Download Calculations with Sig

Calculations with Sig Figs
• When you make calculations using data with a specific level of uncertainty, it is important that you also report your answer with the appropriate level of uncertainty (i.e., the appropriate number of significant digits).
Calculations with Sig Figs
Calculations with Sig Figs
• You should use all available digits, both significant and insignificant, during intermediate calculations, and round to the nearest significant digit only when reporting the final result.
Calculations with Sig Figs
• Note that significant digits are only important when reporting your final answer. Calculations with Sig Figs
• There are different rules for determining the appropriate number of significant digits in the results of different mathematical operations.
We'll look at each of the cases we will use, but first lets check and make sure we are all on the same page so far.
Title Page
Which of the following numbers has the fewest significant digits?
1 a) 1.20000
b) 1.2350 x 105
c) 0.00235
d) 9,684
A
B
C
D
Which of the following numbers has the fewest significant digits?
{
6
a) 1.20000
b) 1.2350 x 105
c) 0.00235
d) 9,684
Which of the following numbers has the fewest significant digits?
5
{
{
6
a) 1.20000
b) 1.2350 x 105
c) 0.00235
d) 9,684
Which of the following numbers has the fewest significant digits?
5
{
{
6
b) 1.2350 x 105
c) 0.00235
d) 9,684
{
a) 1.20000
3
Which of the following numbers has the fewest significant digits?
5
{
{
6
c) 0.00235
d) 9,684
3
{
b) 1.2350 x 105
{
a) 1.20000
4
Which of the following numbers has the fewest significant digits?
5
{
{
6
c) 0.00235
d) 9,684
3
{
b) 1.2350 x 105
{
a) 1.20000
4
NOTES...LIVE AND IN COLOR!!!
I would suggest you take notes like this:
Sig­fig calculation rules
NO ROUNDING UNTIL THE END!!!
1)
ex:
2)
ex:
3)
ex:
4)
ex:
I would suggest you take notes like this:
Sig­fig calculation rules
NO ROUNDING UNTIL THE END!!!
Example:
1) Multiplication & Division Rule:
Round to the least TOTAL sig­figs:
6.5315
x 4.5
29.39175 = 29
2) Addition & Subtraction Rule:
Round to the least decimal places AFTER the decimal point:
123.15
+ 45.123
Example:
168.273 = 168.27
3) When in doubt, write it out:
Avoid scientific notation unless the exponent is the same.
Example:
(1.2 x 10­5) + 1.2
0.000012
(1.2 x 10­5) = 0.000012
+1.2
1.200012 = 1.2
4) Math Functions don't count:
Only count actual data in determining how many sig­figs to round to.
Calculate the average of 1.26, 1.676, and 1.34556.
1.26
1.676
= 1.43
4.28156 ÷ 3 = 1.427186
+ 1.34556
4.28156
1) Multiplication & Division Rule:
Round to the least TOTAL sig­figs
When quantities are multiplied or divided, the number of significant figures in the answer is equal to the number of significant figures in the quantity with the smallest number of significant figures. Example:
1.23 * 4567.89
When quantities are multiplied or divided, the number of significant figures in the answer is equal to the number of significant figures in the quantity with the smallest number of significant figures. Example:
1.23 * 4567.89
1.23 has three significant digits
When quantities are multiplied or divided, the number of significant figures in the answer is equal to the number of significant figures in the quantity with the smallest number of significant figures. Example:
1.23 * 4567.89
4567.89 has six significant digits.
When quantities are multiplied or divided, the number of significant figures in the answer is equal to the number of significant figures in the quantity with the smallest number of significant figures. Example:
1.23 * 4567.89
The result will have the smaller of these ­ three significant digits. When quantities are multiplied or divided, the number of significant figures in the answer is equal to the number of significant figures in the quantity with the smallest number of significant figures. Example:
1.23 * 4567.89
Your calculator produces 5618.5047 as a result
Round it to three significant digits and report: 5.62 x 103
OR
5,620
Title Page
6.5315
x 4.5
1 A
B C a) 23.39175
b) 29.392
c) 29.4
d) 29
D
6.5315
x 4.5
29.39175
Your calculator should give you
a) 23.39175
b) 29.392
c) 29.4
d) 29
6.5315
x 4.5
Your calculator should give you
29.39175
4.5 has fewer sig figs (2), so we round our answer to two sig figs
29
a) 23.39175
b) 29.392
c) 29.4
d) 29
2) Addition & Subtraction Rule:
Round to the least decimal places AFTER the decimal point
When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
1.234 + 567.89
When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
1.234 + 567.89
1.234 has three digits right of the decimal point
When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
1.234 + 567.89
567.89 has two When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
1.234 + 567.89
The result will have the smaller of these ­ two digits right of the decimal point. When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
This is easier to see if you line up the figures in a column:
1.234
+ 567.89
When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
This is easier to see if you line up the figures in a column:
1.234
+ 567.89
569.124
When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.
Example:
This is easier to see if you line up the figures in a column:
1.234
+ 567.89
569.124
We then round to two significant digits right of the decimal point and report 569.12.
Title Page
123.15
+ 45.123
1 A
B a) 168.27
b) 168.273
c) 168.3
d) 168
C D 123.15
+ 45.123
Your calculator should give you
168.273
a) 168.27
b) 168.273
c) 168.3
d) 168
123.15
+ 45.123
Your calculator should give you
168.273
123.15 has fewer decimal places (2), so we round our answer to two decimal places
168.27
a) 168.27
b) 168.273
c) 168.3
d) 168
3) When in doubt, write it out:
Avoid scientific notation unless the exponent is the same.
Be especially careful with numbers which are given in scientific notation.
Example:
1.2 + (3.45 x 10­4) Be especially careful with numbers which are given in scientific notation.
Example:
1.2 + (3.45 x 10­4) The best way to solve this problem is to write the numbers in a column in ordinary notation:
1.2
+ 0.000345
Be especially careful with numbers which are given in scientific notation.
Example:
1.2 + (3.45 x 10­4) The best way to solve this problem is to write the numbers in a column in ordinary notation:
1.2
+ 0.000345
1.200345
Be especially careful with numbers which are given in scientific notation.
Example:
1.2 + (3.45 x 10­4) The best way to solve this problem is to write the numbers in a column in ordinary notation:
1.2
+ 0.000345
1.200345
Only one digit right of the decimal is significant. Report your result as 1.2
You may also convert all numbers into scientific notation with the same exponent:
Example:
(1.23x105) + (4.56x106) + (7.89x107)
You may also convert all numbers into scientific notation with the same exponent:
Example:
(1.23x105) + (4.56x106) + (7.89x107)
1.23 x 105
45.6 x 105
+ 789. x 105
You may also convert all numbers into scientific notation with the same exponent:
Example:
(1.23x105) + (4.56x106) + (7.89x107)
1.23 x 105
45.6 x 105
+ 789. x 105
835.83 x 105
You may also convert all numbers into scientific notation with the same exponent:
Example:
(1.23x105) + (4.56x106) + (7.89x107)
1.23 x 105
45.6 x 105
+ 789. x 105
5
835.83 x 10
The full answer would be 835.83 x 105, but the last two digits are not significant. You may also convert all numbers into scientific notation with the same exponent:
Example:
(1.23x105) + (4.56x106) + (7.89x107)
1.23 x 105
45.6 x 105
+ 789. x 105
5
835.83 x 10
The full answer would be 835.83 x 105, but the last two digits are not significant. Report your result as 836. x 105 or 8.36 x 107.
Title Page
(1.2 x 10­5) + 1.2
1 a) 1.200012
b) 1.2
c) 2.4
d) 2.4 X 10­5
A
B
C
D
(1.2 x 10­5) + 1.2
(1.2 x 10­5) = 0.000012
a) 1.200012
b) 1.2
c) 2.4
d) 2.4 X 10­5
(1.2 x 10­5) + 1.2
(1.2 x 10­5) = 0.000012
0.000012
+1.2
a) 1.200012
b) 1.2
c) 2.4
d) 2.4 X 10­5
(1.2 x 10­5) + 1.2
(1.2 x 10­5) = 0.000012
0.000012
+1.2
1.200012
a) 1.200012
b) 1.2
c) 2.4
d) 2.4 X 10­5
(1.2 x 10­5) + 1.2
(1.2 x 10­5) = 0.000012
0.000012
+1.2
1.200012
a) 1.200012
b) 1.2
c) 2.4
d) 2.4 X 10­5
4) Math Functions don't count:
Only count actual data in determining how many sig­figs to round to.
Title Page
Calculate the average of 1.26, 1.676, and 1.34556.
average = (X1 + X2 + ... + XN) / N
a) 1
b) 1.427
c) 1.43
d) 1.42719
1 A
B
C
D
Calculate the average of 1.26, 1.676, and 1.34556.
average = (X1 + X2 + ... + XN) / N
a) 1
b) 1.427
c) 1.43
d) 1.42719
Calculate the average of 1.26, 1.676, and 1.34556.
average = (X1 + X2 + ... + XN) / N
1.26
1.676
+ 1.34556
4.28156
a) 1
b) 1.427
c) 1.43
d) 1.42719
Calculate the average of 1.26, 1.676, and 1.34556.
average = (X1 + X2 + ... + XN) / N
1.26
1.676
+ 1.34556
4.28156
4.28156 / 3 = 1.427186
a) 1
b) 1.427
c) 1.43
d) 1.42719
CONGRATULATIONS!
BONUS MATERIAL FOR
ENRICHMENT & REVIEW
OUTSIDE OF CLASS!
If you take the time to study and follow along with the following examples, then you are a physics super
!!!
Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
E
M
D
A
S
Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
a
r
e
n
t
h
e
s
e
s
E
M
D
A
S
Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
a
r
e
n
t
h
e
s
e
s
E
x
p
o
n
e
n
t
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M
D
A
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Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
a
r
e
n
t
h
e
s
e
s
E
x
p
o
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e
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t
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M
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t
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D
A
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Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
a
r
e
n
t
h
e
s
e
s
E
x
p
o
n
e
n
t
s
M
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p
l
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D
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v
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d
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A
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Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
a
r
e
n
t
h
e
s
e
s
E
x
p
o
n
e
n
t
s
M
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p
l
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D
i
v
i
d
e
A
d
d
S
Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
a
r
e
n
t
h
e
s
e
s
E
x
p
o
n
e
n
t
s
M
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l
t
i
p
l
y
D
i
v
i
d
e
A
d
d
S
u
b
t
r
a
c
t
Multiple Operations:
Compute the number of significant digits to retain in the same order as the operations: Do you remember:
P
E
M
D
A
S
E
O
P
L
E
A
T
A
N
Y
O
N
U
T
S
N
D
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P
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Multiple Operations:
Multiple Operations:
•
It is important to keep insignificant digits during the intermediate calculations, •
And round to the correct number of significant digits only when reporting the final answer. Multiple Operations:
Multiple Operations:
•
It is important to keep insignificant digits during the intermediate calculations, •
And round to the correct number of significant digits only when reporting the final answer. In the following example, insignificant digits that are retained in intermediate calculations are shown in italics. Example:
(22.56)2 * (1.89 + 115.7553)
Example:
(22.56)2 * (1.89 + 115.7553)
First take the square of what is inside the first set of parentheses. Because the base number has four significant digits, the result will have four significant digits. For now, make a note of this, but write down the entire number.
four SIG figs
= (508.9536) * (1.89 + 115.7553)
Example:
four SIG figs
= (508.9536) * (1.89 + 115.7553)
Now we must do the addition before the multiplication, because of the parentheses. 1.89
+ 115.7553
117.6453
Example:
four SIG figs
= (508.9536) * (1.89 + 115.7553)
The sum will have two significant figures right of the decimal, but again we'll write down the whole number, since we still have the product to perform.
two SIG figs right of decimal
= (508.9536) * (117.6453) Example:
two SIG figs right of decimal
= (508.9536) * (117.6453) To calculate the product, we use all of the figures, but note that the first term (from the square) has four significant figures while the second term has five. Since this is a product, we keep only the lower number, or four total.
four SIG figs
(508.9536) * (117.6453) = 59875.99895808
Example:
four SIG figs
= (508.9536) * (117.6453) = 59875.99895808
So report your final result as:
= 5.988 x 104
NOTE: If you can do a problem like this, then you
understand everything you need to know about
performing calculations with significant figures.