8 x 10-4 in decimal notation. 0.00018 3) Express 4.58 x 106 in decimal notation. 4,580,000 On the graphing calculator, scientific notation is done with the button. 4.58 x 106 is typed 4.58 6

  10. 4) Use a calculator to evaluate: 4.5 x 10-5 1.6 x 10-2 Type 4.5 -5 1.6 -2 You must include parentheses if you don’t use those buttons!! (4.5 x 10 -5) (1.6 x 10 -2) 0.0028125 Write in scientific notation. 2.8125 x 10-3

  11. 5) Use a calculator to evaluate: 7.2 x 10-9 1.2 x 102On the calculator, the answer is: 6.E -11 The answer in scientific notation is 6 x 10 -11 The answer in decimal notation is 0.00000000006

  12. 6) Use a calculator to evaluate (0.0042)(330,000).On the calculator, the answer is 1386. The answer in decimal notation is 1386 The answer in scientific notation is 1.386 x 103

  13. 7) Use a calculator to evaluate (3,600,000,000)(23).On the calculator, the answer is: 8.28 E +10 The answer in scientific notation is 8.28 x 10 10 The answer in decimal notation is 82,800,000,000

  14. Write (2.8 x 103)(5.1 x 10-7) in scientific notation. • 14.28 x 10-4 • 1.428 x 10-3 • 14.28 x 1010 • 1.428 x 1011

  15. Write in PROPER scientific notation.(Notice the number is not between 1 and 10) 8) 234.6 x 109 2.346 x 1011 9) 0.0642 x 104 on calculator: 642 6.42 x 10 2

  16. Scientific notation • We use scientific notation to describe very large and very small numbers • The format for scientific notation is X x 10y • X is a number between 1-10 • Y is a negative number if number is smaller than 1 and positive if number is larger than 1. • Y signifies how many places we will move the decimal

  17. Write 531.42 x 105 in scientific notation. • .53142 x 102 • 5.3142 x 103 • 53.142 x 104 • 531.42 x 105 • 53.142 x 106 • 5.3142 x 107 • .53142 x 108

  18. Scientific notation • We use scientific notation to describe very large and very small numbers • The format for scientific notation is X x 10y • X is a number between 1-10 • Y is a negative number if number is smaller than 1 and positive if number is larger than 1. • Y signifies how many places we will move the decimal

  19. Units • Units – part of the measurement that tells us what scale or standard is being used to represent the results of the measurement • Different units of measure • English System • Metric system • International system (SI) [based on metric]

  20. More units • SI units – you will be required to know these!!!! • Mass = kg (kilogram) • Length = m (meter) • Time = s (second) • Temp = K (kelvin)

  21. Prefixes and metric system You are expected to Memorize the information from the table above 1 Megameter = 1 x 106 meters 1 kilometer = 1 x 103 meters

  22. How to use prefixes • milli- means 0.001 or 10-3 • Put a 1 in front of the unit. • Put an equal sign after the unit. • rewrite the unit • So, millimeter becomes: • 1 millimeter = millimeter • Lastly, on the unit to the right of the equal sign remove the prefix and substitute with value that prefix represents.

  23. How to use prefixes • 1 millimeter= millimeter • remove the prefix on the unit to the right of the equal sign thus: • 1 millimeter = ________meter • Now substitute the value of the prefix in: • 1 millimeter = __0.001__meters or • 1 millimeter = ____10-3__meters

  24. expectations • I expect you to memorize or know the following equality: • 1 in. = 2.54 cm • Length = measure of distance • Volume = measure of 3-d space • Mass = quantity of matter

  25. Uncertainty in measurement • When taking a measurement, there is always certain digits and estimated digits. • The last digit of ALL measurements are estimated. • This block is between 3 and 4 cm. I could record it as 3.6 cm. • Here the 6 would be estimated 1 2 3 4

  26. Significant figures • The concept of significant figures helps us in calculations with estimations. • First we must learn the rules for S.F. • 1. non-zero #s are always significant • 2. zeros • 1. leading zeros are NEVER significant (0.000036 = 2 S.F.) • 2. captive zeros are ALWAYS significant (0.003034 = 4 SF) • 3. Trailing zeros are significant if there is a decimal (0.003034500 = 7 SF) • 3. Exact numbers are counted as infinite SF • 3 eggs = infinite SF

  27. Sig Fig trick • Draw US. • Atlantic ocean = decimal absent • Pacific ocean = decimal present

  28. Arithmetic with S.F. • Addition/subtraction: least # places after the decimal • Multiplication/division: least # of s.f. • Rounding

  29. s.f. add/subtract • Rule: use the least number of places • Example: 7.5527 + 6.273 + 2.8 = 16.6257 (in calculator)= 16.6 (Final answer in correct # of s.f.) • Example 51.3-12 = 39.3 (in calculator) = 39 (Final answer in correct # of s.f.)

  30. s.f. mult/div • Rule= use least # of s.f. for answer • Example: 63.35 x 63.7 x 6040 = 24373785.8 (in calculator) = 24400000 (final answer in correct s.f.) • Example: 30 / 2.0 = 15 (in calculator) = 20 (final answer in correct # of s.f.)

  31. Chapter 5 (cont.) Problem solving and dimensional analysis

  32. objective and definitions • objective: Learn how dimensional analysis can be used to solve various problems • conversion factor: ratio that relates 2 units ex. 2.54 cm/ 1 in. • equivalence statement: numerical relationship between 2 units ex. 2.54 cm = 1 inch • There are 2 conversion factors for each equivalence statement 1 in./2.54 cm and 2.54 cm/ 1 in. • dimensional analysis – using conversion factors to change from 1 unit to another • units – part of measurement tells us which scale/standard is being used

  33. Converting from one unit to another • To carry out dimensional analysis, you must multiply by the conversion factor that puts the units you want to cancel on bottom and units you desire on top. • Example: An Italian bicycle has its frame size given in 62 cm. What is the frame size in inches? • Start the problem with what is given.

  34. Example problem • 62 cm x 1in.=24 inches 2.54 cm • Be sure to include proper # of sig figs in answer.

  35. Multistep Dimensional Analysis • Most problems will require multiple conversion factors. Ex. The length of a marathon is 26.2 miles How many km is that? 1 mile = 1760 yards 1 meter = 1.094 yards

  36. multistep problem (cont.) • 26.2 miles 1 km 1 meter 1760 yards x = x x 1000 m 1.094 yd 1 mile • On calculator you type 26.2 x 1760 / 1.094 / 1000 = • You should get 42.149908592321755027422303473492 km

  37. Dimensional analysis tips • 1. Always include units. (remember a measurement always contains 2 parts: a number AND a unit) • 2. Cancel units as you carry out your calculations • 3. Check that your final answer has correct units • 4. Check that your final answer has correct sig figs • 5. Think about whether your answer makes sense.

  38. Everyday examples • Gas! Your gas tank holds 16 gallons. Gas is $3.11/ gal. at the nearest gas station. How much money are you going to have to collect from your couch to fill up your gas tank? $3.11 16 gal. $49.8 = x gal.

  39. Everyday examples • If $1.00 is equal to 0.7373 euros. What is the value in $ of a 100 euro bill(Answer in 5 s.f.)? • Apples cost $2.00 per lb. There are 8 apples in a lb. of apples. How many apples can you buy for $20? • Text messages on Verizon are $.05/ txt. You typically text 120/ month. Verizon comes out with a plan for unlimited texts for 1 month that costs $5/month. Is this a good deal for you(Explain)?

  40. Unit systems • Metric: mass = kg • English = ?? • Metric: distance = meter • English = ?? • Metric: Volume = L, cm3 • English: Volume = ?? • Metric: time = seconds (s) • English: time=?? • Metric: temperature= ?? • English: temperature = ??

  41. Chapter 5 Density

  42. objective and definitions • Objective: To define density and its units • Density: amt. of matter in a given volume • D= mass/volume • Mass in units of g • volume in units of mL or cm3 (L for gases) • 1 mL = 1 cm3

  43. Density • Density is the amount of matter present in a given volume of substance. It is the ratio of the mass of an object to its volume. Density = mass volume M This triangle can help you! Block the unit you need to solve for and the rest of the triangle will tell you how to get your answer. V D

  44. Common units • Common density units include: g/mL, g/cm3, g/L (gases only). • C.c. stands for cubic centimeter and is often used in the medical field. 1 cc=1mL • 1 cm3 = 1mL

  45. Example problem • You find that 23.50 mL of a certain liquid weighs 35.062g. What is the density? 1.492 g/mL

  46. Water displacement • One way to determine an object’s volume is a technique called water displacement. • In this technique an object is submerged into a given volume of water. The change in volume is the object’s volume.

  47. example • A cube of metal weighs 1.45 kg and displaces 542 mL of water. What is the density of the metal? 0.00268 kg/mL

  48. Density in the real world • A material will float on the surface of a liquid if the material has a density less than that liquid. • Which has a higher density water or ice? • Fish don’t die in the winter!!!!! • Which has a higher density Pb or water?

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