Chapter 2b A Mathematical Toolkit

Measurement

Systeme Internationale d� Unite� s/Metric System

Accuracy and Precision

Significant Figures

Visualizing Data/Graphing

The Problem

Area of a rectangle = length x widthWe measure:Length = 14.26 cm Width = 11.70 cm

Punch this into a calculator and we find the area as:14.26 cm x 11.70 cm = 166.842 cm2

But there is a problem here! This answer makes it seem like our measurements

were more accurate than they really were. By expressing the answer this way we

imply that we estimated the thousandths position, when in fact we were less precise than that!

Significant figures (“sig figs”): the digits in a measurement that are reliable (or precise). The greater the number of sig figs, the more precise that measurement is.

A more precise instrument will give more sig figs in its measurements.

Significant Figures

Every measurement has some degree of uncertainty because the last digit is assumed to be estimated.

Significant Figures

To help keep track of (and communicate to others) the precision and accuracy of our measurements, we use Significant Significant FiguresFigures

These are the digits in any measurement that are known with certainty plus one known with certainty plus one digit that is uncertaindigit that is uncertain (but usually assumed to be accurate ± 1)

Rules for Significant Figures

1. Digits from 1-9 are always significant. 2. Zeros between two other significant digits

are always significant 3. One or more additional zeros to the right of

both the decimal place and another significant digit are significant.

4. Zeros used solely for spacing the decimal point (placeholders) are not significant.

Counting Significant FiguresCounting Significant Figures

RULE 1. All non-zero digits in a measured number are RULE 1. All non-zero digits in a measured number are significant. Only a zero could indicate that rounding significant. Only a zero could indicate that rounding occurred.occurred.

Number of Significant Figures

38.15 cm38.15 cm 44

5.6 ft5.6 ft 22

65.6 lb65.6 lb ______

122.55 m122.55 m ___

Leading ZerosLeading Zeros

RULE 2. Leading zeros in decimal numbers RULE 2. Leading zeros in decimal numbers

areare NOTNOT significantsignificant..Number of Significant Figures

0.008 mm0.008 mm 11

0.0156 oz0.0156 oz 33

0.0042 lb0.0042 lb ________

0.000262 mL0.000262 mL ____

Sandwiched ZerosSandwiched Zeros

RULE 3. Zeros between nonzero numbers are RULE 3. Zeros between nonzero numbers are

significant. (They can not be rounded unless significant. (They can not be rounded unless

they are on an end of a number.)they are on an end of a number.)Number of Significant Figures

50.8 mm50.8 mm 33

2001 min2001 min 44

0.702 lb0.702 lb ________

0.00405 m0.00405 m ____

Trailing ZerosTrailing Zeros

RULE 4. Trailing zeros in numbers without RULE 4. Trailing zeros in numbers without

decimals are NOT significant. They are only decimals are NOT significant. They are only

serving as place holders.serving as place holders.

Number of Significant Figures

25,000 in.25,000 in. 22

200. yr200. yr 33

48,600 gal48,600 gal ________

Examples

EXAMPLES # OF SIG. DIG. COMMENT

453 3 All non-zero digits are always significant.

50574.06

43

Zeros between two significant digits are significant.

5.00106.00114.050

356

Additional zeros to the right of decimal and a significant digit are significant.

0.007 1 Placeholders are not significant

12000 2Trailing zeros in numbers with no decimal point are not significant (= placeholder)

Learning CheckLearning Check

A. Which answers contain 3 significant figures?A. Which answers contain 3 significant figures?

1) 0.47601) 0.4760 2) 0.00476 2) 0.00476 3) 4760 3) 4760

B. All the zeros are significant in B. All the zeros are significant in

1) 0.00307 1) 0.00307 2) 25.300 2) 25.300 3) 2.050 x 10 3) 2.050 x 1033

C. 534,675 rounded to 3 significant figures isC. 534,675 rounded to 3 significant figures is

1) 535 1) 535 2) 535,000 3) 5.35 x 10 2) 535,000 3) 5.35 x 1055

Learning CheckLearning Check

In which set(s) do both numbers In which set(s) do both numbers

contain thecontain 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

State the number of significant figures in State the number of significant figures in each of the following:each of the following:

A. 0.030 mA. 0.030 m 1 1 2 2 3 3

B. 4.050 LB. 4.050 L 2 2 3 3 4 4

C. 0.0008 gC. 0.0008 g 1 1 2 2 4 4

D. 3.00 mD. 3.00 m 1 1 2 2 3 3

E. 2,080,000 beesE. 2,080,000 bees 3 3 5 5 7 7

Learning CheckLearning Check

Practice

How many significant digits in the following?

Number # Significant Digits

1.4682 5

110256.002 9

0.000000003 1

114.00000006 11

110 2

120600 4

When are digits “significant”?“PACIFIC”

Decimal point is PRESENT. Count digits from left side, starting with the first nonzero digit.

The “Atlantic-Pacific” Rule

40603.23 ft2

0.01586 mL

= 7 sig figs

= 4 sig figs

PACIFIC

PACIFIC

When are digits “significant”?“ATLANTIC”

Decimal point is ABSENT. Count digits

from right side, starting with the first

nonzero digit.

40600 ft2

1000 mL

3 sig figs =

1 sig fig =

ATLANTIC

ATLANTIC

0.00932Decimal point present → “Pacific” → count digits

from left, starting with first nonzero digit = 3 sig figs

4035Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit = 4 sig figs

27510Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit = 4 sig figs

Examples

Write the following measurements in scientific notation, then record the number of sig figs.

1. 789 g2. 96,875 mL3. 0.0000133 J4. 8.915 atm5. 0.94°C

7.89*102 g9.6875*104 mL

1.33*10-5 J8.915 atm9.4*10-1 °C

3 sig figs5 sig figs3 sig figs4 sig figs2 sig figs

The ProblemArea of a rectangle = length x width

We measure:Length = 14.26 cm Width = 11.70 cm

Punch this into a calculator and we find the area as:14.26 cm x 11.70 cm = 166.842 cm2

But there is a problem here! This answer makes it seem like our measurements were more

accurate than they really were. By expressing the answer this way we imply that we estimated the

thousandths position, when in fact we were less precise than that! A much better answer would be that the area is 166.84 cm2

because that keeps the same accuracy as our original measurements.

Significant Numbers in CalculationsSignificant Numbers in Calculations

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

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

Significant figures are needed for final Significant figures are needed for final answers fromanswers from

1) adding or subtracting1) adding or subtracting

2) multiplying or dividing2) multiplying or dividing

Multiplication and Division with Significant DigitsThe rule for multiplying or dividing significant digits is that the answer must have only as many significant digits as the original measurement with the least number of significant digits.Our measurements, 14.26 and 11.70 each have four significant digits. Our calculator told us the answer was 166.842, but we need to round it off. Do we round up or do we round it down?

166.842

If our original measurements had been 14.26 and 11.7, what happens?

166.842

How many significant digits would the answer to each of these have?

Problem #Sig. Digits in Result?114.6 cm x 2.0004 cm 4

0.0006 cm x 14.63 cm 1

12.901 cm2 / 6.23 cm 3

166.84

167

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 figuresthe 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) 60 3) 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

Addition and Subtraction with Significant DigitsThe rule for adding or subtracting with significant digits is that the answer must have only as many digits past the decimal point as the measurement with the least number of digits past the decimal.

How many significant digits would the answer to each of these have?

Problem #Digits Past the Decimal?

114.6g + 2.0004g 1

0.0006g + 14.63g 2

12.901g - 6.23g 2

Adding and SubtractingAdding and Subtracting

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

25.2 25.2 one decimal place one decimal place

+ 1.34+ 1.34 two decimal placestwo decimal places

26.5426.54

answer 26.5 answer 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

Rounding After you have determined to what decimal place (or how many digits) your reported answer must be rounded, Look at digit following specified rounding value. If it is 5 or greater, then round up. If not, truncate (cut off the rest of the numbers).

Round to the nearest tenth

6.7512 6.7777 6.7499 6.9521

6.86.86.77.0

Rounding Rules If the first digit to be dropped is less than 5, that digit and all

digits that follow it are simply dropped. Thus, 62.312 rounded off to three sig. figures becomes 62.3

If the first digit to be dropped is greater than 5 or a 5 followed by digits other than 0, the excess digits are all dropped and the last retained digit is increased in value by one unit. Thus, 62.36 rounded off to three sig. figures becomes 62.4.

If the first digit to be dropped is a 5 not followed by any other digit or a 5 followed only by zeros, an odd-even rule applies. If the last retained digit is odd, that digit is increased in value by one unit after dropping the 5 and any zeros that follow it. If the last retained digit is even, its value is not changed, and the 5 and any zeros that follow are simply dropped. Thus 62,150 and 62.450 rounded to 3 sig. figures become 62.2 (odd rule) and 62.4 (even rule).

Reading Vernier Calipers

Introduction

These are the main features of a typical vernier caliper:

Jaws (for outside measurements)

Small jaws (for inside measurements)Metric vernier scaleMetric fixed scale

English vernier scaleEnglish fixed scale

Beam

Depth gauge

applet

Reading a Caliper: metric

You only need to make two readings: one from the fixed scale and one from the vernier portion.

Reading a caliper: metric

Start by obtaining a measurement from the fixed scale...

This is the fixed scale used for the metric readings.

Reading a caliper: metric

Use the zero line on the vernier to locate your position on the fixed scale.

Reading a caliper: metric

So based upon the two readings (one from the fixed scale, and one from the ruler) the length must be 63 mm + .50 mm = 63.50 mm

63 mm.50 mm+

63.50 mm

2.3 Visualizing Data

A Proper Graph

Plotting Line Graphs Identify Independent and Dependent Variable.

Independent variable gets plotted on x-axis (time is usually on x-axis)

Determine range of independent variable Decide whether origin (0,0) is a valid data point Spread data as much as possible, use a

consistent scale Number and label x-axis

Plotting Line Graphs Repeat previous steps for y-axis, except plotting

the dependent variable Plot all data points on the graph Draw “Best fit” line or curve. Line does NOT

have to go through each point, but does have to approximate the “trend” of the data

Give your graph a title, usually an expression of Independent vs. dependent variables (

Linear Relationships Whenever data

results in a straight-line graph, it is referred to as a linear relationship.

Follows general equation Y = mx + bWhere m = slope b = y intercept

m = rise/run or Δy/ Δx

Slope m = 1N/1.5 cm Y = 0

Non-Linear Relationships Quadratic

relationship Y = ax2 + bx + c

Y varies as a function of the square of x

Non-Linear Relationships Inverse

Relationships Y = a/x Y varies as a

function of the inverse of x

Factor-label method of problem solving

Conversion FactorsConversion Factors

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

Example: 1 in. = 2.54 cm

Factors: 1 in. and 2.54 cm

2.54 cm 1 in.

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 hrBy using dimensional analysis / factor-label method, By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the side up, and the UNITS are calculated as well as the

numbers!numbers!

Cancel

Sample Problem

You have $7.25 in your pocket You have $7.25 in your pocket in quarters. How many quarters in quarters. How many quarters do you have?do you have?

7.25 dollars 4 quarters7.25 dollars 4 quarters

1 dollar1 dollar

X= 29= 29 quartersquarters

Learning Check

Write conversion factors that relate Write conversion factors that relate each of the following pairs of units:each of the following pairs of units:

1. Liters and mL1. Liters and mL

2. Hours and minutes2. Hours and minutes

3. Meters and kilometers3. Meters and kilometers

Learning Check

A rattlesnake is 2.44 m long. How A rattlesnake is 2.44 m long. How long is the snake in cm?long is the snake in cm?

a) 2440 cma) 2440 cm

b)b) 244 cm244 cm

c)c) 24.4 cm24.4 cm

Learning Check

How many seconds are in 1.4 days?

Unit plan: days hr min seconds

1.4 days x 24 hr x ?? 1 day

Solution

Unit plan: days hr min seconds

1.4 day x 24 hr x 60 min x 60 sec

1 day 1 hr 1 min

= 1.2 x 105 sec

Wait a minute!

What isWhat is wrongwrong with the followingwith the following setup?setup?

1.4 day x 1.4 day x 1 day 1 day x x 60 min 60 min x x 60 sec 60 sec

24 hr 1 hr 1 min24 hr 1 hr 1 min

English and Metric ConversionsEnglish and Metric Conversions

If you know ONE conversion for each type If you know ONE conversion for each type of measurement, you can convert of measurement, you can convert anything!anything!

You mustYou must memorizememorize and use these and use these conversions:conversions: Mass: 454 grams = 1 poundMass: 454 grams = 1 pound Length: 2.54 cm = 1 inchLength: 2.54 cm = 1 inch Volume: 0.946 L = 1 quartVolume: 0.946 L = 1 quart

Learning Check

An adult human has 4.65 L of blood. How An adult human has 4.65 L of blood. How many gallons of blood is that?many gallons of blood is that?

Unit plan: L qt gallon Equalities: 1 quart = 0.946 L

1 gallon = 4 quarts

Your Setup: