Transcript 1 . 00 6
Uncertainty:
not surely known; doubtful;
varying
Significant: full of meaning; important
Uncertainty in Measurements
Measurements
made (by ruler, graduated
cylinder, etc.) require estimation
imagine distance
Above is a ruler, where the labeled
numbers (i.e. “1” and “2”) are
considered “certain numbers.”
http://www.kidsnewsroom.org/resources/sol/TX/G08S06/20graphicaa.
gif
Above is a graduated cylinder,
where the labeled numbers (i.e.
“40” and “50”) are “certain.”
http://www.kidsnewsroom.org/resources/sol/TX/G08S06/20graphica
a.gif
Uncertainty in Measurements
continued…
Certain
numbers: same regardless of who
made them; estimated
Uncertain numbers: estimated values;
depend on device
NOTE!: measurements always have some
level of uncertainty
Example
Why
is there always uncertainty in a
measurement?
In a measuring device, there are only a limited
amount of “tick marks” that indicate the
certain numbers. There can always be a
measurement between two indicated tick
marks on a measuring device, and another
measurement within that. Since the
measurement can never be exact, you would
need to estimate. This estimation makes the
measurement uncertain.
Rules for Significant Numbers
Significant figures (sig figs): numbers recorded
(all certain numbers and one estimated
uncertain number)
Nonzero integers ALWAYS count; include {…-3,2,-1,1,2,3…}
NOTE!: integers are whole numbers and NOT
decimals/fraction
Exact numbers: determined by
counting/definition (ex: 5 pencils); unlimited sig
figs
Flow Chart on Sig Figs
http://www.rpi.edu/dept/phys/Dept2/APPhys1/sigfigs/sigfig/node152.html
Types of Zeros
Below is a chart of zeros and examples of each
Zeros
Captive zeros: between nonzero digits; ALWAYS count as significant numbers
1 . 00 6
captive zeros
Leading zeros: precede nonzero digits; NEVER count as significant numbers
0.00 6
leading zeros
Trailing Zeros: follow nonzero digits at the end; counts as significant number if
they follow a period; does not count without the period
1 0.00
10
Count
does not count
Example
How
many sig figs are present in “100?”
Why do the zeros do/do not count?
There is only one sig fig in this value, because
the zeros are considered “trailing.” In the
case of trailing zeros, they can be considered
significant or not. They are only counted as
significant if and only if they follow a decimal
point. In this case, however, there is no
decimal point; therefore, they do not count.
Rules for Rounding Off
than 5 preceding number stays
the same
≥ 5 preceding number increases by one
NOTE!: carry extra digits to the final result
and round
When rounding use only the first number
to the right of the last sig fig
Less
Example
Round
3.564 to the nearest hundredth.
3.56. Since “4” is less than 5, the “6” is left as is.
Rules for Multiplying/Dividing
Significant Numbers
Number
of sig figs act as the “limiting
factor”
Number with smallest number of sig figs
acts as “limiting factor”
1.08 x 5.6
3 sig figs 2 sig figs
Example
1.4
x 5.78. Find the answer and give the
limiting factor.
8.1. The limiting factor is “1.4”, because
compared to “5.78,” which has 3 sig figs, it has
only 2 sig figs. limits to 2 sig figs in answer
Rules for Adding/Subtracting
Significant Numbers
Decimal
places counted as “limiting factor”
Number with the least decimal places is
“limiting factor”
1.08 x 5.6
2 places
1 place
Example
How
many sig figs should the answer of
6.3421 – 2.543 – 2.1 contain? What is the
limiting factor? Solve.
1.7. The limiting factor is 2.1, because it has only
one decimal place.
Quiz
1. In the picture below, estimate the measurement of the
purple line in inches. Are the estimated values
certain/uncertain?
Quiz continued…
2. When counting 30 apples, how many sig
figs are there? Why?
3. Round 5.6345 to the nearest thousandth.
4. What is meant by the term limiting factor
when multiplying/dividing sig figs?
5. How many sig figs should the answer of
5.1 + 2.34 + 1.23 contain? Give the
limiting factor and solve.
Answers
1. The measurement of the line is about
1.55 inches. The estimated part is
uncertain, because there is no mark on the
ruler that physically displays a
measurement beyond the hundredth
place. Therefore, the estimated part is
uncertain, since you are only “imagining”
tick marks between the 5th and 6th tick
mark.
Answers continued…
2. There are unlimited significant figures, because
this measurement is done by counting.
3. 5.635. Since the number after “4” is “5”, which
is ≥ 5, the “4” is increased by one.
4. When multiplying/dividing, look at the number
with the least number of sig figs. This term limits
the answer to that amount of sig figs.
5. 8.7; the “5.1” limits the answer to 2 sig figs.
Reference
http://becauseican.co.za/wp-
content/uploads/2008/04/ruler_0_10.jpg
http://www.kidsnewsroom.org/resources/s
ol/TX/G08S06/20graphicaa.gif
http://www.rpi.edu/dept/phys/Dept2/APPhy
s1/sigfigs/sigfig/node152.html