Transcript Scientific Notation

Scientific Notation
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Copyright © 2011 Lynda Greene Aguirre
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Table of Contents
Click on links to skip to each section listed
1. Concepts & Definitions
2. Change Scientific to Standard Notation
3. Change Standard to Scientific Notation
4. Special Cases
5. Practice Problems
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Concepts and Definitions
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Why use Scientific Notation?
In scientific applications, there are often very
large or very small numbers such as these
Very large number: 53,000,000,000,000,000
Very small number: 0.000000000000418
Note: The numbers above are written in STANDARD NOTATION
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Why use Scientific Notation?
Trying to do calculations using such long
numbers can be time consuming.
So a different way of writing these
numbers was developed which
is commonly called scientific notation.
Also called “exponential notation”
in some textbooks
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Why use Scientific Notation?
The process (in a nutshell) is to shorten the
number by changing the long string of leading
or trailing zeros into a power of 10.
53,000,000,000,000,000
“leading”
zeros
0.000000000000418
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“trailing”
zeros
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Scientific Notation
Question: Why use a power of 10?
Positive powers create “trailing” digits
Note: Each new zero
actually represents a
decimal place, it
might not be the
number “zero” when
you are using it in
calculations.
The power on
the ten is the
same as the
number of
zeros on the
right.
Answer: You can use it to move the
decimal point to the right.
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Scientific Notation
Question: Why use a power of 10?
Negative Powers create “leading” digits
If you line up the
“1”s, notice that
each negative
power moves the
decimal point to the
left of its previous
position
Answer: You can use it to move the
decimal point to the left.
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What does Scientific Notation look like?
Positive or negative sign
indicates BIG or SMALL
number
Format:
THE COEFFICIENT
Multiplication
symbol
THE EXPONENT
Note: If the coefficient has a decimal point after the
first digit, it is called “normalized”.
The position of the decimal point does not
change any mathematical procedures
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The following numbers are all written in scientific notation
A missing decimal
Parentheses are
point means it’s to
Negative
sometimes used to
the right of the power means indicate multiplication
coefficient.
it’s a very
(2 means 2.0)
small number Many calculators show
scientific notation this way
Zeros in between non-zero
digits are included in the
coefficient
The decimal might
not be after the
first digit
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EE3 means 10
to the third
power
10
Is it a Positive or Negative Power?
Big numbers with TRAILING ZEROS have a
positive power (usually not shown).
430,000,000
Trailing Zeros
will have
no sign on the
power
No sign HERE means
it’s positive
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Positive or Negative Sign?
Small numbers with LEADING ZEROS have a
negative power
0.00000043
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Leading Zeros
will have a
negative power
on the “10”
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Scientific Notation
Moving the decimal point
Decimal
moved
from
HERE
5 digits to the right
To HERE
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Note: A decimal
point to the
right of any
number does
not have to be
included
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Changing from Scientific
Notation into Standard
Notation
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Step by Step Process
1. Write
Positive power down the
move right
coefficient
The power of ten tells you how far to move
2. Move
decimal
point two
places to
the right
You can drop the
decimal point if it’s
to the right of the
last digit
Note: You may need to fill in some zeros if there
are not enough digits in the original number
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Step by Step Process
POSITIVE
power
Write down
the
coefficient
Move the
decimal point
six places to
the right
Fill in some
if there are not
enough digits in
the original
number
Drop the decimal point if it’s behind the last digit
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Step by Step Process
POSITIVE
power
Write down
the
coefficient
Move the
decimal point
four places to
the right
Fill in some
if there are not
enough digits in
the original
number
Drop the decimal point if it’s behind the last digit
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Step by Step Process
POSITIVE
power
Write down
the
coefficient
Move the
decimal point
two places to
the right
No need to fill in
There are
enough digits in
the original
number
Keep the decimal point because it’s not behind the last digit
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Step by Step Process
Write down
the
coefficient
NOTE: Leave room on the left
side to move the decimal
Look at
the
NEGATIVE
power
Move the
decimal point
three places
to the LEFT
Fill in some
if there are not
enough digits in
the original
number
Keep the decimal point because it’s not behind the last digit
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Step by Step Process
NEGATIVE
power
Write down
the
coefficient
Move the
decimal point
three places
to the LEFT
Fill in
Keep the decimal point because it’s not behind the last digit
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Step by Step Process
Write down
the
coefficient
NOTE: Leave room on the left
to move the decimal
NEGATIVE
power
Move the
decimal point
four places to
the LEFT
Fill in
Keep the decimal point because it’s not behind the last digit
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Step by Step Process
NEGATIVE
power
Write down
the
coefficient
Move the
decimal two
places to the
LEFT
Fill in
Keep the decimal point because it’s not behind the last digit
Practice Problems
Table of contents
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Step by Step Process
NEGATIVE
power
1. Write
down the
coefficient
Move the
decimal point
one place to
the LEFT
3. No need to fill
in
There
are enough digits
in the original
number
Keep the decimal point because it’s not behind the last digit
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Changing from Scientific
Notation into Standard
Notation
Special Cases
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Zero power
ZERO
power
1. Write
down the
coefficient
This is the answer
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2. Decimal
point doesn’t
need to be
moved
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No Decimal on Coefficient
Note: If the coefficient does not have a
decimal point visible, it is “understood” to
be to the right of it.
REWRITE THE COEFFICIENT TO SHOW THE
POSITION OF THE DECIMAL POINT
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No Decimal on Coefficient
POSITIVE
power
1. Write
down the
coefficient
Move 3
places to the
right
Fill in
Drop the decimal point if it’s behind the last digit
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Calculator Format
Note: If a different format of scientific
notation is used, you can change the format
or work the problem in its original form
YOU CAN REWRITE THE PROBLEM IN NORMALIZED FORM
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Calculator Format
POSITIVE
power
1. Write
down the
coefficient
Move 3
places to the
right
Fill in
Drop the decimal point if it’s behind the last digit
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Different Multiplication Format
Note: If a different format of scientific
notation is used, you can change the format
or work the problem in its original form
YOU CAN REWRITE THE PROBLEM IN NORMALIZED FORM
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Different Multiplication Format
Example:
POSITIVE
power
1. Write
down the
coefficient
Move 3
places to the
right
Fill in
Drop the decimal point if it’s behind the last digit
Practice Problems
Table of contents
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Changing from Standard
Notation into Scientific
Notation
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Review
Number
Type
SMALL
NUMBERS
BIG
NUMBERS
Location of zeros
(decimal places)
left side of the
coefficient
right side of the
coefficient
Sign of the
Power
negative power
positive power
Error Prevention Tip:
Knowing where the zeros belong will prevent students from
moving the decimal in the wrong direction
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Standard to Scientific Notation
Now that we know how to do this process in one
direction, it is simple to reverse it.
The coefficient includes
everything except the
trailing zeros
The zeros are on the
right hand side, so the
power will be positive
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Standard to Scientific Notation
1. Write the coefficient in normalized form
(i.e. decimal point after the first digit)
2. Write “x 10” next to it
(i.e.the power of ten)
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Standard to Scientific Notation
3. This is a big number, so the power will be positive
Note: You can omit the “+” sign if you like,
but it’s not wrong to write it down
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Count decimal places
Decimal point was not showing,
which means it’s behind the last digit
4. Count the number of digits between the new
position and its old position
To HERE
There are 8 digits between the
new and old positions
This gives us the power: +8
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Decimal
moved
from
HERE
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Putting the pieces together
1. Write the coefficient in Normalized Form
(decimal point after the first digit)
2. Write “x 10” next to it
3. Sign of the power?
zeros on the right?-positive
zeros on the left?-negative
4. How far did the decimal point move?
(count digits between old and new decimal points)
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Putting the pieces together
Extra digits are
to the RIGHT of
the coefficient
1. Write the coefficient in Normalized Form
+6
3. Sign of
the
2. Write “x 10” next to it
power?
4. How far did the decimal point move?
There are 6 digits between the
new and old positions
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Putting the pieces together
Extra digits are
to the LEFT of
the coefficient
1. Write the coefficient in Normalized Form
-4
3. Sign of
the
2. Write “x 10” next to it
power?
4. How far did the decimal point move?
There are 4 digits between the
new and old positions
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Putting the pieces together
Extra digits are
to the RIGHT of
the coefficient
1. Write the coefficient in Normalized Form
+8
3. Sign of
the
2. Write “x 10” next to it
power?
4. How far did the decimal point move?
There are 8 digits between the
new and old positions
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Putting the pieces together
Extra digits are
to the LEFT of
the coefficient
1. Write the coefficient in Normalized Form
-2
3. Sign of
the
2. Write “x 10” next to it
power?
4. How far did the decimal point move?
Practice Problems
Table of contents
There are 2 digits between the new and
old positions
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Practice Problems
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Practice Problems A
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Practice Problems B
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End of Tutorial