### Scientific Notation

```Scientific Notation- Why?
 Also used to maintain the correct
number of significant figures.
 An alternative way of writing numbers
that are very large or very small.
 characteristic will be positive
 Ex: 6.022X1023
 602200000000000000000000
Method to express really big or small numbers.
Format is
Mantissa
Decimal part of
original number
6.02 x
Base Power
x
Decimal
you moved
23
10
We just move the decimal point around.
602000000000000000000000
Using the Exponent
Key
on a Calculator
EE
EXP
EE or EXP means “times 10 to the…”
How
How to
to type
type out
out 6.02
6.02 xx10
102323::
6
0
.
2
EE
2
3
Don’t do it like this…
6
WRONG!
0
.
yx
2
2
3
…or like this…
6
.
0
WRONG!
2
x
1
…or like this:
6
.
0
0
EE
2
3
TOO MUCH WORK.
2
x
1
0
yx
2
3
1.2 x 105
Example:
 2.8 x 1013
Type this calculation in like this:
1
.
2
EE
5
2
.
8
EE
1

3
=
Calculator gives… 4.2857143 –09
or… 4.2857143 E–09
This is NOT written… 4.3–9
But instead is written… 4.3 x 10–9
or
4.3 E –9
Converting Numbers to
Scientific Notation
0.00002205
1
2
3
4
2.205 x
-5
10
5
In scientific notation, a number is separated into two parts.
The first part is a number between 1 and 10.
The second part is a power of ten.
Scientific Notation- How
 To convert TO scientific notation,
 move decimal to left or right until you
have a number between 1 & 10.
 Count # of decimal places moved
 If original is smaller than 1 than
characteristic will be negative
 If original is larger than 1 the
 If original number is negative, don’t
forget to put the – back on the front!
Example:
 If you move the decimal
the characteristic will be
 If you move the decimal
the characteristic will be
to the left
positive
to the right
negative
 Convert 159.0 to scientific notation
 1.59 x 102
 Convert -0.00634
 -6.34 x 10-3
1. 17600.0
2. 0.00135
1.76 x 104
1.35 x 10-3
1.02 x 101
3.
4.
5.
6.
7.
8.
10.2
-67.30
4.76
- 0.1544
301.0
-0.000130
-6.730 x 101
4.76 x 100
-1.544 x 10-1
3.010 x 102
-1.30 x 10-4
- keep trailing
Expand Scientific Notation
 If characteristic is positive move
decimal to the right
 If the characteristic is negative move
the decimal to the left
 Ex: 8.02 x 10-4
 0.000802
 -9.77 x 105
 -977,000
7.5 x 10-6  - 8.7 x 10-4
= -6.525 x 10-9
4.35 x 106  1.23 x 10-3
= 5.3505 x 103 or 5350.5
report -6.5 x 10-9 (2 sig. figs.)
report 5.35 x 103 (3 sig. figs.)
5.76 x 10-16  9.86 x 10-4
= 5.84178499 x 10-13
report 5.84 x 10-13 (3 sig. figs.)
8.8 x 1011  3.3 x 1011
= 2.904 x 1023
report 2.9 x 1023 (2 sig. figs.)
6.022 x 1023  - 5.1 x 10-8 = -3.07122 x 1016
report -3.1 x 1016 (2 sig. figs.)
Correcting Scientific Notation
 The mantissa needs to have one place holder to
the left of the decimal (3.67 not 36.7), look at the
absolute value
 Count how many decimals places you move and
then you will increase or decrease the
characteristic accordingly
 If you must INCREASE the mantissa, DECREASE
the characteristic
 If you must DECREASE the mantissa, INCREASE
the characteristic
 Be careful with negative characteristics!
 If you decrease 10-3 by two the new value is 10-5
Confused? Example
 To correct 955 x 108
 Convert 955 to 9.55 – (move decimal left 2 times).
 Did we increase or decrease 955?
 955 is larger than 9.55 so we decreased it -so we
must increase 8 by 2.
 955 x 108 becomes 9.55 x 1010
 -9445.3 x 10-6
 Convert -9445.3 to -9.445 (move decimal left 3
times).
 Did we increase or decrease -9445.3? (absolute
value)
 We decreased the absolute value by 3 decimal
places, so we must increase the characteristic
 955 x 108 becomes 9.55 x 1010
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
36.7 x 101
-0.015 x 103
75.4 x 10-1
-14.5 x 102
0.123 x 104
97723 x 10-2
851.6 x 10-3
94.2 x 10-4
-0.012 x 103
966 x 10-1
3.76 x 102
-1.5 x 10-5
7.54 x 100
-1.45 x 103
1.23 x 103
9.7723 x 102
8.516 x 10-3
9.42 x 10-4
-1.2 x 101
9.66 x 10-1
- keep trailing
Rule for Multiplication
Calculating with Numbers Written in Scientific
Notation
1. MULTIPLY the mantissas
3. Correct the result to proper scientific notation when needed
Sample Problem: (4 x 10-3) (3 x 10-3)
(4) x (3) = 12
(-3) + (4) = 1 or 101
1.2 x 102
Rule for Division
Calculating with Numbers Written in Scientific
Notation
1. DIVIDE the mantissas
2. SUBTRACT the characteristic of the denominator from
the characteristic of the numerator
3. Correct the result to proper sci. notation if needed
Sample Problem: Divide 7.2 x 10-4 by -8 x 105
.
(7.2) . (-8) = -0.9
(-4) - 5) = -1 or 10-9
-0.9 x 10-9
Correct: -9 x 10-10
1.
2.
3.
4.
(2 x 104)(3 x 10-3)
(5 x 10-3)(4 x 10-4)
(6 x104)(-7 x 10-5)
(-4.5 x 10-2)(2 x 10-7)
1.
2.
3.
4.
5.
(8 x 10-5) / (2 x 10-3)
(4 x 103) / (8 x 10-3)
(6 x 10-7)/(3 x 108)
(4.5 x 104) / (9.0 X 10-12)
((2 X 103)(4X10-2)) /
((6 X 10-9)(4 X 105))
6 x 101
20 X 10-7  2 x 10-6
-42 x 10-1  4.2 x 100
-0 x 10-9
4 x 10-2
5.0 x 105
2 x 101
5 x 1015
3.3 x 104
- keep trailing
Calculating with Numbers Written in Scientific
Notation
In order to add or subtract numbers written in scientific
notation, you must express them with the same power of 10.
(Same characteristic). Then correct to proper scientific notation.
Sample Problem: Add 5.8 x 103 and 2.16 x 104
27.4 x 103
2.74 x 104
Exercise: Add 8.32 x 10-7 and 1.2 x 10-5
1.28 x 10-5
(5.8 x 103) + (21.6 x 103) =
```