MATH for SCIENCEMATH for SCIENCE Scientific Notation Scientific Notation

Scientists ~Scientists ~A.A. Deal with:Deal with:

Some very large numbers Some very large numbers Some extremely small numbersSome extremely small numbers

These numbers can be quite cumbersome to work with. To These numbers can be quite cumbersome to work with. To make it easier scientists frequently use “Scientific make it easier scientists frequently use “Scientific

Notation.”Notation.”B.B. Scientific Notation:Scientific Notation:

A numerical shorthand frequently used for writing very large and A numerical shorthand frequently used for writing very large and extremely small numbers.extremely small numbers.

C.C. Converting Decimal format to Scientific Notation Converting Decimal format to Scientific Notation format: format:

Scientific Notation sets up numbers with: Scientific Notation sets up numbers with: a. Only the leading, non-zero digit/number to the left of the decimal point a. Only the leading, non-zero digit/number to the left of the decimal point in the units place. in the units place. b. All the remaining numbers are placed to the right of the decimal point.b. All the remaining numbers are placed to the right of the decimal point.c. Then, that number is multiplied by 10c. Then, that number is multiplied by 10nn..

d. The power/exponent “d. The power/exponent “nn” will correspond to:” will correspond to:

1.1. the number of places.the number of places.

2. 2. the direction the decimal point was moved.the direction the decimal point was moved.

e. The power “e. The power “nn” is:” is:

1.1. positive (+)positive (+) when the original number is when the original number is greater than 1greater than 1

2.2. negative (-)negative (-) when the original number is when the original number is less than 1.less than 1.

f.f. For numbers greater than 1:For numbers greater than 1:

1.1. count the number of places the decimal point count the number of places the decimal point was was

moved to the left until you have only one non-moved to the left until you have only one non-zero zero

number/digit to the left of the decimal point.number/digit to the left of the decimal point.

2.2. that number becomes the power/exponent that that number becomes the power/exponent that goes to goes to

the upper right of thethe upper right of the 10 10nn..

g. g. Examples:Examples:

## Moving the Decimal Pt. Moving the Decimal Pt. AnswerAnswer

i. 98765i. 98765 9 9..87658765 99..8765 x8765 x 101044

4 3 2 14 3 2 1

ii. 123ii. 123 1 1..2323 11..23 x23 x 10 1022

2 12 1

iii. 4680iii. 4680 4 4..680680 44..680 x680 x 101033

3 2 13 2 1

h.h. For numbers less than 1:For numbers less than 1: i.i. count the number of places the decimal point was count the number of places the decimal point was

moved to moved to the right until you have only one non-zero the right until you have only one non-zero number/digit to number/digit to the left of the decimal point.the left of the decimal point.

ii.ii. count the number of places the decimal point was count the number of places the decimal point was moved to moved to the right until you have only one non-zero the right until you have only one non-zero number/digit to number/digit to the left of the decimal point.the left of the decimal point.

iii.iii. Examples:Examples:

## Moving the Decimal Pt.Moving the Decimal Pt. AnswerAnswer

00..0001200012 0.00010.0001..22 11..2 x 102 x 10 --

44

1 2 3 41 2 3 4

00..00000003450000000345 0.000000030.00000003..4545 33..45 x 1045 x 10 -8-8

1 2 3 4 5 6 7 81 2 3 4 5 6 7 8

00..067067 0.060.06..77 6.7 x 106.7 x 10 --

22

1 21 2

D.D. Converting Scientific Notation Converting Scientific Notation format to Decimal format:format to Decimal format:

1.1. For numbers withFor numbers with 10 10+n+n : : a.a. Move the decimal point to the right to make the Move the decimal point to the right to make the number bigger (greater than 1). number bigger (greater than 1). b. When you move the decimal point and there b. When you move the decimal point and there

are no are no numbers left, fill the counting loops in with zeros.numbers left, fill the counting loops in with zeros.

2. Examples:2. Examples:## Moving the Decimal Moving the Decimal

Pt.Pt. AnswerAnswer77..43 x 1043 x 1055 7.437.43000000..

743,000743,000.. 1 2 3 4 51 2 3 4 5

22..153 x 10153 x 1022 2.152.15..33 215215..33 1 21 2

66..8 x 108 x 1044 6.86.8000000.. 68,00068,000.. 1 2 3 41 2 3 4

3. For numbers with3. For numbers with 10 10-n-n ::

Move the decimal point to the left to Move the decimal point to the left to

make the number smaller (less than make the number smaller (less than 1).1).

4. Examples:4. Examples:

## Moving the Decimal Pt.Moving the Decimal Pt. AnswerAnswer 33..75 x 1075 x 10 -2-2 0033..7575

00..03750375 2 12 1

88..4 x 104 x 10 --55 ..000000008.48.4 00..000084000084 5 4 3 2 15 4 3 2 1

11..26 x 1026 x 10 --33 ..00001.261.2600..0012600126

3 2 1 3 2 1

II. Computations with Scientific Notation ~II. Computations with Scientific Notation ~ When multiplying or dividing with two or When multiplying or dividing with two or

more numbers in more numbers in Scientific Notation format, the process is done in Scientific Notation format, the process is done in

two stages.two stages.A.A. Multiplication:Multiplication:

1.1. Stage 1 has 2 steps:Stage 1 has 2 steps: a.a. Step 1:Step 1: Multiply the two leading Multiply the two leading

numbers together.numbers together. b.b. Step 2:Step 2: Multiply the base 10 Multiply the base 10

numbers together.numbers together. (Remember, this means you just add the (Remember, this means you just add the

powers/exponents.)powers/exponents.) c.c. Example:Example:

(2.5 x 10(2.5 x 1033) (5.0 x 10) (5.0 x 1022))(2.5 x 5.0) (10(2.5 x 5.0) (1033 x 10 x 1022))12.5 x 1012.5 x 1055

2.2. Stage 2 has 2 steps:Stage 2 has 2 steps:These two steps are determined by which format, decimal These two steps are determined by which format, decimal

or Scientific or Scientific Notation, is required for the answer.Notation, is required for the answer.

Decimal FormatDecimal Format Scientific Notation Format Scientific Notation FormatStep 3: Move the decimal point the number Step 3: Take the decimally Step 3: Move the decimal point the number Step 3: Take the decimally

formatted firstformatted first of places and the direction indicated number and change it to of places and the direction indicated number and change it to by the x 10by the x 10nn exponent. exponent. Scientific Notation. Scientific Notation.Step 4: Fill in the blank loops/spaces with Step 4: Multiply the number from Step 4: Fill in the blank loops/spaces with Step 4: Multiply the number from

step 3 step 3 zeros.zeros. with the base 10 number from step with the base 10 number from step

1212..5 x 105 x 1055 1212..5 x 105 x 1055

12.512.500000000.. (1(1..25 x 1025 x 1011) (10) (1055)) 1 2 3 4 51 2 3 4 5

1,250,0001,250,000.. 11..25 x 1025 x 1066

B.B. Examples:Examples:

1.1. (3.3 x 10 (3.3 x 10 -2-2) (4.5 x 10) (4.5 x 1055)) (3.3 x 4.5) (10 (3.3 x 4.5) (10 -2-2 x 10 x 1055) ) 1414..85 x 1085 x 1033

Decimal FormatDecimal Format Scientific Notation FormatScientific Notation Format1414..85 x 1085 x 1033 1414..85 x 1085 x 1033

14.8514.8500.. (1(1..485 x 10485 x 1011) (10) (1033)) 1 2 31 2 3

14,85014,850.. 11..485 x 10485 x 1044

2. (8.2 x 102. (8.2 x 10-3-3) (3.6 x 10) (3.6 x 10-2-2)) (8.2 x 3.6) (10(8.2 x 3.6) (10-3-3 x 10 x 10-2-2)) 2929..52 x 1052 x 10-5-5

Decimal FormatDecimal Format Scientific Notation FormatScientific Notation Format2929..52 x 1052 x 10-5-5 2929..52 x 1052 x 10-5-5

..00000029.5229.52 (2(2..952 x 10952 x 1011) (10) (10-5-5)) 5 4 3 2 15 4 3 2 1

00..00029520002952 22..952 x 10952 x 10-4-4

3.3. (6.95 x 10 (6.95 x 1044) (2.3 x 10) (2.3 x 10-7-7))

(6.95 x 2.3) (10(6.95 x 2.3) (1044 x 10 x 10-7-7))

1515..985 x 10985 x 10-3-3

Decimal FormatDecimal Format Scientific Notation Scientific Notation FormatFormat

1515..985 x 10985 x 10-3-3 1515..985 x 10985 x 10-3-3

..0015.98515.985 (1(1..5985 x 105985 x 1011) (10) (10-3-3)) 3 2 13 2 1

00..015985015985 11..5985 x 105985 x 10-2-2

C.C.Division:Division:1.1. Stage 1 has 2 steps:Stage 1 has 2 steps:

a.a. Step 1: Divide the two leading numbers, thenStep 1: Divide the two leading numbers, thenb.b. Step 2: Divide the base 10 numbers Step 2: Divide the base 10 numbers

(Remember: this means you just subtract the (Remember: this means you just subtract the exponents/powers.)exponents/powers.)

2. Stage 2: Convert the result of stage 1 to either or both 2. Stage 2: Convert the result of stage 1 to either or both decimal decimal format &/or Scientific Notation.format &/or Scientific Notation.

D. Examples:D. Examples:1.1. 96.24 x 1096.24 x 10-3-3 → → 96.2496.24 x x 1010-3-3 → 80.2 x 10 → 80.2 x 10-3 – (-5)-3 – (-5) = 80.2 x 10 = 80.2 x 1022 = 8.02 x 10 = 8.02 x 1033 or 8020 or 8020

1.2 x 101.2 x 10-5-5 1.2 10 1.2 10-5 -5

2.2. 8.2 x 108.2 x 1055 → → 8.28.2 x x 101055 → 1.2 x 10 → 1.2 x 1033 or 1,200 or 1,2006.0 x 106.0 x 1022 6.0 10 6.0 1022

3.3. 1.92 x 101.92 x 1044 → → 1.921.92 x x 101044 → 0.3048 x 10 → 0.3048 x 1077 = (3.048 x 10 = (3.048 x 10-1-1) (10) (1077) = 3.048 x 10) = 3.048 x 1066 6.3 x 106.3 x 10-3-3 6.3 10 6.3 10-3-3 or or 3,048,000 3,048,000

E.E. Addition & Subtraction:Addition & Subtraction:1.1. To add or subtract any number in Scientific Notation, To add or subtract any number in Scientific Notation, each each number MUST:number MUST:

a.a. Be converted back to decimal format.Be converted back to decimal format.b.b. Line up the decimal point.Line up the decimal point.c.c. Then, add or subtract the numbers.Then, add or subtract the numbers.

F.F. Examples:Examples: 1.1. 1.4 x 10 1.4 x 1033 + 3.0516 x 10 + 3.0516 x 1044 + 9.723 x 10 + 9.723 x 1022

1.4 x 101.4 x 1033 14001400..3.0516 x 103.0516 x 1044 30516 30516..9.723 x 109.723 x 1022 + + 972 972..33

32,88832,888..3333..28883 x 1028883 x 1044

2. 4.0125 x 102. 4.0125 x 1033 - 6.375 x 10 - 6.375 x 1022

4.0125 x 104.0125 x 1033 40124012..556.375 x 106.375 x 1022 - 637- 637..55

33753375..0033..3750 x 103750 x 1033

3. 3. 1.3842 x 101.3842 x 1022 + 4.965 x 10 + 4.965 x 1011 + 8.6 x 10 + 8.6 x 10-2-2

1.3842 x 101.3842 x 1022 138138..4242

4.965 x 104.965 x 1011 49 49..6565

8.6 x 108.6 x 10-2-2 + + ..086086

188188..15615611..88156 x 1088156 x 1022

4.4. 7.385 x 10 7.385 x 10-2-2 - 8.126 x 10 - 8.126 x 10-3-3

7.385 x 107.385 x 10-2-2 0 0..0738607386

8.126 x 108.126 x 10-3-3 - 0- 0..008126008126

00..06572406572466..5724 x 105724 x 10-2-2