Counting s vs Measured s l Counting numbers

Counting #’s vs. Measured #’s l 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 l MEASURED NUMBERS – ALWAYS INVOLVE AN ESTIMATE WITH AN UNCERTAINTY IN THE LAST MEASURED DIGIT l 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 l Uncertainty l Example: – Range of possible error 13. 76 g +. 01 g Means the true value lies within range 13. 77 g 13. 76 g 13. 75 g Link to uncertainty animation

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

IMPORTANCE OF SIGNIFICANT DIGITS l The conclusions that you can draw from data cannot exceed the accuracy your measuring device can actually measure l 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 l Digital Electronic Device – record all of the numbers exactly as they appear on the screen. Example: l Screen reads: 1000. 00 g l 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 l ESTIMATE ONE PLACE TO THE RIGHT OF THE SMALLEST MARKED UNIT l EXAMPLE: Smallest marked unit =. 1 cm → Estimate to nearest. 01 l a 3 centimeters b 4 c 5

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

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

Significant Figure Rules – Zero’s as Space-holders l It is sometimes necessary to insert zero to locate a decimal even though a place has not been accurately measured. l 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 l Can’t report # as 5_ _ _ l Can’t report # as 5 (very different than ½ million!) l Convention: Use zero’s to take help locate decimals even though we haven’t actually measured those places

Rules for Significant Digits l 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) l Nonzero digits – ALWAYS significant Example: 15. 68 m. L = 4 sig fig.

LEADING ZERO’S LEADING – 0’s in front of all nonzero digits l LEADING ZERO’S are NEVER SIGNIFICANT l 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 l CAPTIVE ZERO’S – 0’s between nonzero digits. l Example: 7. 08 g (3 SF) l CAPTIVE ZERO’S are ALWAYS SIGNIFICANT.

TRAILING ZERO’S l TRAILING ZERO’S – 0’s at the end of a number (i. e. to the right of all nonzero digits) l Example: 100 m. L l TRAILING ZERO’S ARE NOT SIGNIFICANT UNLESS MARKED BY A DECIMAL. l Examples: 100 m. L = 1 SF; 100. m. L = 3 SF 1 x 102 m. L = 1 SF; 1. 0 x 102 m. L = 2 SF; 1. 00 x 102 m. L = 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 l

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

Significant Figures Problem Set HW 4 l l l l 1 a) 5. 432 g ANS: 4 SF (all nonzero digits) 1 b) 40. 319 g ANS: 5 SF (captive zero is significant) 1 c) 3 pencils ANS: Counting # (infinite sig fig’s, sig. fig rules do not apply) 1 d) 0. 189 lb ANS: 3 SF ; 0. 189 lb (leading zero are never significant)

Significant Figures Problem Set HW 4 l 1 e) 300 kg l ANS: 1 SF ; 300 (trailing zero, no decimal) 1 f) 300. kg 1 f) ANS: 3 SF; 300. kg ; (trailing zero with decimal) 0. 000235 g 1 g) ANS: 3 SF; 0. 000235 g (leading zeros) l l

Significant Figures Problem Set HW 4 l l l 1 h) 2500. 0 cm 1 h) ANS: 5 SF; 2500. 0 (trailing zero with decimal are significant) 1 i) 0. 002300 mg ANS: 4 SF; 0. 002300 (leading zero not significant, trailing zero with decimal are) 1 j) 3. 450 x 103 m ANS: 4 SF; 3. 450 x 103 (trailing zero is significant, only first number between in scientific determines SF).