1. 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 cm

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

3. 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.

4. 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!

5. Importance of Sig Fig’s cont.
If arrested person’s value =
0.08% % → 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 %

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

7. 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

8. Reading a Meniscus Read at eye-level, from the bottom of the meniscus6 7

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

10. 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

11. 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

12. 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: mL = 4 sig fig.

13. 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) → kg (Still 3 SF)
Accuracy of balance didn’t change; still 3 SF

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

15. 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; mL = 3 SF
1 x 102 mL = 1 SF; 1.0 x 102 mL = 2 SF;
1.00 x 102 mL = 3 SF

16. 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 feet = feet (2 SF)
12 in

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

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

19. 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) g
1g) ANS: 3 SF; g (leading zeros)

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