Section 2.2Section 2.2

Scientific Notation and Scientific Notation and Dimensional AnalysisDimensional Analysis

ObjectivesObjectives

• Express numbers in scientific Express numbers in scientific notationnotation

• Use dimensional analysis to convert Use dimensional analysis to convert between units.between units.

Scientific NotationScientific Notation

• It is easier to work with very large or It is easier to work with very large or very small numbers very small numbers whenwhen they are in they are in scientific notation.scientific notation.

• Scientific notationScientific notation expresses a expresses a number as 2 factors: a number number as 2 factors: a number between 1 and 10 times ten raised between 1 and 10 times ten raised to a power or exponent.to a power or exponent.

Converting intoConverting intoScientific NotationScientific Notation

• The decimal must be moved to create a The decimal must be moved to create a number between 1 and 10number between 1 and 10

• The number of places that the decimal is The number of places that the decimal is moved are counted.moved are counted.– If the decimal is moved to the LEFT, the number If the decimal is moved to the LEFT, the number

of places is converted to a POSITIVE exponent.of places is converted to a POSITIVE exponent.– If the decimal is moved to the RIGHT, the number If the decimal is moved to the RIGHT, the number

of places is converted to a NEGATIVE exponent.of places is converted to a NEGATIVE exponent.

• 1, 625 g = 1.625 x 101, 625 g = 1.625 x 1033 g g

• 0.00283 m = 2.83 x 100.00283 m = 2.83 x 10-3-3 m m

Practice ProblemsPractice Problems• Express the following in scientific Express the following in scientific

notationnotation– 0.540 s0.540 s– 540,000 s540,000 s

• Express the following in standard Express the following in standard notationnotation– 9.87 x 109.87 x 10-5-5 g g– 7.2 x 107.2 x 10-1-1 g g

Doing Math inDoing Math inScientific NotationScientific Notation• Adding & SubtractingAdding & Subtracting

– the exponents must be the samethe exponents must be the same– add or subtract the first factor and add or subtract the first factor and

keep the power of ten the samekeep the power of ten the same

• Practice ProblemsPractice Problems– 2.7 x 102.7 x 1077 + 1.7 x 10 + 1.7 x 1088

– 4.5 x 104.5 x 10-12-12 - 3.9 x 10 - 3.9 x 10-13-13

Doing Math inDoing Math inScientific NotationScientific Notation• Multiplying & DividingMultiplying & Dividing

– In multiplication, multiply the first In multiplication, multiply the first factors and add the exponentsfactors and add the exponents

– In division, divide the first factors and In division, divide the first factors and subtract the exponentssubtract the exponents

• Practice ProblemsPractice Problems– 3.5 x 103.5 x 10-3-3 x 9.6 x 10 x 9.6 x 101010

– 4.8 x 104.8 x 10-12-12/5.2 x 10/5.2 x 1088

Dimensional AnalysisDimensional Analysis• Dimensional analysisDimensional analysis is a method for is a method for

problem-solving that focuses on the problem-solving that focuses on the units used to describe matter.units used to describe matter.

• Dimensional analysis uses Dimensional analysis uses conversion factorsconversion factors.. A A conversion conversion factorfactor is a ratio of is a ratio of equivalent equivalent valuesvalues that express the that express the same same quantity in different unitsquantity in different units..

Conversion FactorsConversion Factors• For example, we know:For example, we know:

1 meter = 101 meter = 1022 cm cm

• This expression relates equivalent This expression relates equivalent values in different units.values in different units.

• If we express this equivalency as a If we express this equivalency as a ratio, we will have a conversion factor.ratio, we will have a conversion factor.

• There are 2 possible ratios that can be There are 2 possible ratios that can be written:written:1 meter1 meter or or 101022 cm cm101022 cm 1 meter cm 1 meter

Practice ProblemsPractice ProblemsWrite the possible conversion factors for Write the possible conversion factors for

the following equivalencies.the following equivalencies.• 1 hr = 60 s1 hr = 60 s• 4 tbs = ¼ cup4 tbs = ¼ cup• 1 liter = 101 liter = 1033 milliliters milliliters• 1 km = 101 km = 1033 m m

Dimensional AnalysisDimensional Analysis• Problem: Convert 48 km to m.Problem: Convert 48 km to m.

1.1. Identify the known (given) unit & the Identify the known (given) unit & the unknown unit.unknown unit.

kmkm is known & is known & mm is unknown is unknown

2.2. Find an equivalent relationship Find an equivalent relationship between the units & write the possible between the units & write the possible CFsCFs

1 km = 101 km = 1033 m; the CFs: m; the CFs: 1 km1 km and and 10103 3

mm 101033 m 1 km m 1 km

Dimensional AnalysisDimensional Analysis

• Problem: Convert 48 km to m.Problem: Convert 48 km to m.3.3. Multiply the given quantity & unit by Multiply the given quantity & unit by

the conversion factor that has the conversion factor that has the the given unit in its denominatorgiven unit in its denominator..

48 km x 48 km x 101033 m m = 4.8 x 10 = 4.8 x 1044 m m 1 km1 km

Dimensional AnalysisDimensional Analysis

48 km x 48 km x 101033 m m = 4.8 x 10 = 4.8 x 1044 m m 1 km1 km

NOTE:NOTE:

The given unit will cancel out when The given unit will cancel out when multiplied by the conversion factor, multiplied by the conversion factor, giving you the unit you need.giving you the unit you need.

The conversion factor itself is equal to 1 The conversion factor itself is equal to 1 so the quantity does NOT change, but so the quantity does NOT change, but the units do!the units do!

Practice ProblemsPractice Problems (Express answers in correct scientific (Express answers in correct scientific notation.)notation.)1.1. Convert 245 ms to s.Convert 245 ms to s.

2.2. How many meters are in 5600 dm?How many meters are in 5600 dm?

3.3. Convert 250 kg to g.Convert 250 kg to g.

4.4. How many milliliters are in 34.8 L?How many milliliters are in 34.8 L?

5.5. How many mm’s are in 0.067 m? How many mm’s are in 0.067 m?

Multiple Conversion FactorsMultiple Conversion Factors

• In some cases you may need to do In some cases you may need to do more than one conversion to end up more than one conversion to end up with the proper unit.with the proper unit.

• For example: Convert 0.0915 km to dm.For example: Convert 0.0915 km to dm.1.1. Known is Known is kmkm, unknown is , unknown is dmdm..

2.2. Since I do not know an equivalency Since I do not know an equivalency between km & dm, I recall what I between km & dm, I recall what I dodo know: know:

1 km = 101 km = 1033 m and 1 dm = 10 m and 1 dm = 10-1-1 m m

Multiple Conversion FactorsMultiple Conversion Factors

(cont.) 1 km = 10(cont.) 1 km = 1033 m and 1 dm = 10 m and 1 dm = 10-1-1 m m

The CFs: The CFs: 1 km1 km , , 101033 m m 1 dm1 dm , , 1010-1-1 m m 101033 m 1 km 10 m 1 km 10-1-1 m 1 dm m 1 dm• a. First, multiply the given by the CF a. First, multiply the given by the CF

involving the given unit.involving the given unit.

0.0915 km x 0.0915 km x 101033 m m = 9.15 x 10 = 9.15 x 1011 m m 1 km 1 km

Multiple Conversion FactorsMultiple Conversion Factors3.3. b. Then, multiply the answer to step 3a b. Then, multiply the answer to step 3a

by one of the other CFs. (Pick the CF in by one of the other CFs. (Pick the CF in which the unit cancels.)which the unit cancels.)

9.15 x 109.15 x 1011 m x m x 1 dm1 dm = 9.15 x 10 = 9.15 x 1022 dm dm 1010-1-1 m m

Note: The multiplication Note: The multiplication cancan be done in one be done in one step with 2 CFs.step with 2 CFs.

0.0915 km x 0.0915 km x 101033 m m x x 1 dm1 dm = 9.15 = 9.15 x x 101022 dm dm 1 km 101 km 10-1-1 m m

Practice ProblemsPractice Problems1.1. How many cups is 3.5 tsp? (3 tsp = 1 How many cups is 3.5 tsp? (3 tsp = 1

tbs. & 4 tbs. = ¼ cup)tbs. & 4 tbs. = ¼ cup)2.2. Convert 300 min into days.Convert 300 min into days.3.3. How many yds. is 89 in.? (1 yd = 3 ft. How many yds. is 89 in.? (1 yd = 3 ft.

& 1 ft. = 12 in.)& 1 ft. = 12 in.)4.4. Mary’s foot is 20.5 cm long. What is Mary’s foot is 20.5 cm long. What is

that in feet? (1 m = 3.3 ft.)that in feet? (1 m = 3.3 ft.)5.5. What is the speed of 65 km/min in What is the speed of 65 km/min in

m/min?m/min?

More Practice ProblemsMore Practice Problems1.1. What is 550 m/s in mi/hr? (1 km = What is 550 m/s in mi/hr? (1 km =

0.62 mi)0.62 mi)

2.2. What is 790 mWhat is 790 m33 in km in km33??

3.3. The density of oxygen is 1.43 x 10The density of oxygen is 1.43 x 10-3-3 g/mL. Express this in kg/mg/mL. Express this in kg/m33..

4.4. Convert 1000 ftConvert 1000 ft22 into in into in22 and yd and yd22..

Word ProblemsWord Problems• Word problems are solved in exactly the same Word problems are solved in exactly the same

way as any other dimensional analysis way as any other dimensional analysis problem.problem.

• The difference is this - you are NOT The difference is this - you are NOT directlydirectly given the known and unknown. Rather, you given the known and unknown. Rather, you must pick them out from a descriptive must pick them out from a descriptive paragraph. paragraph.

• Also, you are often given one or more Also, you are often given one or more conversion factors in the problem. You must conversion factors in the problem. You must also be able to pick them out.also be able to pick them out.

Word ProblemsWord Problems John goes to the local hardware store and finds that John goes to the local hardware store and finds that

they carry a large selection of widgets. The widgets he they carry a large selection of widgets. The widgets he needs are 8 for $5.25. How many widgets can he buy needs are 8 for $5.25. How many widgets can he buy for $20.00? for $20.00?

• As with all word problems, you should make a list and As with all word problems, you should make a list and identify the information that is given to you.identify the information that is given to you.

8 widgets = $5.25 This is a conversion factor.8 widgets = $5.25 This is a conversion factor.

$20.00 This is the KNOWN.$20.00 This is the KNOWN.

# of widgets This is the UNKNOWN.# of widgets This is the UNKNOWN.

• Now that everything is identified, you solve the problem Now that everything is identified, you solve the problem as you would any other dimensional analysis problem.as you would any other dimensional analysis problem.