Scientific Notation and Dimensional

Analysis (2.2)

• Cornell Notes

– 5 questions

– 5-sentence summary

Scientific Notation

• Makes writing very small numbers and very

large numbers easier

– For example, .000000000000000000000000021

or 60,220,000,000,000,000,000,000

• Uses base-10 numbers written with exponents

– 102 = 100

– 105 = 100,000

– 10-3 = .001

How write in SCIENTIFIC NOTATION

FORMAT

1) Take decimal and move it so that it is behind

only ONE DIGIT

2) Decimal number maybe only be between 1

and 10

2) Count how many spaces you moved the

decimal

Move LEFT � exponent of 10 is POSITIVE

Move RIGHT �exponent of 10 is NEGATIVE

Convert Back to Standard Number

Format

• Move the number of spaces depending on the

exponent of the base 10

POSITIVE� move RIGHT

NEGATIVE � move LEFT

Metric Prefixes in Scientific Notation/Base-10

• Tera (T): 1012 =

1,000,000,000,000

• Giga (G): 109 =

1,000,000,000

• Mega (M): 106 =

1,000,000

• Kilo (k): 103 = 1,000

• Hecto (h): 102 = 100

• Deca (D or da): 101 = 10

• Base (no prefix): 100 = 1

• Deci (d): 10-1 = 0.1

• Centi (c): 10-2 = 0.01

• Milli (m): 10-3 = 0.001

• Micro (μ): 10-6 =

0.000001

• Nano (n): 10-9 =

0.000000001

• Pico (p): 10-12 =

0.000000000001

How write in SCIENTIFIC NOTATION

FORMAT

1) Take decimal and move it so that it is behindonly one digit

- 40 � 4.0

- 9835.29 � 9.83529

- 0.0000782 � 7.82

2) Count how many spaces you moved the decimal and that is the exponent of 10

Moved LEFT � POSITIVE

Moved RIGHT � NEGATIVE- 40 � 4.0 x 101

- 9835.29 � 9.83529 x 103

- 0.0000782 � 7.82 x 10-5

Scientific Notation Operations

• ADDITION and SUBTRACTION

1) Convert each value so that the exponent is the same

2) Add the decimal numbers together with the same

base-10 exponent

3) Adjust base-10 number so decimal number is between

1-10 (if needed)

4) If adjusting decimal to the RIGHT, decrease the

exponent (by number of shifts)

5) If adjusting decimal to the LEFT, increase the exponent

Example

• (9.37×104) + (9.91×103)

• (9.37×104) + (0.991×104)

• (9.37 + 0.991) × 104

• 10.361×104

• 1.0361×105

• OR

• (93.7×103) + (9.91×103)

• (93.7 + 9.91) × 103

• 103.61×103

• 1.0361×105

Scientific Notation Operations

• MULTIPLICATION and DIVISION

1) Multiply or divide decimal numbers together

2) Then, multiply or divide base-10 numbers

3) Multiplying base-10 numbers: ADD exponents

4) Dividing base-10 numbers: SUBTRACT exponents

5) Adjust base-10 number so decimal number is between

1-10 (if needed)

6) If adjusting decimal to the RIGHT, decrease the

exponent (by number of shifts)

7) If adjusting decimal to the LEFT, increase the exponent

Example

MULTIPLICATION

• (2.91×10-3) × (7.60×105)

• (2.91×7.60) × (10-3×105)

• (22.116) × (10-3+5)

• 22.116×102

• 2.2116×103

DIVISION

•

• (1.20÷4.80) × (104÷10-3)

• (0.25) × (104-(-3))

• 0.25×107

• 2.5×106

4

-3

(1.20×10 )

(4.80×10 )

Using scientific notation with metric

• Example (exponents in CF are always positive):

____ Gm = ____ m � 1 Gm = 109 m

• Example:

4.05 μL = ____ hL � 4.05×10-8 hL

• Example:

1.38×10-5 Tg = ____ mg � 1.38×1010 mg

12 3-5 1010 g 10 mg

1.38×10 Tg × × =1.38×10 mg1 Tg 1 g

Dimensional Analysis

What is dimensional analysis?

• A method of problem-solving focused on units

used to describe matter (mass, volume,

density, speed, temperature and much more).

• Use of conversion factors (C.F.), a ratio or

fraction of equal values in different units, is

important

– For example “12 inches = 1 foot” is a conversion

factor

Conversion Factors

• Written in ratio form to equal 1

– Examples

– Ratios can be written with either unit on the top

or bottom in fraction form

– Can only use values that are equivalent (equal)

like “4 quarters = 1 dollar”

1 foot 12 inches or

12 inches 1 foot

4 quarters 1 dollar or

1 dollar 4 quarters

Conversion Factors for Metric Prefix

Units

• Make conversion factors of any prefix equal to base

(whether g, m or L)

• For example,

– ______ km = _______ m

– ______ ng = _______ g

• C.F.’s should be from base � prefix or prefix � base

• Enter “1” for the larger unit (prefix or base)

• Enter the base-10 value for the smaller unit but the

exponent should ALWAYS BE POSITIVE

1

1

103

10-9

2 Representative Methods

1 foot24 inches × = feet

12 inches

FRACTION METHOD

BOX METHOD

Dimensional Analysis and Factor

Labeling

• Factor-Labeling – using the units to cross

cancel out one unit for another

Steps of Dimensional Analysis

• Sample problem: Convert 24 feet � yards

1. Begin with INITIAL VALUE

2. Find your conversion factor and ratios

3. Multiply initial value by conversion factor so that the

starting value unit is on the bottom of the ratio so that

cross cancel out

24 feet

3 ft = 1 yd1 yd

3 ft

3 ftor

1 yd

1 yd24 feet

3 ft×

Steps of Dimensional Analysis

continued4. Cross cancel out the units that are diagonal from each

other

5. Multiply initial value with all numbers on top

6. Then divide by all numbers on the bottom, one by one

7. Write the remained (uncrossed out) unit next to the

number answer

1 yd24 feet

3 ft×

1 yd 24×124 feet × = = 8

3 ft 3

1 yd24 feet × = 8 yd

3 ft

Multi-Step Dimensional Analysis

• Sometimes you have to use more than one

conversion factor to convert units, especially from

one metric prefix unit to another (“prefix to prefix”

• Sample: Convert 0.592 kL � cL

1. Set up all the conversion factors you’ll need,

comparing each prefix to the base3

3

3

1 kL 10 L1 kL = 10 L or

10 L 1 kL

-2-2

-2

1 cL 10 L1 cL = 10 L or

10 L 1 cL

Multi-Step Dimensional Analysis

continued

2. Start with INITIAL, then set up the ratios so that the units cross cancel out (bottom unit matches the previous ratio’s top unit)

3. Initial and all top numbers are multiplied, then divide bottom numbers

4. Solve, then write new unit.

310 L

0.592 kL1 kL

× -2

1 cL

10 L×

3 3

-2 -2

10 L 1 cL 0.592 10 10.592 kL × × =

1 kL 10 L 1 10

× ×

×

4= 5.92×10 cL