numbers that have been measured.
I can . . .
4 marbles
3 marbles
+
=
7 marbles
Addition and Subtraction have always been done with COUNTED numbers.�i.e. EXACT values
Multiplication and Division have always been done with COUNTED numbers.�i.e. EXACT values
7 squares
4 squares
7 x 4 = 28 squares
http://www.boreas-online.com/catalog/pics/Green_Apple.jpg
15 Apples ÷ 3 baskets = 5 apples per basket
http://www.eurocosm.com/Application/images/wicker-baskets/garden-vegetable-basket-lg.jpg
We now need to learn how to�add and subtract,�multiply and divide �MEASURED numbers.
i.e. Numbers with ERROR
The rules ARE different!
http://baixaki.ig.com.br/site/w214.htm
Thirty-five years ago,�I learned that the dinosaurs�went extinct 65,000,000 years ago
65,000,000 years� + 35 years
= 65,000,035 years
13,800,000,000 ± 100,000,000 years old
We are 93,000,000 miles from the sun!�(491,040,000,000 feet)
= 491,040,000,000 – 10�= 491,039,999,990
http://www.wizards.de/~frank/astro/
0
1
0.781 m
0
1
0.36 m
0.36 m x 0.781 m = 0.28116 m2
Area = Base x Height
± .01
± .001
± .00001
Multiplying and Dividing
Measured Numbers
Significant�Figures��“Sig Figs”
0
1
2
3
4
2.47
± .01
Certain
Guess
All three numbers are important.
There are three “Significant Figures”.
2.47 kilometers
2,470 meters
2,470,000 millimeters
.000008 kilometers
.008 meters
8 millimeters
3 sig figs
3 sig figs
3 sig figs
1 sig fig
1 sig fig
1 sig figs
Zeros are NOT “significant figures” when they are merely place holders.
0
1
0
1
0
1
0.7 ± .1
0.67 ± .01
0.675 ± .001
1 sig fig
2 sig figs
3 sig figs
0
1
0.4 ± .1
0.40 ± .01
0.400 ± .001
0
1
0
1
1 sig fig
2 sig figs
3 sig figs
Zeros ARE significant when they make the measurement more precise!
2.06 ± .01
0
1
2
3
4
3 sig figs
Zeros ARE significant �when they are “sandwiched”
http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Oreo.jpg/800px-Oreo.jpg
23,560,000��6,700��2,004,600��0.000182��.0060045��94.000
82,000,020��4,000��3,004,600.00200��5,000.0��.000,000,045��1,001
Identify the Number that is a Guess
Count the number of “Sig Figs”
A chain is only as strong�as its weakest link!
Rule for�multiplying and dividing�MEASURED numbers
Rule for�multiplying and dividing�MEASURED numbers
Round your answer to the�same number of SIG FIGS�as the LEAST number of SIG FIGS.
11 x 40 = 440
2 Sig Figs
= 400
1 Sig Fig
1 Sig Fig
0
1
0.781 m
0
1
0.36 m
0.36 m x 0.781 m = 0.28116 m2
2 Sig Figs
= 0.28 m2
3 Sig Figs
2 Sig Figs
Area = Base x Height
Round your answer to the�same number of SIG FIGS�as the LEAST number of SIG FIGS.
= 36,250 cm/s
= 811.0627719 mi/hr
2(145) cm�.008 s
=
= 800 mi/hr
Distance
Time
÷
1200 ÷ 335 = 3.582089…
2 Sig Figs
= 3.6
3 Sig Figs
2 Sig Figs
Rule for�multiplying and dividing�MEASURED numbers
Round your answer to the�same number of SIG FIGS�as the LEAST number of SIG FIGS.
3.456 x 95 = 328.32
= 330
Round to the correct number of sig figs.
Adding and Subtracting
Measured Numbers
2.5 ± .1
2.6 ± .1
2.5 ± .3
2.3 ± .3
2.5 ± .1
Accuracy
Precision
Degree of�Precision
The position of the guess.
45,000
.03
Thousands position
Hundredths position
Degree of Precision
.7 .70
Which number is more precise?
0
1
0
1
Degree of Precision
8,280 8,000
Which number is more precise?
9,000
8,000
7,000
9,000
8,000
7,000
however,
+
=
2 in
4.25 in
6.25 in
+
=
2.00 in
4.25 in
6.25 in
6 in
23,560,000��6,700��2,004,600��0.000182��.0060045��94.000
82,000,020��4,000��3,004,600.00200��5,000.0��.000,000,045��1,001
Identify the Number that is a Guess
10,000’s
100’s
100’s
1,000,000th
10,000,000th
Find the DEGREE OF PRECISION
1,000th
Rule for�adding and subtracting�MEASURED numbers
A chain is only as strong�as its weakest link!
Rule for�adding and subtracting�MEASURED numbers
Round your answer to the�same DEGREE of PRECISION�as the LEAST PRECISE number.
4.3 - 4.187 = .113
Least precise
= .1
7,600 + 6,489 = 14,089
Least precise
= 14,100
Rule for�adding and subtracting�MEASURED numbers
Round your answer to the�same DEGREE of PRECISION�as the LEAST PRECISE number.
3.426 + 9.5 = 12.926
= 12.9
Round to the correct degree of precision.
Counted numbers are considered to be EXACT �i.e. an infinite number of sig figs.
Example:
Ninety pencils, each one is 12.9 grams.
90 x 12.9 = 1,161 grams total mass
3 sig figs
sig figs
∞
3 sig figs
= 1,160 grams total mass
Exactly 12 inches in 1 foot.
Conversions WITHIN a measurement system are considered to be EXACT �i.e. an infinite number of sig figs.
Exactly 3 ft in 1 yard.
Exactly 5280 ft in 1 mile.
Exactly 100 centimeters in 1 meter.
Exactly 1,000 meters in 1 kilometer.
Exactly? 2.54 centimeters in 1 inch.
Approximately 1.6 kilometers in 1 mile.
July 1, 1959
How many inches is 2,160 ft?
2,160 x 12 = 25,920 in
3 sig figs
sig figs
∞
3 sig figs
= 25,900 in
Conversions WITHIN a measurement system are considered to be EXACT �i.e. an infinite number of sig figs.
256 x 39.0625 = 10,000
3 sig figs
6 sig figs
Using Scientific Notation to Properly Show Sig Figs.
= 1.00 x 104
3 sig figs
TWO RULES
Addition and Subtraction�IDENTIFY�DEGREE OF PRECISION
Multiplication and Division�COUNT�SIGNIFICANT FIGURES
1)
2)
numbers that have been measured.
I can . . .