1. Section 2: Scientific Notation and Dimensional Analysis
Scientists use scientific methods to systematically pose and test solutions to questions and assess the results of the tests. K
What I Know W
What I Want to Find Out L
What I Learned

2. Essential Questions Why use scientific notation to express numbers?
How is dimensional analysis used for unit conversion?
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Scientific Notation and Dimensional Analysis

3. Vocabulary Review New quantitative data scientific notation
dimensional analysis
conversion factor
Scientific Notation and Dimensional Analysis
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4. Scientific Notation
Scientific notation can be used to express any number as a number between 1 and 10 (known as the coefficient) multiplied by 10 raised to a power (known as the exponent).
Carbon atoms in the Hope Diamond = 4.6 x 1023
4.6 is the coefficient and 23 is the exponent.
Count the number of places the decimal point must be moved to give a coefficient between 1 and 10.
The number of places moved equals the value of the exponent.
The exponent is positive when the decimal moves to the left and negative when the decimal moves to the right.
800 = 8.0 × 102
= 3.43 × 10–5
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Scientific Notation and Dimensional Analysis

5. Scientific Notation Addition and subtraction
Exponents must be the same.
Rewrite values to make exponents the same.
Example: x x 1017, you must rewrite one of these numbers so their exponents are the same. Remember that moving the decimal to the right or left changes the exponent.
2.840 x x 1018
Add or subtract coefficients.
Example: x x 1017 = 3.2 x 1018
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Scientific Notation and Dimensional Analysis

6. Scientific Notation Problem Response SOLVE FOR THE UNKNOWN
Move the decimal point to give a coefficient between 1 and 10. Count the number of places the decimal point moves, and note the direction.
Move the decimal point six places to the left.
Move the decimal point eight places to the right.
Write the coefficients, and multiply them by 10n where n equals the number of places moved. When the decimal point moves to the left, n is positive; when the decimal point moves to the right, n is negative. Add units to the answers.
a × 106 km
b. 2.8 × 10-8 g/cm3
Use with Example Problem 2.
Problem
Write the following data in scientific notation.
a. The diameter of the Sun is 1,392,000 km.
b. The density of the Sun’s lower atmosphere is g/cm3 .
Response
ANALYZE THE PROBLEM
You are given two values, one much larger than 1 and the other much smaller than 1. In both cases, the answers will have a coefficient between 1 and 10 multiplied by a power of 10.
Scientific Notation and Dimensional Analysis
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7. Scientific Notation EVALUATE THE ANSWER
The answers are correctly written as a coefficient between 1 and 10 multiplied by a power of 10. Because the diameter of the Sun is a number greater than 1, its exponent is positive. Because the density of the Sun’s lower atmosphere is a umber less than 1, its exponent is negative.
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Scientific Notation and Dimensional Analysis

8. Scientific Notation Multiplication and division
To multiply, multiply the coefficients, then add the exponents.
Example: (4.6 x 1023)(2 x 10-23) = 9.2 x 100
To divide, divide the coefficients, then subtract the exponent of the divisor from the exponent of the dividend.
Example: (9 x 107) ÷ (3 x 10-3) = 3 x 1010
Note: Any number raised to a power of 0 is equal to 1: thus, 9.2 x 100 is equal to 9.2.
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Scientific Notation and Dimensional Analysis

9. Multiplying and Dividing Numbers in Scientific Notation
SOLVE FOR THE UNKNOWN
Problem A:
State the problem.
a. (2 × 103) × (3 × 102)
Multiply the coefficients.
2 × 3 = 6
Add the exponents.
3 + 2 = 5
Combine the parts.
6 × 105
Problem B:
b. (9 × 108) ÷ (3 × 10-4)
Divide the coefficients.
9 ÷ 3 = 3
Use with Example Problem 3.
Problem
Solve the following problems.
a. (2 × 103) × (3 × 102)
b. (9 × 108) ÷ (3 × 10-4)
Scientific Notation and Dimensional Analysis
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10. Multiplying and Dividing Numbers in Scientific Notation
EVALUATE THE ANSWER
To test the answers, write out the original data and carry out the arithmetic. For example, Problem a becomes 2000 × 300 = 600,000, which is the same as 6 × 105.
SOLVE FOR THE UNKNOWN
Problem B continued:
Subtract the exponents.
8 − (−4) = = 12
Combine the parts.
3 × 1012
Scientific Notation and Dimensional Analysis
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11. Dimensional Analysis
Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move, or convert, from one unit to another.
A conversion factor is a ratio of equivalent values having different units.
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Scientific Notation and Dimensional Analysis

12. Dimensional Analysis Writing conversion factors
Conversion factors are derived from equality relationships, such as 1 dozen eggs = 12 eggs.
Percentages can also be used as conversion factors. They relate the number of parts of one component to 100 total parts.
Using conversion factors
A conversion factor must cancel one unit and introduce a new one.
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Scientific Notation and Dimensional Analysis

13. USING CONVERSION FACTORS
KNOWN
UNKNOWN
Length = 6 Egyptian cubits
length= ? g
7 palms = 1 cubit
1 palm = 4 fingers
1 finger = mm
1 m = 1000 mm
Use with Example Problem 4.
Problem
In ancient Egypt, small distances were measured in Egyptian cubits. An Egyptian cubit was equal to 7 palms, and 1 palm was equal to 4 fingers. If 1 finger was equal to mm, convert 6 Egyptian cubits to meters.
Response
ANALYZE THE PROBLEM
A length of 6 Egyptian cubits needs to be converted to meters.
Scientific Notation and Dimensional Analysis
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14. USING CONVERSION FACTORS
EVALUATE THE ANSWER
Each conversion factor is a correct restatement of the original relationship, and all units except for the desired unit, meters, cancel.
Scientific Notation and Dimensional Analysis
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15. Review Essential Questions Vocabulary
Why use scientific notation to express numbers?
How is dimensional analysis used for unit conversion?
Vocabulary
scientific notation
dimensional analysis
conversion factor
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Scientific Notation and Dimensional Analysis