Chapter 2b A Mathematical Toolkit Measurement Système Internationale d̀Unité́s/Metric System Accuracy and Precision Significant Figures Visualizing Data/Graphing

The Problem Area 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!

Significant Figures Every measurement has some degree of uncertainty because the last digit is assumed to be estimated. 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 To help keep track of (and communicate to others) the precision and accuracy of our measurements, we use Significant Figures These are the digits in any measurement that are known with certainty plus one digit that is uncertain (but usually assumed to be accurate ± 1)

Rules for Significant Figures Digits from 1-9 are always significant. Zeros between two other significant digits are always significant One or more additional zeros to the right of both the decimal place and another significant digit are significant. Zeros used solely for spacing the decimal point (placeholders) are not significant.

Counting Significant Figures RULE 1. All non-zero digits in a measured number are significant. Only a zero could indicate that rounding occurred. Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb ___ 122.55 m ___

Leading Zeros RULE 2. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____

Sandwiched Zeros RULE 3. Zeros between nonzero numbers are significant. (They can not be rounded unless they are on an end of a number.) Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____

Trailing Zeros 25,000 in. 2 Number of Significant Figures 200. yr 3 RULE 4. Trailing zeros in numbers without decimals are NOT significant. They are only serving as place holders. Number of Significant Figures 25,000 in. 2 200. yr 3 48,600 gal ____

Examples EXAMPLES # OF SIG. DIG. COMMENT 453 3 All non-zero digits are always significant. 5057 4.06 4 Zeros between two significant digits are significant. 5.00 106.00 114.050 5 6 Additional zeros to the right of decimal and a significant digit are significant. 0.007 1 Placeholders are not significant 12000 2 Trailing zeros in numbers with no decimal point are not significant (= placeholder)

Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4760 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 105

Learning Check In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and 22.00 2) 400.0 and 40 3) 0.000015 and 150,000

Learning Check State the number of significant figures in each of the following: A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 D. 3.00 m 1 2 3 E. 2,080,000 bees 3 5 7

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 = 7 sig figs PACIFIC PACIFIC 0.01586 mL = 4 sig figs

When are digits “significant”? “ATLANTIC” Decimal point is ABSENT. Count digits from right side, starting with the first nonzero digit. 3 sig figs = 40600 ft2 ATLANTIC 1 sig fig = 1000 mL ATLANTIC

Examples 0.00932 Decimal point present → “Pacific” → count digits from left, starting with first nonzero digit = 3 sig figs 4035 Decimal point absent → “Atlantic” → count digits from right, starting with first nonzero digit = 4 sig figs 27510

Write the following measurements in scientific notation, then record the number of sig figs. 96,875 mL 0.0000133 J 8.915 atm 0.94°C 3 sig figs 7.89*102 g 5 sig figs 9.6875*104 mL 1.33*10-5 J 3 sig figs 4 sig figs 8.915 atm 2 sig figs 9.4*10-1 °C

The Problem Area 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 Calculations A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

Multiplication and Division with Significant Digits The 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? 166.842 166.84 If our original measurements had been 14.26 and 11.7, what happens? 166.842 167 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

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

Learning Check A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60 C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

Addition and Subtraction with Significant Digits The 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 Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 one decimal place + 1.34 two decimal places 26.54 answer 26.5 one decimal place

Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 + 19.6 + 2.1 = 1) 256.75 2) 256.8 3) 257 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 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 6.8 6.7 7.0

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: Small jaws (for inside measurements) Depth gauge Metric vernier scale Metric fixed scale Beam Jaws (for outside measurements) English vernier scale English fixed scale

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 + b Where 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 Factors Fractions in which the numerator and denominator are EQUAL quantities expressed in different units 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 60 min = 150 min 1 hr Cancel By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers!

Sample Problem You have $7.25 in your pocket in quarters. How many quarters do you have? 7.25 dollars 4 quarters 1 dollar X = 29 quarters

Learning Check 1. Liters and mL 2. Hours and minutes Write conversion factors that relate each of the following pairs of units: 1. Liters and mL 2. Hours and minutes 3. Meters and kilometers

Learning Check a) 2440 cm b) 244 cm c) 24.4 cm A rattlesnake is 2.44 m long. How long is the snake in cm? a) 2440 cm b) 244 cm c) 24.4 cm

How many seconds are in 1.4 days? 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

What is wrong with the following setup? Wait a minute! What is wrong with the following setup? 1.4 day x 1 day x 60 min x 60 sec 24 hr 1 hr 1 min

English and Metric Conversions If you know ONE conversion for each type of measurement, you can convert anything! You must memorize and use these conversions: Mass: 454 grams = 1 pound Length: 2.54 cm = 1 inch Volume: 0.946 L = 1 quart

An adult human has 4.65 L of blood. How many gallons of blood is that? Learning Check An adult human has 4.65 L of blood. How many gallons of blood is that? Unit plan: L qt gallon Equalities: 1 quart = 0.946 L 1 gallon = 4 quarts Your Setup: