Transcript Significant Figures in Mathematical Operations

Significant Figures (digits)
= reliable figures obtained by
measurement
= all digits known with certainty plus
one estimated digit
Taking the measurement
• Is always some uncertainty
• Because of the limits of the instrument you
are using
EXAMPLE: mm ruler
Is the length of the line between 4 and 5 cm? Yes, definitely.
Is the length between 4.0 and 4.5 cm? Yes, it looks that way.
But is the length 4.3 cm? Is it 4.4 cm?
Let’s say we are certain that it is 4.3 cm or 43mm, but not at
long as 4.4cm.
So – we need to add one more digit to ensure the
measurement is more accurate.
Since we’ve decided that it’s closer to 4.3 than 4.4 it may be
recorded at 4.33 cm.
• It is important to be honest when reporting
a measurement, so that it does not appear
to be more accurate than the equipment
used to make the measurement allows.
• We can achieve this by controlling the
number of digits, or significant figures,
used to report the measurement.
As we improve the sensitivity of the
equipment used to make a measurement,
the number of significant figures increases.
Postage Scale
3g
1g
1 significant
figure
Two-pan
balance
2.53 g
0.01 g
3 significant
figures
Analytical
balance
2.531 g
0.001g
4 significant
figures
Which numbers are
Significant?
How to count them!
Non-Zero integers
• Always count as significant figures
1235 has 4 significant digits
Zeros – there are 3 types
Leading zeros (place holders)
The first significant figure in a
measurement is the first digit other than
zero counting from left to right
0.0045g
(4 is the 1st sig. fig.)
“0.00” are place holders.
The zeros are not significant
Captive zeros
Zeros within a number at always significant
–
30.0809 g
All digits are significant
Trailing zeros – at the end of numbers but to the right of the
decimal point
2.00 g - has 3 sig. digits (what this means is
that the measuring instrument can measure exactly to
two decimal places.
100 m has 1 sig. digit
Zeros are significant if a number contains decimals
Exact Numbers
Are numbers that are not obtained by
measuring
Referred to as counting numbers
EX : 12 apples, 100 people
Exact Numbers
Also arise by definition
1” = 2.54 cm
or
12 in. = 1 foot
Are referred to as conversion factors that
allow for the expression of a value using
two different units
Significant Figures
Rules for sig figs.:
•Count the number of digits in a measurement from left to
right:
•Start with the first nonzero digit
•Do not count place-holder zeros.
•The rules for significant digits apply only to
measurements and not to exact numbers
Sig figs is short for significant figures.
Determining Significant Figures
State the number of significant figures in the following measurements:
2005 cm
4
0.050 cm
2
25,000 g
2
0.0280 g
3
25.0 ml
3
50.00 ml
4
0.25 s
2
1000 s
1
0.00250 mol
3
1000. mol
4
Rounding Numbers
• To express answer in correctly
• Only use the first number to the right of the
last significant digit
Rounding
• Always carry the extra digits through to the
final result
• Then round
EX:
Answer is 1.331 rounds to 1.3
OR
1.356 rounds to 1.4
Significant Figures
Rounding off sig figs (significant figures):
Rule 1: If the first non-sig fig is less than 5, drop all non-sig
fig.
Rule 2: If the first sig fig is 5, or greater that 5, increase the
last sig fig by 1 and drop all non-sig figs.
Round off each of the following to 3 significant figures:
12.514748
12.5
192.49032
192
0.6015261
14652.832
0.602
14,700
Math Problems w/Sig Figs
When combining measurements with
different degrees of accuracy and
precision, the accuracy of the final
answer can be no greater than the least
accurate measurement.
Adding and Subtracting Sig.
Figures
This principle can be translated into a simple
rule for addition and subtraction:
When measurements are added or
subtracted, the answer can contain no
more decimal places than the least
accurate measurement.
Significant Figures
Adding and subtracting sig figs - your
answer must be limited to the value
with the greatest uncertainty.
Line up decimals and Add
150.0 g H2O (using significant figures)
0.507 g salt 150.5 g solution
150.5 g solution
150.0 is the least precise so the answer will
have no more than one place to the right
of the decimal.
Example
Answer will have the same number of decimal
places as the least precise measurement used.
12.11 cm
18.0 cm
1.013 cm
31.132 cm
9.62 cm
71.875 cm
Correct answer would be 71.9 cm – the last sig fig is “8”,
so you will round using only the first number to the right of
the last significant digit which is “7”.
Significant Figures
Multiplication and division of sig figs - your answer must
be limited to the measurement with the least number of
sig figs.
5.15
X 2.3
11.845
so 11.845
is rounded to 12
3 sig figs
2 sig figs
only allowed 2 sig figs
5 sig fig
2 sig figs
Multiplication and Division
Answer will be rounded to the same number
of significant figures as the component
with the fewest number of significant
figures.
4.56 cm x 1.4 cm = 6.38 cm2
= 6.4 cm2
28.0 inches
2.54 cm
x 1 inch = 71.12 cm
Computed measurement is 71.12 cm
Answer is 71.1 cm
When both addition/subtraction and
multiplication/division appear in the same problem
• In addition/subtraction the number of significant digits is
limited by the value of greatest uncertainty.
• In multiplication/division, the number of significant digits is
limited by the value with the fewest significant digits.
• Since the rules are different for each type of operation, when
they both occur in the same problem,
– complete the first operation and establish the correct
number of significant digits.
– Then proceed with the second and set the final answer
according to the correct number of significant digits based
on that operation
(1.245 + 6.34 + 8.179)/7.5
• Add
1.245 + 6.34 + 8.179
Then divide by 7.5 =
=