Scientific Notation and Dimensional Analysis (2.2)

Transcription

1 Scientific Notation and Dimensional Analysis (2.2) Cornell Notes 5 questions 5-sentence summary

2 Scientific Notation Makes writing very small numbers and very large numbers easier For example, or 60,220,000,000,000,000,000,000 Uses base-10 numbers written with exponents 10 2 = = 100, =.001

3 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

4 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

5 Metric Prefixes in Scientific Notation/Base-10 Tera(T): = 1,000,000,000,000 Giga (G): 10 9 = 1,000,000,000 Mega (M): 10 6 = 1,000,000 Kilo (k): 10 3 = 1,000 Hecto(h): 10 2 = 100 Deca(D or da): 10 1 = 10 Base (no prefix): 10 0 = 1 Deci(d): 10-1 = 0.1 Centi(c): 10-2 = 0.01 Milli(m): 10-3 = Micro (μ): 10-6 = Nano(n): 10-9 = Pico (p): =

6 How write in SCIENTIFIC NOTATION FORMAT 1) Take decimal and move it so that it is behind only one digit ) Count how many spaces you moved the decimal and that is the exponent of 10 Moved LEFT POSITIVE Moved RIGHT NEGATIVE x x x 10-5

7 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, decreasethe exponent (by number of shifts) 5) If adjusting decimal to the LEFT, increase the exponent

8 Example ( ) + ( ) ( ) + ( ) ( ) OR ( ) + ( ) ( )

9 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, decreasethe exponent (by number of shifts) 7) If adjusting decimal to the LEFT, increase the exponent

10 Example MULTIPLICATION ( ) ( ) ( ) ( ) (22.116) ( ) DIVISION 4 ( ) -3 ( ) ( ) ( ) (0.25) (10 4-(-3) )

11 Using scientific notation with metric Example (exponents in CF are always positive): Gm= m 1 Gm= 10 9 m Example: 4.05 μl = hl hl Example: Tg= mg mg g 10 mg Tg = mg 1 Tg 1 g

12 Dimensional Analysis

13 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

14 Conversion Factors Written in ratio form to equal 1 Examples 1 foot 12 inches or 12 inches 1 foot 4 quarters 1 dollar or 1 dollar 4 quarters 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

15 Conversion Factors for Metric Prefix Units Make conversion factors of any prefix equal to base (whether g, m or L) For example, 1 km = 10 3 m 1 ng= 10-9 g C.F. s should be from base prefix or prefix base Enter 1 for the largerunit (prefix or base) Enter the base-10 value for the smallerunit but the exponent should ALWAYS BE POSITIVE

16 2 Representative Methods FRACTION METHOD 1 foot 24 inches = feet 12 inches BOX METHOD

17 Dimensional Analysis and Factor Labeling Factor-Labeling using the units to cross cancel out one unit for another

18 Steps of Dimensional Analysis Sample problem: Convert 24 feet yards 1. Begin with INITIAL VALUE 24 feet 2. Find your conversion factor and ratios 3 ft = 1 yd 1 yd 3 ft 3 ft or 1 yd 3. Multiply initial value by conversion factor so that the starting value unit is on the bottomof the ratio so that cross cancel out 1 yd 24 feet 3 ft

19 Steps of Dimensional Analysis continued 4. Cross cancel out the units that are diagonal from each other 1 yd 24 feet 3 ft 5. Multiply initial value with all numbers on top 6. Then divide by all numbers on the bottom, one by one 1 yd feet = = 8 3 ft 3 7. Write the remained (uncrossed out) unit next to the number answer 1 yd 24 feet = 8 yd 3 ft

20 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 kl cl 1. Set up all the conversion factors you ll need, comparing each prefix to the base kl 10 L 1 kl = 10 L or 3 10 L 1 kl cl 10 L 1 cl = 10 L or L 1 cl

21 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) kl 3 10 L -2 1 kl 1 cl 10 L 3. Initial and all top numbers are multiplied, then divide bottom numbers 4. Solve, then write new unit L 1 cl kl = 1 kl 10 L = cl