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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:

C.
A numerical shorthand frequently used for writing very large and extremely
small numbers.
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.
2.
the number of places.
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:
#
i. 98765
Moving the Decimal Pt.
9.8765
Answer
9.8765 x 104
ii. 123
1.23
1.23 x 102
iii. 4680
4.680
4.680 x 103
4321
21
321
h.
i.
For numbers less than 1:
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
1234
0.0000000345
0.00000003.45
3.45 x 10
0.067
0.06.7
6.7 x 10
12345678
12
-8
-2
D.
Converting Scientific Notation format
to Decimal format:
1.
For numbers with 10+n :
a.
b.
2.
Move the decimal point to the right to make the
number bigger (greater than 1).
When you move the decimal point and there are no
numbers left, fill the counting loops in with zeros.
Examples:
#
Moving the Decimal Pt.
Answer
7.43 x 105
743,000.
7.43000.
12345
2.153 x 102
2.15.3
215.3
6.8 x 104
6.8000.
68,000.
12
1234
3. For numbers with 10-n :
Move the decimal point to the left to
make the number smaller (less than 1).
4. Examples:
#
3.75 x 10 -2
Moving the Decimal Pt.
03.75
Answer
0.0375
21
8.4 x 10 -5
.00008.4
0.000084
1.26 x 10 -3
.001.26
0.00126
54321
321
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
12.50000.
12.5 x 105
(1.25 x 101) (105)
12345
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)
14.85 x 103
Decimal Format
14.85 x 103
14.850.
Scientific Notation Format
14.85 x 103
(1.485 x 101) (103)
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
Decimal Format
29.52 x 10-5
.00029.52
Scientific Notation Format
29.52 x 10-5
(2.952 x 101) (10-5)
0.0002952
2.952 x 10-4
123
54321
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
15.985 x 10-3
.015.985
Scientific Notation Format
15.985 x 10-3
(1.5985 x 101) (10-3)
0.015985
1.5985 x 10-2
321
C.
Division:
1.
Stage 1 has 2 steps:
a.
b.
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.
D.
Stage 2: Convert the result of stage 1 to either or both decimal
format &/or Scientific Notation.
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
188.1561.88156 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