Scientific Notation
Objectives:
1.
2.
Convert measurements to scientific notation.
Perform calculations involving scientific notation.
3.1 Scientific Notation
1
Quiz
Answer these questions. Show all your work
1.  Write in scientific notation: 32000000
2.  Write in scientific notation: 0.000467
3.  Express 5.43 x 10-3 as a standard number
4. (4.5 x 10-14) x (5.2 x 103) = ?
5  (6.1 x 105) ÷ (1.2 x 10-3) = ?
6  (3.74 x 10-3)4 = ?
3.1 Scientific Notation
2
Quiz
Answer these questions. Show all your work
1.  Write in scientific notation: 32000000
2.  Write in scientific notation: 0.000467
3.  Express 5.43 x 10-3 as a standard number
4. (4.5 x 10-14) x (5.2 x 103) = ?
5  (6.1 x 105) ÷ (1.2 x 10-3) = ?
6  (3.74 x 10-3)4 = ?
1. 3.2 x 107;
2. 4.67 x 10-4;
3. 0.00543
4. 2.3 x 10-10
5 5.1 x 108
6. 1.96 x 10-10
3.1 Scientific Notation
3
Do Now
1.
Scientists have found that there are
602,000,000,000,000,000,000,000 atoms in 1.008g
of element hydrogen
Numbers that are extremely large are hard to handle!
a)  How do you use a shorthand method to
express this very large number?
b) How do you name this shorthand method
(shorthand notation)?
3.1 Scientific Notation
4
Do Now
2.
Scientists have found that the mass of a water
molecule is 0.000,000,000,000,000,000,029,91 g
Numbers that are extremely small are hard to handle!
a)  How do you use a shorthand method to
express this very small number?
b) How do you name this shorthand method
(shorthand notation)?
3.1 Scientific Notation
5
What is Scientific Notation?
•  Scientific notation is the way that scientists easily
handle very large numbers or very small numbers
•  It is based on powers of the 10
3.1 Scientific Notation
6
General Form
Scientific notation always has the following form:
X 10
coefficient
exponent
1 ≤ M < 10
•
Coefficient: a number between 1 and 9.999…
•
Exponent: a positive or negative integer
3.1 Scientific Notation
7
Expressing Large Numbers
•  Write 501000 m in scientific notation.
501000 m can be written as 501000. m
Step 1: shift the decimal point to the left until you arrive at a
number between 1 and 9.999… . Drop the tailing zeros.
501000.
5
4 3
2 1
Decimal point move to the left
5.01
This number is the coefficient part
Step 2: count the number of places the decimal point moved
(in here 5 places). That number is the exponent part
5.01 x
10 5 m
exponent
coefficient
3.1 Scientific Notation
8
Expressing Large Numbers
•  Write 501000 m in scientific notation.
501000.
5
4 3
2 1
Decimal point move to the left
5.01 x 10 5 m
exponent
coefficient
•  A positive exponent shows that the decimal
point is shifted that number of places to the left
3.1 Scientific Notation
9
Learning Check
1.  Scientists estimate that there are more than 200 billion
stars in the Milky Way galaxy.
•  How do you write this number
in scientific notation?
200 000 000 000. = 2 x 1011
•  Because you have moved the
decimal point 11 places to the
left, the exponent (power of
ten) is positive 11
•  A positive exponent shows that the decimal point is
shifted that number of places to the left
3.1 Scientific Notation
10
Learning Check
2.  The diameter of the Sun is 696,000,000 m.
Write this number in scientific notation.
•  Begin by moving the decimal point to the left until you
arrive at a number between 1 and 9.999….
In this case, 6.96, which is the coefficient part of the
notation
•  Count how many times you had to move the decimal
point to get 6.96
In this case, you have to move the decimal point 8
places to the left. This number is the exponent part.
•  So, in scientific notation the diameter of the Sun is
6.96 x 108 m
3.1 Scientific Notation
11
Expressing Small Numbers
•  Write the number 0.00002205 g in scientific notation
Step 1: shift the decimal point to the right until you arrive at
a number between 1 and 9.999… .
0.00002205
1
2
3
4
2.205
5
Step 2: count the number of places the decimal point moved
(here 5 places). That number is the exponent part
2.205 x 10-5 g
•  Because you have to moved the decimal point 5 place
to the right, the exponent (power of ten) is negative 5.
3.1 Scientific Notation
12
Expressing Small Numbers
•  Write the number 0.00002205 g in scientific
notation
0.00002205
1
2
3
4
2.205
5
2.205 x 10-5 g
•  A negative exponent shows that the decimal
point is shifted that number of places to the right
3.1 Scientific Notation
13
Learning Check
3.  The radius of a hydrogen atom is 0.000000000053 m
Write this number in scientific notation.
•  Begin by moving the decimal point to the right until you
arrive at a number between 1 and 9.999….
In this case, 5.3, which is the coefficient part of the
notation
•  Count how many times you had to move the decimal
point to get 5.3
In this case, you have to move the decimal point 11
places to the right.
•  So, in scientific notation, the radius of a hydrogen
atom is 5.3 x 10-11m
3.1 Scientific Notation
14
Learning Check
4.  Diameter of a proton:
0.000 000 000 000 002 m
Express this number in scientific notation
2 x 10-15 m
•  A negative exponent shows that the decimal point is
shifted that number of places to the right
3.1 Scientific Notation
15
Understanding Scientific Notation
•
A positive exponent indicates a number greater than 1
Examples:
•
A negative exponent indicates a number between 0 and
1 (not a number less than 0)
Examples:
•
2000 = 2 x 103
501 = 5.01 x 102
5,216,000 = 5.216 x 106
0.005
0.0701
0.00086
= 5 x 103
= 7.01 x 102
= 8.6 x 10-4
Numbers less than 0 are indicated by putting a negative
sign before the coefficient (not in the exponent)
Examples:
- 12500 = - 1.25 x 10 4
- 0.09024 = - 9.024 x 10 - 2
3.1 Scientific Notation
16
How Does Scientific Notation Work?
•
To understand how this shorthand notation works,
consider the large number 30,000,000
•
Mathematically, this number is equal to:
30,000,000 = 3 x 10 x 10 x 10 x 10 x 10 x 10 x 10
check this out on your calculator
•
We can abbreviate this chain of numbers by writing all
the 10’s in an exponential form
3 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 3 x 10 7
•
Note that 10 7 is the same as 10 x 10 x 10 x 10 x 10 x 10 x 10
3.1 Scientific Notation
17
How Scientific Notation Works?
•
•
To understand how this shorthand notation works, consider
the small number 0.0009
9
9
Mathematically, this
0.0009 =
=
number is equal to:
10000 10 ×10 ×10 ×10
check this out on your calculator
•
We can abbreviate this chain of numbers by writing all the
10’s in an exponential form:
9
9
= 4
10 ×10 ×10 ×10 10
Note that 10 4 is the same as 10 x 10 x 10 x 10
•  Because dividing by a number is equivalent multiplying by the
reciprocal of that number 9
−4
=
9
×
10
10 4
In Scientific Notation 0.0009 becomes 9 x 10 - 4
3.1 Scientific Notation
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Reasons for Using Scientific Notation
•  It takes a lot less time and space to report very
large or very small numbers
e.g., the number of hydrogen atoms in 1g of hydrogen
can be reported as 6.02 x 1023 rather than
602,000,000,000,000,000,000,000
•  It also helps to represent measurements in
correct number of significant figures
3.1 Scientific Notation
19
Learning Check
5.  Change to scientific notation.
12,340 = 1.234 x 104
0.369 = 3.69 x 10-1
0.008 = 8 x 10-3
7,080,200,000 = 7.0802 x 109
3.1 Scientific Notation
20
Learning Check
6  Change to standard form
a)  1.87 x 10–5 = 0.0000187
b)  3.7 x 108
= 370,000,000
c)  7.88 x 101 = 78.8
d)  2.164 x 10–2 = 0.02164
3.1 Scientific Notation
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Rule for Multiplication
When multiplying numbers in scientific notation, multiply the
coefficients and add the exponents.
Example: (3.2 x 10 - 7 m ) X (2.1 x 10 5 m )
(3.2) x (2.1) = 6.72
(10 - 7 ) x (10 5 ) = 10 ( - 7 + 5 ) = 10 - 2
6.7 x 10 - 2 m 2
Practice : (9.6 x 10 7 m ) X (1.5 x 10 4 m ) = 14.4 x 10 1 1
= 1.4 x 10 1 2m2
3.1 Scientific Notation
22
Learning Check
•  Multiply the coefficients and
•  add the exponents
7. (3.0 x 108 m) X (5.0 x 102 m)
8. (2.1 x 103 m) X (4.0 x 10-7 m)
Answer # 7: (3.0 x 5.0) x 108+2 = 15 x 1010 = 1.5 x 1011m2
Answer # 8: (2.1 x 4.0) x 103 +(- 7 ) = 8.4 x 10 - 4 m 2
3.1 Scientific Notation
23
Rule for Division
When dividing numbers in scientific notation, divide the
coefficient in the numerator by the coefficient in the
denominator. Then subtract the exponent in the denominator
from the exponent in the numerator.
6.4 ×10 6
Example:
2
1.7 ×10
6 .4
= 3.8
1 .7
10 6
( 6 - 2 ) = 10 4
=
10
10 2
Practice:
3.8 x 104
2.4 ×10 −7
-21
=
0.77
x
10
3.1×1014
3.1 Scientific Notation
= 7.7 x 10-22
24
Learning Check
•  Divide the coefficients and,
•  Subtract the exponents
9.
11
1
.
5
×
10
11
8
1.5 ×10 ÷ 3.0 ×10 =
3.0 ×108
1.5
=
×10 (11−8)
3.0
= 0.50 ×10 3
= 5.0 x 102
3.1 Scientific Notation
25
Rule for Addition and Subtraction
In order to add or subtract numbers written in scientific
notation, you must express them with the same power of 10.
1.  Find a common exponent (You may choose the larger
exponent to match)
2.  Then add or subtract coefficients leaving the exponent
unchanged
Example: (5.8 x 10 3 m L ) + (2.16 x 10 4 m L )
Since 10 3 and 10 4 not the same exponent
change 5.8 x 10 3 to .58 x 10 4
= (.58 x 104) + (2.16 x 104) = 2.7 x 104 mL
Practice: (8.32 x 107g) – (1.2 x 105g)
3.1 Scientific Notation
= 8.3 x 107g
26
Learning Check
•  Find a common exponent – the exponents must be the
same (You may choose the larger to match)
•  Then add or subtract coefficients leaving the exponent
unchanged
10. (2.0 x 10 3 ) + (2.0 x 10 4 )
11. (3.0 x 10 - 2 ) + (2.0 x 10 2 )
10.
Change 2.0 x 10 3 to 0.2 x 10 4
0.2 x 10 4 + 2.0 x 10 4 = 2.2 x 10 4
11.
Change 3.0 x 10- 2 to 0.0003 x 10 2
0.0003 x 10 2 + 2.0 x 10 2 = 2.0 x 10 2
3.1 Scientific Notation
27
Using the Exponent Key
on a Calculator
EE
EXP
EE or EXP means “times 10 to the…”
3.1 Scientific Notation
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Using Calculators
Example - Change to standard form: 5 x 103
5 x 10 3
= 5 x (10 x 10 x 10)
= 5 x 1000
= 5000
5  EXP 3
=
= 5000
Learning Check
12. Use calculator to change to standard form
a)  3.7 x 10 8
= 370,000,000
b)  7.88 x 10 1
= 78.8
3.1 Scientific Notation
29
Using Calculators
Example - Change to standard form: 5 x 103
7.2 x 10 - 2
7.2
7.2
=
=
10 ×10 100
7.2 EXP [+/-] 2
=
=0.072
= 0.072
Learning Check
13. Use calculator to change to standard form
a)  1.87 x 10–5
= 0.0000187
b)  2.164 x 10–2
= 0.02164
3.1 Scientific Notation
30
Using Calculators
Example: (5.44 x 107) ÷ (8.1 x 104)
How to enter this on a calculator:
..
5.44 EXP 7
8.1 EXP 4
= 671.6049383
=
rounded to 6.7 x 102
Learning Check
14. Use calculator to solve this problem
(6.2 x 10-3) x (1.5 x 101)
= 9.3 x 10-2
3.1 Scientific Notation
31
Using Calculators
−3
6
(
3
.
8
×
10
)
×
(
1
.
2
×
10
)
Example:
8.0 ×10 4
How to enter this on a calculator:
3.8
EXP
[+/-] 3
X
1.2
EXP
6
.
.
8.0
EXP
4
=
= 0.057
5.7 x 10 - 2
Learning Check
15. Use calculator to solve this problem
6.6 ×10 6
(8.8 ×10 −2 ) × (2.5 ×103 )
= 3.0 x 104
3.1 Scientific Notation
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