Chapter 2 A Mathematical Toolkit

Measurement

Systeme Internationale d� Unite� s/Metric System

Accuracy and Precision

Significant Figures

Visualizing Data/Graphing

Objectives

2.1 The Measure of Science Define the SI standards of measurement Use common metric prefixes Estimate measurements and solutions to

problems Perform arithmetic operations using scientific

notation

Objectives

2.2 Measurement Uncertainty Distinguish between accuracy and precision Indicate the precision of measured quantities

with significant digits Perform arithmetic operations with significant

digits

Objectives

2.3 Visualizing Data Graph the relationship between independent

and dependent variables Recognize linear and direct relationships and

interpret the slope of a curve Recognize quadratic and inverse

relationships

What is measurement? Defined as a comparison of an unknown quantity to a known

Standard. The measurement instrument must be standardized against the known standard

Every measurement has a value and a unit

Standard kilogram of mass, officially known as the “International prototype of the kilogram” composed of platinum-iridium alloy, stored under glass in a vacuum since 1889

Standards of MeasurementStandards of Measurement

When we measure, we use a measuring When we measure, we use a measuring tool to compare some dimension of an tool to compare some dimension of an object to a standard.object to a standard.

For example, at one time the For example, at one time the standard for length was the king’s standard for length was the king’s

foot. What are some problems foot. What are some problems with this standard?with this standard?

Historical standard platinum iridium meter bar

The meter now is defined as the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. The speed of light is c = 299,792,458 m/s

SI measurementSI measurement Le Systeme international d'unitesLe Systeme international d'unites The only countries that have not The only countries that have not officiallyofficially

adopted SI are Liberia (in western Africa) adopted SI are Liberia (in western Africa) and Myanmar (Burma), but may now and Myanmar (Burma), but may now may using metric regularlymay using metric regularly

Metrication is a process that does not Metrication is a process that does not happen all at once, but is rather a happen all at once, but is rather a process that happens over time.process that happens over time.Why???? Why????

Among countries with non-metric usage, Among countries with non-metric usage, the U.S. is the the U.S. is the only country significantly only country significantly holding outholding out.. The U.S. officially adopted SI The U.S. officially adopted SI in 1866.in 1866.

Information from U.S. Metric Association

Measurement In ActionMeasurement In ActionOn 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s atmosphere 100 km lower than planned and was destroyed by heat.

1 lb = 1 N1 lb = 4.45 N

“This is going to be the cautionary tale that will be embedded into introduction to the metric system in elementary school, high school, and college science courses till the end of time.”

Base UnitsBase Units of Measurement in of Measurement in SISI

LengthLength

MassMass

TimeTime

TemperatureTemperature

______________________________

__________________________________

__________________________

______________________________

7 total SI Base Units

Electrical charge Coulomb Amount of substance Mole Luminous intensity Candela

Derived Units Units which are “calculated” using base units or

other derived units

Frequency Hertz S-1

Area square meter m2

Pressure Pascal N/m2

Energy/Work Joule kg·m2/s2

Speed meter/sec m/s

And many others…..

Name

Symbol

Quantity Expression in terms of other units

Expression in terms of SI base units

hertz Hz Frequency 1/s s−1

newton

N Force, Weight m∙kg/s2 m∙kg∙s−2

joule J Energy, Work, Heat N∙m m2∙kg∙s−2

watt W Power, Radiant flux J/s m2∙kg∙s−3

pascal

Pa Pressure, Stress N/m2 m−1∙kg∙s−2

lumen lm Luminous flux cd∙sr = 1/cd cd

lux lx Illuminance lm/m2 = 1/(m2∙cd) m−2∙cd

coulomb

C Electric charge or flux s∙A s∙A

volt V Electrical potential difference, Electromotive force

W/A = J/C m2∙kg∙s−3∙A−1

ohm Ω Electric resistance, Impedance, Reactance

V/A m2∙kg∙s−3∙A−2

farad F Electric capacitance C/V m−2∙kg−1∙s4∙A2

Other Derived Units

Some Tools for Some Tools for MeasurementMeasurement

Which tool(s) would Which tool(s) would you use to measure:you use to measure:

A. temperatureA. temperature

B. volumeB. volume

C. timeC. time

D. weightD. weight

Learning Check

Match L) length M) mass V) volumeMatch L) length M) mass V) volume

____ A. A bag of tomatoes is 4.6 kg.____ A. A bag of tomatoes is 4.6 kg.

____ B. A person is 2.0 m tall.____ B. A person is 2.0 m tall.

____ C. A medication contains 0.50g aspirin.____ C. A medication contains 0.50g aspirin.

____ D. A bottle contains 1.5 L of water.____ D. A bottle contains 1.5 L of water.

Learning CheckLearning Check

What are some U.S. units that are used What are some U.S. units that are used to measure each of the following?to measure each of the following?

A. length A. length

B. time B. time

C. weightC. weight

D. temperatureD. temperature

SolutionSolution

Some possible answers areSome possible answers are

A. length-- inch, foot, yard, mile A. length-- inch, foot, yard, mile

B. volume-- cup, teaspoon, gallon, pint, quart B. volume-- cup, teaspoon, gallon, pint, quart

C. weight--C. weight-- ounce, pound (lb), ton ounce, pound (lb), ton

D. temperature-- D. temperature-- F, RankineF, Rankine

Metric PrefixesMetric Prefixes

Kilo- means 1000 of that unitKilo- means 1000 of that unit

1 kilometer (km) = 1000 meters (m)1 kilometer (km) = 1000 meters (m)

Centi- means 1/100 of that unitCenti- means 1/100 of that unit

1 meter (m) = 100 centimeters (cm)1 meter (m) = 100 centimeters (cm)

1 dollar = 100 cents1 dollar = 100 cents

Milli- means 1/1000 of that unitMilli- means 1/1000 of that unit

1 Liter (L) = 1000 milliliters (mL)1 Liter (L) = 1000 milliliters (mL)

You are responsible for “Giga” through “Nano”

Learning Check

Select the unit you would use to measure Select the unit you would use to measure

1. Your height 1. Your height

a) millimeters a) millimeters b) metersb) meters c) kilometers c) kilometers

2. Your mass 2. Your mass

a) milligramsa) milligrams b) grams b) grams c) kilograms c) kilograms

3. The distance between two cities 3. The distance between two cities

a) millimetersa) millimeters b) meters b) meters c) kilometers c) kilometers

4. The width of an artery4. The width of an artery

a) millimetersa) millimeters b) meters b) meters c) kilometers c) kilometers

Scientific Notation

Scientific notation consists of Scientific notation consists of two parts:two parts:

A number between 1 and 10A number between 1 and 10

A power of 10A power of 10

N x 10N x 10xx

Scientific Notation “Writing a number as a power of 10.” Why? It makes very large and very small numbers

more manageable to write and use. Also, all digits in scientific notation (1-10) are

considered to be significant and are clearly shown (no question about significance) Example 1200 mg has 2 sig.figs, written as 1.2 X 103

.00000230g has 3 sig. figs written as 2.30 X 10-6

Rule of thumb: Use when number is greater than 1000 or smaller than 0.001 Or, you may always use it!

1. Move the decimal point in the original number so that it is located to the right of the first nonzero digit.

2. Multiply the new number by 10 raised to the proper power that is equal to the number of places the decimal moved.

3. If the decimal point moves: To the left, the power of 10 is positive. To the right, the power of 10 is negative.

Writing in scientific notation

ExamplesExamples

Given: 289,800,000Given: 289,800,000 Use: 2.898 (moved 8 places)Use: 2.898 (moved 8 places) Answer:Answer: 2.898 x 102.898 x 1088

Given: 0.000567Given: 0.000567 Use: 5.67 (moved 4 places)Use: 5.67 (moved 4 places) Answer:Answer: 5.67 x 105.67 x 10-4-4

Learning Check

Express these numbers in Express these numbers in Scientific Notation:Scientific Notation:

1)1) 405789405789

2)2) 0.0038720.003872

3)3) 30000000003000000000

4)4) 22

5)5) 0.4782600.478260

To change scientific notation to To change scientific notation to standard form…standard form…

Simply move the decimal point to the Simply move the decimal point to the right for positive exponent 10. right for positive exponent 10.

Move the decimal point to the left for Move the decimal point to the left for negative exponent 10.negative exponent 10.

(Use zeros to fill in places.)(Use zeros to fill in places.)

ExampleExample

Given: 5.093 x 10Given: 5.093 x 1066

Answer:Answer: 5,093,0005,093,000 (moved 6 (moved 6 places to the right)places to the right)

Given: 1.976 x 10Given: 1.976 x 10-4-4

Answer:Answer: 0.00019760.0001976 (moved 4 (moved 4 places to the left)places to the left)

2.2 Measurement Uncertainty

Measurement needs to be precise and accurate

Precision: How closely multiple measurements of the

same quantity come to each other. This will depend on the measuring device.

For example, a thermometer that shows degrees in tenths is more precise than one that only shows single degrees.

Measurement and numbers Measurements consist of two parts

The number itself (the quantity)The number itself (the quantity) The units (the nature of the quantity measured)The units (the nature of the quantity measured)

There are two kinds of numbers Counted or definedCounted or defined - exact, not subject to estimate.

Ex: number of eggs in a carton. MeasuredMeasured - always carries some amount of

uncertainty because measurement involves estimation. The size of uncertainty depends on the precision of the measuring device AND the skill of the person using the device

Estimation in measurement

When we measure, the quantity rarely falls exactly on the calibration marks of the scale we are using.

Because of this we are estimating the last digit of the measurement.

For instance, we could measure “A” above as about 2.3 cm. We are certain of the digit “2”, but the “.3” part is a guess - an estimate.

What is your estimate for B and C?

1 cm 2 cm 3 cm 4 cm 5 cm

A B C

Higher Precision

A measuring device with more marks on the scale is more precise. I.e., we are estimating less, and get a more accurate reading. Here we are estimating the hundredths place instead of the tenths.

Here, we can measure A as 1.25 cm. Only the last digit is uncertain.

Usually we assume the last digit is accurate ± 1. How would you read B, C, and D?

1 cm 2 cm 3 cm

A B C D

Reading a Meter stickReading a Meter stick

. l. l22. . . . I . . . . I. . . . I . . . . I33 . . . .I . . . . I . . . .I . . . . I44. . cm. . cmFirst digit (known)First digit (known) = 2 = 2 2.?? cm2.?? cm

Second digit (known)Second digit (known) = 0.7 = 0.7 2.7? cm2.7? cm

Third digit (estimated) between 0.05- 0.07Third digit (estimated) between 0.05- 0.07

Length reportedLength reported == 2.75 cm 2.75 cm

oror 2.74 cm2.74 cm

oror 2.76 cm2.76 cm

Zero as a Measured NumberZero as a Measured Number

. l3. . . . I . . . . I4 . . . . I . . . . I5. . cm

What is the length of the line?What is the length of the line?

First digit First digit 5.?? cm5.?? cm

Second digit Second digit 5.0? cm5.0? cm

Last (estimated) digit is Last (estimated) digit is 5.00 cm5.00 cm

Accuracy Refers to how close a measurement

comes to the true or accepted value. This depends on both the measuring

device and the skill of the person using the measuring device.

This can be determined by comparing the measured value to the known or accepted value.

A graduated cylinder:

50

100 mL Beaker

50 mL Graduated cylinder

A beaker:

41.0

41.2 mL (3 sig figs = very precise)

40. mL (2 sig figs = not as precise)

Accurate or Precise?

Precise!(but not accurate)

What is the temperature at which water boils?

•Measurements: 95.0°C, 95.1°C, 95.3°C

•True value: 100°C

Accurate or Precise?

Accurate!(it’s hard to be accurate without being precise)

What is the temperature at which water freezes?

•Measurements: 1.0°C, 1.2°C, -5.0°C

•True value: 0.0°C

Accurate or Precise?

Not Accurate & Not Precise(don’t quit your day job)

What is the atmospheric pressure at sea level?

•Measurements: 10.01 atm, 0.25 atm, 234.5 atm

•True value: 1.00 atm

Accurate or Precise?

Accurate & Precise(it’s time to go pro)

What is the mass of one Liter of water?

•Measurements: 1.000 kg, 0.999 kg, 1.002 kg

•True value: 1.000 kg

Three Three targets with targets with three arrows three arrows each to each to shoot.shoot.

Can you hit the bull's-eye?Can you hit the bull's-eye?

Both accurate and precise

Precise but not accurate

Neither accurate nor precise

How do How do they they compare?compare?

Can you define accuracy and precision?Can you define accuracy and precision?

Accuracy or Precision?When deciding on accuracy, precision, both, or

neither….it is quantitative data (numerical), not qualitative (descriptive)

1) The recipe calls for 25 chocolate chips per cookie. The cookies analyzed have 34, 35, and 32 respectively.

2) The percent NaCl is 99%, 99%, and 98%.

3) The number of grams of KF required is 0.04 g. The amounts used were 0.038, 0.039, 0.041, and 0.040.

4) To win, Henry must earn 500 points. In his three trials, he earned 400, 480, and 395 points.