Counting #’s vs. Measured #’s Counting numbers – when we can exactly count the # of objects and there is no UNCERTAINTY in the values Example: Exactly 29 students in the room, no question about fractions of a person MEASURED NUMBERS – ALWAYS INVOLVE AN ESTIMATE WITH AN UNCERTAINTY IN THE LAST MEASURED DIGIT Example: The # of digits in your height depends on how many marks on the ruler Person’s height with different rulers: 174 cm vs. 174.2 cm

Uncertainty in a Measurement Uncertainty – Range of possible error Example: 13.76 g + .01 g Means the true value lies within range 13.77 g 13.75 g 13.76 g Link to uncertainty animation

Definition of Significant Digits: Digits in a measurement that are meaningful given the accuracy of the measuring device All of the places in a measurement that are certain plus 1 estimated place.

IMPORTANCE OF SIGNIFICANT DIGITS The conclusions that you can draw from data cannot exceed the accuracy your measuring device can actually measure Example: In Colorado, Blood alcohol of 0.08% = DUI If a breathalyzer with uncertainty of .01% were used → potentially big legal problem!

Importance of Sig Fig’s cont. If arrested person’s value = 0.08% + 0.01% → means range of true values is: 0.07% (INNOCENT) 0.08 % (GUILITY) 0.09% (GUILITY) In practice, reduce # of ambiguous results by using more accurate instrument e.g. uncertainty of + 0.0001%

Rules for Recording Significant Figures Digital Electronic Device – record all of the numbers exactly as they appear on the screen. Example: Screen reads:1000.00 g Record: 1000.00 g Uncertainty = + 0.01 g Incorrect: 1000 g Implies uncertainty is + 1 g

RULES FOR RECORDING SIGNIFICANT FIGURES IN A NON-DIGITAL DEVICE DETERMINE THE SMALLEST MARKED UNIT ESTIMATE ONE PLACE TO THE RIGHT OF THE SMALLEST MARKED UNIT EXAMPLE: Smallest marked unit = .1 cm → Estimate to nearest .01 3 4 5 centimeters a b c

Reading a Meniscus Read at eye-level, from the bottom of the meniscus 6 7

Rules for Interpreting Significant Figures In a Recorded Measurement When is a Significant Figure NOT Significant? Answer: When it is a space-holding zero!

Significant Figure Rules – Zero’s as Space-holders It is sometimes necessary to insert zero to locate a decimal even though a place has not been accurately measured. Example: Newspaper Headline: 500,000 ATTEND FREE CONCERT IN CENTRAL PARK In reality, this # is an estimate, the exact # of people who attended is unknown

Zero’s as spaceholders Can’t report # as 5_ _ _ _ _ _ Can’t report # as 5 (very different than ½ million!) Convention: Use zero’s to take help locate decimals even though we haven’t actually measured those places

Rules for Significant Digits COUNTING # - numbers whose values are exactly known with no estimate. Significant figure rules don’t apply Example: 7 calculators = infinite number of significant figures (rules don’t apply, write down as many places as you want) Nonzero digits – ALWAYS significant Example: 15.68 mL = 4 sig fig.

LEADING ZERO’S LEADING – 0’s in front of all nonzero digits LEADING ZERO’S are NEVER SIGNIFICANT Example: Weigh object in grams: 9.67 g (3 SF) Convert mass to kg by dividing measurement by 1000: 9.67 g (3 SF) → 0.00967 kg (Still 3 SF) Accuracy of balance didn’t change; still 3 SF

CAPTIVE ZERO’S CAPTIVE ZERO’S – 0’s between nonzero digits. Example: 7.08 g (3 SF) CAPTIVE ZERO’S are ALWAYS SIGNIFICANT.

TRAILING ZERO’S TRAILING ZERO’S – 0’s at the end of a number (i.e. to the right of all nonzero digits) Example: 100 mL TRAILING ZERO’S ARE NOT SIGNIFICANT UNLESS MARKED BY A DECIMAL. Examples: 100 mL = 1 SF; 100. mL = 3 SF 1 x 102 mL = 1 SF; 1.0 x 102 mL = 2 SF; 1.00 x 102 mL = 3 SF

Conversion Factors CONVERSION FACTORS – exact definitions → infinite number of significant figures In conversion problems: final SF in calculation match = SF original measurement Example: 6.0 in ( 2 sig figures) → convert to feet → final answer 2 sig figures 6.0 in 1.0 feet = 0.50 feet (2 SF) 12 in

SIG FIGURE’S RULE SUMMARY COUNTING #’S and Conversion factors – INFINITE NONZERO DIGIT’S: ALWAYS ZERO’S: LEADING : NEVER CAPTIVE: ALWAYS TRAILING :SOMETIMES Decimal present? YES; SIGNIFICANT NO; NOT SIGNIFICANT

Significant Figures Problem Set HW 4- 1a) 5.432 g ANS: 4 SF (all nonzero digits) 1b) 40.319 g ANS: 5 SF (captive zero is significant) 1c) 3 pencils ANS: Counting # (infinite sig fig’s, sig. fig rules do not apply) 1d) 0.189 lb ANS: 3 SF ; 0.189 lb (leading zero are never significant)

Significant Figures Problem Set HW 4- 1e) 300 kg ANS: 1 SF ; 300 (trailing zero, no decimal) 1f) 300. kg 1f) ANS: 3 SF; 300. kg ; (trailing zero with decimal) 0.000235 g 1g) ANS: 3 SF; 0.000235 g (leading zeros)

Significant Figures Problem Set HW 4- 1h) 2500.0 cm 1h) ANS: 5 SF; 2500.0 (trailing zero with decimal are significant) 1i) 0.002300 mg ANS: 4 SF; 0.002300 (leading zero not significant, trailing zero with decimal are) 1j) 3.450 x 103 m ANS: 4 SF; 3.450 x 103 (trailing zero is significant, only first number between in scientific determines SF).