8.4 The Scientific Notation Objective The student will be able to express numbers in scientific and...
-
Upload
kristopher-whitehead -
Category
Documents
-
view
220 -
download
5
Transcript of 8.4 The Scientific Notation Objective The student will be able to express numbers in scientific and...
8.4 The Scientific Notation
ObjectiveThe student will be able to express numbers in scientific and decimal notation.
Helping us write really tiny or
really big numbers
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.
Mathematicians are Lazy!!!
They decided that by using powers of 10,
they can create short versions of tiny and
really big numbers.
Scientific NotationScientific Notation
A number is expressed in
scientific notation when it is in
the form
a x 10n
where a is between 1 and 10
and n is an integer (such as -12, -5, -1, 0, 4, 9, 17, etc.)
Rules to Scientific NotationParts: a x 10n
1. Coefficient (the a) – must be a number between 1 and 10
2. Exponent (the n) – a power of 10
3.4 x 106
Easier than writing 3,400,000
When we convert a decimal number to the scientific notation, if we move the decimal point to get the a in direction to the
Left Positive exponentRight Negative exponent
Numbers Greater Than 10 (big)
1. Find the number by moving the decimal point that is between 1 and 10
45,300,000 4.53
2. Write a positive exponent which is equal to the number of places you moved the decimal point to the left.
4.53 x 107
Numbers Less Than 1 (tiny)
1. Find the number by moving the decimal point that is between 1 and 10
0.000291 2.91
2. Write a negative exponent which is equal to the number of places you moved the decimal point to the right.
2.91 x 10-4
Write the width of the universe in scientific notation.
210,000,000,000,000,000,000,000miles
Where is the decimal point now?
After the last zero.
Where would you put the decimal to
make this number be between 1 and10?
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 is2.1 x 1023
Example 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
Additional Example 1: Writing Numbers in Scientific Notation
Think: The number is less than 1, so the exponent will be negative.
A. 0.00709 Think: The decimal needs to move 3 places to get a number between 1 and 10.
7.09 103
Write the number in scientific notation.
So 0.00709 written in scientific notation is 7.09 10–3.
A. 14 104
Multiply.
14.0 0 0 0Since the exponent is a positive 4, move the decimal point 4 places to the right.
Additional Example 1: Multiplying by Powers of 10
140,000
B. 3.6 105
0 0 0 0 3.6Since the exponent is a negative 5, move the decimal point 5 places to the left.
0.000036
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
Your Turn
1) When convert from scientific notation to decimal notation, if the power in the exponent is negative, then move the decimal point in the coefficient equal to the number of places to the LEFT.
2) When convert from scientific notation to decimal notation, if the poser in the exponent is positive, then move the decimal point in the coefficient equal to the number of places to the RIGHT.
Convert from Scientific Notation to Decimal Notation
Example 2 Express 1.8 x 10-4 in decimal notation.
0.00018
Example 3 Express 4.58 x 106 in decimal notation.
4,580,000
On the graphing calculator, scientific notation is done with the “2nd” and “LOG” buttons.
4.58 x 106 is typed 4.58 “2nd” and “LOG” buttons 6
A. 14 104
Write in Decimal Notation
14.0 0 0 0Since the exponent is a positive 4, move the decimal point 4 places to the right.
Additional Example 2 & 3: Convert Scientific Notation to Decimal Notation
140,000
B. 3.6 105
0 0 0 0 3.6Since the exponent is a negative 5, move the decimal point 5 places to the left.
0.000036
Example 4 Use a calculator to evaluate: 4.5 x 10-5
1.6 x 10-2
Type 4.5 -5 1.6 -2
You must include parentheses if you don’t use those buttons!!
(4.5 x 10 -5) (1.6 x 10 -2)
0.0028125Write in scientific notation.
2.8125 x 10-3
Example 5 Use a calculator to evaluate: 7.2 x 10-5
1.2 x 102
On the calculator, the answer is:6.E -7
The answer in scientific notation is 6 x 10-7
The answer in decimal notation is 0.0000006
Example 6 Use a calculator to evaluate (0.0042)(330,000).
On the calculator, the answer is
1386.
The answer in decimal notation is
1386
The answer in scientific notation is
1.386 x 103
Example 7 Use a calculator to evaluate (3,600,000,000)(23).
On the calculator, the answer is:
8.28 E +10
The answer in scientific notation is
8.28 x 10 10
The answer in decimal notation is
82,800,000,000
Write in PROPER scientific notation.(Notice the coefficient MUST be between 1 and 10) Example 8 Write 234.6 x 109 in scientific notation.
2.346 x 1011
Example 9 Write 0.0642 x 104 in scientific notation
on calculator: 642
6.42 x 10 2
Write (2.8 x 103)(5.1 x 10-7) in scientific notation.
1. 14.28 x 10-4
2. 1.428 x 10-3
3. 14.28 x 1010
4. 1.428 x 1011
Write 531.42 x 105 in scientific notation.
1. .53142 x 102
2. 5.3142 x 103
3. 53.142 x 104
4. 531.42 x 105
5. 53.142 x 106
6. 5.3142 x 107
7. .53142 x 108
Convert the following numbers into correct scientific notation:
35.9 x 103
556.67 x 104
22.7 x 10-3
0.0348 x 10-1
1845 x 105
123.4 x 1023
0.00345 x 107
3.59 x 104
5.5667 x 106
2.27 x 10-2
3.48 x 10-3
1.845 x 108
1.234 x 1025
3.45 x 104
Application Example Representing Large
Numbers 93,000,000 miles from the Earth to the Sun (sunlight takes 8 minutes to reach us)
93,000,000 = 9.3 x 10,000,000= 9.3 x 10 x 10 x 10 x 10 x 10 x 10 x 10= 9.3 x 107 (Decimal point moved 7 digits to the left)
Number between 1 and 10
Appropriate power of ten
Application Example
• Representing Small Numbers
0.000167To obtain a number between 1 and 10 we must move the decimal point to the right.
0.000167 = 1.67 10-4
10-4 = 1/10000 (one ten-thousandth)
A certain cell has a diameter of approximately 4.11 10-5 meters. A second cell has a diameter of 1.5 10-5 meters. Which cell has a greater diameter?
4.11 10-5 1.5 10-5
Compare the exponents.
Example 10: Comparing Numbers in Scientific Notation
Compare the values between 1 and 10.
The first cell has a greater diameter.
4.11 > 1.5
Notice that 4.11 10-5 > 1.5 10-5.
A star has a diameter of approximately 5.11 103 kilometers. A second star has a diameter of 5 104 kilometers. Which star has a greater diameter?
5.11 103 5 104 Compare the exponents.
The second star has a greater diameter.
Notice that 3 < 4. So 5.11 103 < 5 104
Example 11: Comparing Numbers in Scientific Notation
A star has a radius of approximately 6.74 104 kilometers. If the surface area of a sphere can be calculated as SA = 4r2. What is the surface area of the star in scientific notation?
SA = 4(6.74 104)2 = 4(6.74)2 (104)2
=570.86 108Is this your answer in Scientific Notation?
Note all the learned knowledge will be called.
Power of a Product and Power of a Power Property
= 570.86 108 = 5.7086 1010
Example 12: Converting the Product of Numbers into Scientific Notation