Unit 2:SCIENTIFIC MEASUREMENT

OBJECTIVES (Don’t Copy!)Convert Between Standard Notation to Scientific Notation Identify Significant Figures & Uncertainty in MeasurementsPerform Operations with Significant Figures

Addition & Subtraction Multiplication & Division

What is Scientific What is Scientific Notation?Notation? a way of expressing really big a way of expressing really big

numbers or really small numbers.numbers or really small numbers.

For some numbers, scientific For some numbers, scientific notation is more concise.notation is more concise.

Scientific notation consists of Scientific notation consists of two parts:two parts: A number between 1 and 10 (the A number between 1 and 10 (the

“coefficient”)“coefficient”)

7.017.01 x 10x 1088

A power of 10A power of 10

Ex:Ex:

Converting from StandardScientific NotationEX: Convert 289,800,000 to scientific notation.EX: Convert 289,800,000 to scientific notation.

_______ x 10_______ x 10______

STEP 1: STEP 1: Moving the decimal, convert this number so that it falls between 1 and 10.Moving the decimal, convert this number so that it falls between 1 and 10.STEP 2: STEP 2: Count the number of places you moved the decimal. This is the exponent. Count the number of places you moved the decimal. This is the exponent. If the number was large to start with, exponent is positive If the number was large to start with, exponent is positive If the number was small to start with, exponent is negative If the number was small to start with, exponent is negative

2.8982.898 8

ExamplesExamples

Given: 0.000567Given: 0.000567

___________ x 10 ___5.675.67 -4-4

STEP 1STEP 1 STEP 2STEP 2

NOTE: This is a very small number, so the exponent is SMALL (negative)

Scientific Notation & Your Calculator

Video Instructions- Using Calculator

Plugging Scientific Notation into My Calculator

Find the button on your calculator that is used to enter SCIENTIFIC NOTATION. OR

Note: If you find one of the symbols ABOVE a key, rather than ON a key, you must push

OR

2nd EE

2nd EXP

EE

EXP

Plugging Scientific Notation Into My Calculator Ex: Plug this number into your

calculator: 8.93 x 1o-13

Write the steps you used below1. Type 8.932. Push ______ button.3. Type_____13_____4. Push ____OR____ button5. What I see on the screen is _________

NOTE: Sometimes these 2 steps can be reversed

EE

+/- (-)

8.93 -13

NOTE: Use your calculator’s keys if they differ from what is written here

Scientific Notation:Adding & Subtracting Ex: 3 x 104 + 2.5 x 105

USE CALCULATOR:

Problem

3 x 104

+ 2.5 x 105

What you type

3 EE 4

+ 2.5 EE 5

What you see on calculator

304 2.505

NOTE: Answers must be in scientific notation!Calculator says: 280000Correct Answer: 2.8 x 105

NOTE: Use your calculator’s keys if they differ from what is written in table

Practice

9.1 x 10-3 + 4.3 x 10-2

ANSWER: 5.21 x 10-2

Calculator says: 0.0521

Scientific Notation:Multiplying&Dividing Ex: (6.1 x 10-3) (7.2 x 109)

USE CALCULATOR:

Problem

6.1 x 10-3

x 7.2 x 109

What you type

6.1 EE 3-

x 7.2 EE 9

What you see on calculator

6.1-03 7.209

NOTE: Use your calculator’s keys if they differ from what is written in table

NOTE: Answers must be in scientific notation!Calculator says: 43920000Correct Answer: 4.392 x 107

Stating a MeasurementStating a Measurement

In every measurement there is aIn every measurement there is a

Number Number followed by a followed by a

Unit Unit from a measuring devicefrom a measuring device

The number should also be as precise as The number should also be as precise as the measuring device.the measuring device.

Ex: Reading a MeterstickEx: Reading a Meterstick

. l. l22. . . . I . . . . I. . . . I . . . . I33 . . . .I . . . . I . . . .I . . . . I44. . cm. . cm

First digit (known)First digit (known) = 2 = 2 2.?? cm2.?? cmSecond digit (known)Second digit (known) = 0.7 = 0.7 2.7? cm2.7? cmThird digit (estimated) between 0.05- 0.07Third digit (estimated) between 0.05- 0.07Length reportedLength reported == 2.75 cm 2.75 cm

oror 2.74 cm 2.74 cm oror 2.76 cm2.76 cm

Significant FiguresSignificant Figures

The numbers reported in a The numbers reported in a measurement are limited by the measurement are limited by the measuring toolmeasuring tool

Significant figures in a Significant figures in a measurement include the known measurement include the known digits digits plus one estimated digitplus one estimated digit

Shortcuts to Sig FigsThe Atlantic-Pacific Rule says:

"If a decimal point is Present, ignore zeros on the Pacific (left) side.

If the decimal point is Absent, ignore zeros on the Atlantic (right) side.

Everything else is significant."

Counting Significant Figures:Counting Significant Figures:Unlimited Sig Figs Unlimited Sig Figs

2 instances in which there are an unlimited # of sig 2 instances in which there are an unlimited # of sig figs.figs.

a)a)CountingCounting. Ex: 23 people in our classroom. . Ex: 23 people in our classroom. b)b)Exactly defined quantities.Exactly defined quantities. Ex: 1hr = 60 Ex: 1hr = 60

min.min.

Both are exact values. There is no uncertainty. Neither of these types of values affect the Neither of these types of values affect the

process of rounding an answerprocess of rounding an answer..

Learning CheckLearning Check

A. Which answers contain 3 significant A. Which answers contain 3 significant figures?figures?1) 0.47601) 0.4760 2) 0.00476 2) 0.00476 3) 4760 3) 4760

B. All the zeros are significant inB. All the zeros are significant in 1) 0.00307 1) 0.00307 2) 25.300 2) 25.300 3) 2.050 x 3) 2.050 x 101033

C. 534,675 rounded to 3 significant figures isC. 534,675 rounded to 3 significant figures is 1) 535 1) 535 2) 535,000 2) 535,000 3) 5.35 x 10 3) 5.35 x 1055

Learning CheckLearning Check

In which set(s) do both numbers In which set(s) do both numbers contain the contain the samesame number of number of significant figures?significant figures? 1) 22.0 and 22.00 1) 22.0 and 22.00

2) 400.0 and 40 2) 400.0 and 40 3) 0.000015 and 150,0003) 0.000015 and 150,000

Rounding With Sig Figs When rounding an answer, determine which is the last significant figure. This is where you will round your number.

If the digit immediately to the right of the last sig fig is less than 5, the value of the last sig fig remains the same. 34, 231 rounded to 3 sig figs

If it is 5 or greater, round up. Ex: 0.09246 rounded to 3 sig figs

34,200

0.0925

Practice Rounding (p 69)

Round off each measurement to the number of sig figs shown in parentheses.

314.721 meters (four) 0.001775 meter (two) 8792 meters (two) 25,599 (four)

NOTE: Sometimes the only way to show sig figs properly is to use scientific notation!

= 2.560 x 10 4

= 314.7 meters

=0.0018 meter

= 8800 meters

Significant Numbers in Calculations Significant Numbers in Calculations

A calculated answer cannot be more precise than A calculated answer cannot be more precise than the measuring tool. the measuring tool.

A calculated answer must match the A calculated answer must match the least precise least precise measurement.measurement.

Significant figures are needed for final answers Significant figures are needed for final answers fromfrom 1) adding or subtracting1) adding or subtracting2) multiplying or dividing2) multiplying or dividing

If you must round to obtain the right # of sig figs, If you must round to obtain the right # of sig figs, do so do so after all calcs are completeafter all calcs are complete

Adding and SubtractingAdding and Subtracting

The answer has the same number of The answer has the same number of decimal places as the measurement with decimal places as the measurement with the fewest decimal places.the fewest decimal places.

25.25.22 one decimal placeone decimal place

+ 1.+ 1.3434 two decimal placestwo decimal places 26.5426.54answer 26.5answer 26.5 one decimal placeone decimal place

Learning CheckLearning Check

In each calculation, round the answer to In each calculation, round the answer to the correct number of significant figures.the correct number of significant figures.A. 235.05 + 19.6 + 2.1 = A. 235.05 + 19.6 + 2.1 =

1) 256.751) 256.75 2) 256.8 2) 256.8 3) 2573) 257

B. 58.925 - 18.2B. 58.925 - 18.2 ==1) 40.7251) 40.725 2) 40.73 2) 40.73 3) 40.73) 40.7

Multiplying and Dividing

Round (or add zeros) to the Round (or add zeros) to the calculated answer until you have calculated answer until you have the the same number of significant same number of significant figures as the measurement with figures as the measurement with the fewest significant figures.the fewest significant figures.

Learning CheckLearning Check

A. 2.19 X 4.2 =A. 2.19 X 4.2 = 1) 91) 9 2) 9.2 2) 9.2 3) 9.1983) 9.198

B. 4.311 ÷ 0.07 =B. 4.311 ÷ 0.07 = 1)1) 61.5861.58 2) 62 2) 62 3) 603) 60

C. C. 2.54 X 0.00282.54 X 0.0028 = = 0.0105 X 0.060 0.0105 X 0.060 1) 11.31) 11.3 2) 112) 11 3) 0.041 3) 0.041

MassM

D VDensity Volume

Cover the variable you are solving for and perform the operation with the given amounts

X

÷

Density = __Mass__ Volume

D = _M_ V

Common Units for Density Probs Volume = L (mL), cm3

NOTE: 1 mL = 1 cm3

Mass = g (kg, mg, etc.), lbs

Density Practice Problem #1 (p 91)QUESTION: A copper penny has a mass of

3.1 g and a volume of 0.35 cm3. What is the density of copper?

SOLUTION: Givens: Unknown:

mass = 3.1 g density = ? Volume = 0.35 cm3

Find equation and solve D = m/v D = 3.1 g /0.35 cm3

D = 8.8571 g/cm3

Density Practice Problem #2 (p 92)What is the volume of a pure silver coin that

has a mass of 14 g? The density of silver (Ag) is 10.5 g/cm3.

Givens: Unknown: Mass = 14 g volume = ? Density = 10.5 g/cm3

Find formula & derive it: D = M/V V = M/D Substitute values & solve V = 14 g 10.5

g/cm3

UNITS OF MEASUREMENTUNITS OF MEASUREMENTUse Use SI unitsSI units — based on the metric — based on the metric

systemsystem

Length Length

MassMass

VolumeVolume

TimeTime

TemperatureTemperature

Meter, mMeter, m

Kilogram, kgKilogram, kg

Seconds, sSeconds, s

Celsius degrees, ˚CCelsius degrees, ˚Ckelvins, Kkelvins, K

Liter, LLiter, L

Metric PrefixesMetric Prefixes

Base unit (100) goes here(g, m, L)

Conversion FactorsConversion Factors

Fractions in which the numerator and denominator are Fractions in which the numerator and denominator are EQUAL quantities expressed in different unitsEQUAL quantities expressed in different units

Example: 1 km = 103 m

Factors: 1 km and 103 m 103 m 1 km

They don’t change the value of the measurement, just the units in which it is expressed.

How to set up conversion factors in the metric system If you always select the larger of the

2 units as your “1” unit, Then the multiplier (on prefixes

table) will have a positive exponent.

Ex 1: Compare Megagrams & grams. Which is larger?

Megagrams

1 Mg = 106 grams

This is our “1” unit

This is the multiplier from our table

Ex: Compare meters & millimeters Which is the larger of the 2 units?

meters This is your “1” unit

1 meter =

NOTE: when you look on your prefixes chart, the “milli-” multiplier is 10-3. When we use the method of making the larger unit the “1” unit, we always have a positive exponent for our multiplier.

103 mm

How many millimeters in 1205 meters?

1205 m = _____mm

1. Identify your given. Place it far left.

2. Identify your unknown. Place it far right. The given units are in the NUMERATOR. Your goal is to get rid of the units of your given. How?

3. Place the units of the given in the denominator of the conversion factor!

m

4. Place units of the unknown in the numerator of the conversion factor. (if you can find a relationship between the 2 units. We can! 1m = 103 mm)

mm

5. Cancel units & do the math! (Sig figs!)

1.205 x106103

1

Conversion FactorsConversion Factors

Fractions in which the numerator and denominator are Fractions in which the numerator and denominator are EQUAL quantities expressed in different unitsEQUAL quantities expressed in different units

Example: 1 hr. = 60 min

Factors: 1 hr. and 60 min60 min 1 hr.

They don’t change the value of the measurement, just the units in which it is expressed.

Ex: Convert your weight from pounds (lbs) to kg.

150 lbs = _____kg

1. Identify your given. Place it far left.

2. Identify your unknown. Place it far right. The given units are in the NUMERATOR. Your goal is to get rid of the units of your given. How? 3. Place the units of

the given in the denominator of the conversion factor!

lbs

4. Place units of the unknown in the numerator of the conversion factor. (if you can find a relationship between the 2 units. We can! 1kg = 2.2lb)

kg

5. Cancel units & do the math! (Sig figs!)

681

2.2

Ex: How many minutes are in 2.5 hours?

Conversion factor

2.5 hr x 2.5 hr x 60 min 60 min = 150 min = 150 min 1 hr1 hr

cancelBy using dimensional analysis, the UNITS ensure that you By using dimensional analysis, the UNITS ensure that you have the conversion right side up, and the UNITS are have the conversion right side up, and the UNITS are calculated as well as the numbers!calculated as well as the numbers!

Learning Check

How many seconds are in 1.4 days?

Unit plan: days hr min seconds

1.4 days x 24 hr x _60min x 60 s =___s 1 day 1 hr 1 min

ANSWER: 120,960 s.FINAL ANSWER (in sig figs) = 120,000 s