MATH for SCIENCE Scientific Notation Scientists ~ A. Deal with: Some very large numbers Some extremely small numbers These numbers can be quite cumbersome to work with. To make it easier scientists frequently use “Scientific Notation.” B. Scientific Notation: A numerical shorthand frequently used for writing very large and extremely small numbers. C. Converting Decimal format to Scientific Notation format: Scientific Notation sets up numbers with: a. Only the leading, non-zero digit/number to the left of the decimal point in the units place. b. All the remaining numbers are placed to the right of the decimal point. c. Then, that number is multiplied by 10n.

d. The power/exponent “n” will correspond to: 1. the number of places. 2. the direction the decimal point was moved. e. The power “n” is: 1. positive (+) when the original number is greater than 1 2. negative (-) when the original number is less than 1. f. For numbers greater than 1: 1. count the number of places the decimal point was moved to the left until you have only one non-zero number/digit to the left of the decimal point. 2. that number becomes the power/exponent that goes to the upper right of the 10n.

g. Examples: # Moving the Decimal Pt. Answer i. 98765 9.8765 9.8765 x 104 4 3 2 1 ii. 123 1.23 1.23 x 102 2 1 iii. 4680 4.680 4.680 x 103 3 2 1

h. For numbers less than 1: i. count the number of places the decimal point was moved to the right until you have only one non-zero number/digit to the left of the decimal point. ii. count the number of places the decimal point was moved to the right until you have only one non-zero number/digit to the left of the decimal point. iii. Examples: # Moving the Decimal Pt. Answer 0.00012 0.0001.2 1.2 x 10 -4 1 2 3 4 0.0000000345 0.00000003.45 3.45 x 10 -8 1 2 3 4 5 6 7 8 0.067 0.06.7 6.7 x 10 -2 1 2

D. Converting Scientific Notation format to Decimal format: 1. For numbers with 10+n : a. Move the decimal point to the right to make the number bigger (greater than 1). b. When you move the decimal point and there are no numbers left, fill the counting loops in with zeros. 2. Examples: # Moving the Decimal Pt. Answer 7.43 x 105 7.43000. 743,000. 1 2 3 4 5 2.153 x 102 2.15.3 215.3 1 2 6.8 x 104 6.8000. 68,000. 1 2 3 4

# Moving the Decimal Pt. Answer 3. For numbers with 10-n : Move the decimal point to the left to make the number smaller (less than 1). 4. Examples: # Moving the Decimal Pt. Answer 3.75 x 10 -2 03.75 0.0375 2 1 8.4 x 10 -5 .00008.4 0.000084 5 4 3 2 1 1.26 x 10 -3 .001.26 0.00126 3 2 1

a. Step 1: Multiply the two leading numbers together. II. Computations with Scientific Notation ~ When multiplying or dividing with two or more numbers in Scientific Notation format, the process is done in two stages. A. Multiplication: 1. Stage 1 has 2 steps: a. Step 1: Multiply the two leading numbers together. b. Step 2: Multiply the base 10 numbers together. (Remember, this means you just add the powers/exponents.) c. Example: (2.5 x 103) (5.0 x 102) (2.5 x 5.0) (103 x 102) 12.5 x 105

2. Stage 2 has 2 steps: These two steps are determined by which format, decimal or Scientific Notation, is required for the answer. Decimal Format Scientific Notation Format Step 3: Move the decimal point the number Step 3: Take the decimally formatted first of places and the direction indicated number and change it to by the x 10n exponent. Scientific Notation. Step 4: Fill in the blank loops/spaces with Step 4: Multiply the number from step 3 zeros. with the base 10 number from step 12.5 x 105 12.5 x 105 12.50000. (1.25 x 101) (105) 1 2 3 4 5 1,250,000. 1.25 x 106

B. Examples: 1. (3.3 x 10 -2) (4.5 x 105) (3.3 x 4.5) (10 -2 x 105) Decimal Format Scientific Notation Format 14.85 x 103 14.85 x 103 14.850. (1.485 x 101) (103) 1 2 3 14,850. 1.485 x 104 2. (8.2 x 10-3) (3.6 x 10-2) (8.2 x 3.6) (10-3 x 10-2) 29.52 x 10-5 29.52 x 10-5 29.52 x 10-5 .00029.52 (2.952 x 101) (10-5) 5 4 3 2 1 0.0002952 2.952 x 10-4

3. (6.95 x 104) (2.3 x 10-7) (6.95 x 2.3) (104 x 10-7) 15.985 x 10-3 Decimal Format Scientific Notation Format 15.985 x 10-3 15.985 x 10-3 .015.985 (1.5985 x 101) (10-3) 3 2 1 0.015985 1.5985 x 10-2

C. Division: 1. Stage 1 has 2 steps: Step 1: Divide the two leading numbers, then Step 2: Divide the base 10 numbers (Remember: this means you just subtract the exponents/powers.) 2. Stage 2: Convert the result of stage 1 to either or both decimal format &/or Scientific Notation. D. Examples: 1. 96.24 x 10-3 → 96.24 x 10-3 → 80.2 x 10-3 – (-5) = 80.2 x 102 = 8.02 x 103 or 8020 1.2 x 10-5 1.2 10-5 2. 8.2 x 105 → 8.2 x 105 → 1.2 x 103 or 1,200 6.0 x 102 6.0 102 3. 1.92 x 104 → 1.92 x 104 → 0.3048 x 107 = (3.048 x 10-1) (107) = 3.048 x 106 6.3 x 10-3 6.3 10-3 or 3,048,000

E. Addition & Subtraction: 1. To add or subtract any number in Scientific Notation, each number MUST: a. Be converted back to decimal format. b. Line up the decimal point. c. Then, add or subtract the numbers. F. Examples: 1. 1.4 x 103 + 3.0516 x 104 + 9.723 x 102 1.4 x 103 1400. 3.0516 x 104 30516. 9.723 x 102 + 972.3 32,888.3 3.28883 x 104 2. 4.0125 x 103 - 6.375 x 102 4.0125 x 103 4012.5 6.375 x 102 - 637.5 3375.0 3.3750 x 103

3. 1.3842 x 102 + 4.965 x 101 + 8.6 x 10-2 1.3842 x 102 138.42 4.965 x 101 49.65 8.6 x 10-2 + .086 188.156 1.88156 x 102 4. 7.385 x 10-2 - 8.126 x 10-3 7.385 x 10-2 0.07386 8.126 x 10-3 - 0.008126 0.065724 6.5724 x 10-2