Download Scientific Notation

Scientific Notation Notes
Physical Science (Freshman Physics)
Objective
The student will be able to:
express numbers in scientific and
decimal notation.
How wide is our universe?
210,000,000,000,000,000,000,000
miles
(22 zeros)
This number is written in decimal
notation. When numbers get this
large, it is easier to write them in
scientific notation.
Scientific Notation
A number is expressed in scientific
notation when it is in the form
a x 10n
where a is between 1 and 10 (there can
be only 1 number in front of the
decimal)
and n is an integer
Write the width of the universe
in scientific notation.
210,000,000,000,000,000,000,000
miles
Where is the decimal point now?
After the last zero.
Where would you put the decimal to
make this number be between 1 and
10?
Between the 2 and the 1
2.10,000,000,000,000,000,000,000.
How many decimal places did you move
the decimal?
23
When the original number is more than
1, the exponent is positive.
The answer in scientific notation is
2.1 x 1023
1) Express 0.0000000902 in scientific
notation.
Where would the decimal go to make the
number be between 1 and 10?
9.02
The decimal was moved how many
places?
8
When the original number is less than 1,
the exponent is negative.
9.02 x 10-8
Write 28750.9 in scientific notation.
1.
2.87509 x 10-5
2.
2.87509 x 10-4
3.
2.87509 x 104
4.
2.87509 x 105
PERFORMING
CALCULATIONS IN
SCIENTIFIC
NOTATION
ADDITION AND SUBTRACTION
6
10
IF the exponents are the
4 x
6
+ _______________
3 x 10 same, we simply add or
subtract the numbers in
6
7 x 10
front and bring the
exponent down
unchanged.
4 x 106 If the exponents are
+ 3 x 105 NOT the same, we must
move a decimal to make
them the same.
Determine which of the numbers has the smaller exponent.
1. Change this number by moving the decimal place to the left and
raising the exponent, until the exponents of both numbers agree.
Note that this will take the lesser number out of standard form.
2. Add or subtract the coefficients as needed to get the new
coefficient.
3. The exponent will be the exponent that both numbers share.
4. Put the number in standard form.
6
10
6
10
4.00 x
4.00 x
6
5
+ .30 x 10
+ 3.00 x 10
Move the decimal on the smaller
number to the left and raise the
exponent !
Note: This will take the lesser number out of standard form.
6
10
6
10
4.00 x
4.00 x
6
5
+ .30 x 10
+ 3.00 x 10
6
4.30 x 10
Add or subtract the coefficients as
needed to get the new coefficient.
The exponent will be the exponent that
both numbers share.
Make sure your final answer is
in scientific notation. If it is
not, convert it to scientific
notation.
A Problem for you…
-6
10
2.37 x
-4
+ 3.48 x 10
Solution…
-6
002.37
2.37 x 10
-4
+ 3.48 x 10
Solution…
-4
0.0237 x 10
-4
+ 3.48
x 10
-4
3.5037 x 10
PERFORMING
CALCULATIONS
IN SCIENTIFIC
NOTATION
MULTIPLYING AND DIVIDING
Rule for Multiplication
When multiplying with scientific notation:
1. Multiply the coefficients together.
2. Add the exponents.
3. The base will remain 10.
(2 x 103) • (3 x 105) =
6 x 108
(4.6x108) (5.8x106) =26.68x1014
Notice: What is wrong with this example?
Although the answer is correct, the number is not
in scientific notation.
To finish the problem, move the decimal one
space left and increase the exponent by
one.
26.68x1014 = 2.668x1015
((9.2 x 105) x (2.3 x 107) =
21.16 x 1012 =
2.116 x 1013
(3.2 x 10-5) x (1.5 x 10-3) =
4.8 • 10-8
Rule for Division
When dividing with scientific notation
1. Divide the coefficients
2. Subtract the exponents.
3. The base will remain 10.
(8 • 106) ÷ (2 • 103) =
4 x 103
Please multiply the following numbers.
(5.76 x 102) x (4.55 x 10-4) =
(3 x 105) x (7 x 104) =
(5.63 x 108) x (2 x 100) =
(4.55 x 10-14) x (3.77 x 1011) =
(8.2 x10-6) x (9.4 x 10-3) =
Please multiply the following numbers.
(5.76 x 102) x (4.55 x 10-4) =
(3 x 105) x (7 x 104) =
(5.63 x 108) x (2 x 100) =
(4.55 x 10-14) x (3.77 x 1011) =
(8.2 x10-6) x (9.4 x 10-3) =
2.62 x 10-1
2.1 x 1010
1.13 x 109
1.72 x 10-2
7.71 x 10-8
Please divide the following numbers.
1. (5.76 x 102) / (4.55 x 10-4) =
2. (3 x 105) / (7 x 104) =
3. (5.63 x 108) / (2) =
4. (8.2 x 10-6) / (9.4 x 10-3) =
5. (4.55 x 10-14) / (3.77 x 1011) =
Please divide the following numbers.
1. (5.76 x 102) / (4.55 x 10-4) = 1.27 x 106
2. (3 x 105) / (7 x 104) = 4.3 x 100 = 4.3
3. (5.63 x 108) / (2 x 100) = 2.82 x 108
4. (8.2 x 10-6) / (9.4 x 10-3) = 8.7 x 10-4
5. (4.55 x 10-14) / (3.77 x 1011) = 1.2 x 10-25
Changing from Standard
Notation to Scientific Notation
Ex. 6800
6800
1. Move decimal to get
a single digit # and
count places moved
3 2 1
68 x 10
2. Answer is a single
digit number times
the power of ten of
places moved.
3
Ex. 4.5 x 10 -3
00045
3 2 1
Changing from Scientific
Notation to Standard Notation
1. Move decimal the same
number of places as the
exponent of 10.
(Right if Pos. Left if Neg.)
9.54x107 miles
If the decimal is moved left the power is positive.
If the decimal is moved right the power is negative.
1.86x107 miles
per second
What is Scientific Notation
(3 x 104)(7 x 10–5)
Multiply two numbers
in Scientific Notation
= (3 x 7)(10 4 x 10–5)
1.
= 21 x 10-1
2.
3.
4.
= 2.1 x 10 0
or 2.1
Put #’s in ( )’s Put
base 10’s in ( )’s
Multiply numbers
Add exponents of 10.
Move decimal to put
Answer in Scientific
Notation
A number expressed in scientific notation is
expressed as a decimal number between 1 and 10
multiplied by a power of 10 e( g, 7000 = 7 x 103 or
0.0000019 = 1.9 x 10 -6)
Why do we use it?
It’s a shorthand way of writing very large or very
small numbers used in science and math and
anywhere we have to work with very large or very
small numbers.
2.0 x 10 2 + 3.0 x 103
6.20 x 10–5
8.0 x 103
6.20
8.0
= 0.775 x
10-5
103
10 -8
= 7.75 x 10–9
DIVIDE USING SCIENTIFIC
NOTATION
.2 x 10 3 + 3.0 x 103
= .2+3 x 103
= 3.2 x
1.
2.
Scientific
Notation
Makes
These
Numbers
Easy
Divide the #’s &
Divide the powers of ten
(subtract the exponents)
Put Answer in Scientific
Notation
103
Addition and subtraction
Scientific Notation
1. Make exponents of 10 the same
2. Add 0.2 + 3 and keep the 103 intact
The key to adding or subtracting numbers
in Scientific Notation is to make sure the
exponents are the same.
2.0 x 10 7 - 6.3 x 105
2.0 x 10 7 -.063 x 10 7
= 2.0-.063 x 10 7
= 1.937 x 10 7
1. Make exponents of 10 the same
2. Subtract 2.0 - .063 and
keep the 107 intact