1.02
Basic Math
Scientific (Exponential) Notation
Dr. Fred Garces
Chemistry 100
Miramar College
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Basic Math
Jan ‘11
Science and the use of large numbers
In science we deal with
either very large numbers.
- For example:
How many Copper atoms are in
a single penny ?
29 500 000 000 000 000 000 000 atoms of copper.
That’s 295 followed by 20 zeros !!!
or = 2.95•1022 atoms of copper.
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Science also use very small numbers
... or we deal with very
small numbers.
For example:
What is the mass (lb.) of a
single copper atom ?
.000 000 000 000 000 000 000 000 23 lb. of copper.
That’s 24 zeros in front of 23 !!!
or = 2.3•10-25 lb. of copper.
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Why use exponential notation ?
The exponential notation system makes it easier to
manipulate very large numbers in numerical
calculations.
Example : 0 .000 000 000 000 000 000 000 164 23
+ 0.000 000 000 000 000 000 008 244 33 = ???
0.000 000 000 000 000 000 000 164 23

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Numbers with lots of zeros
cannot be accommodated by
simple calculators unless
exponential notation is used.
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Basic Math
Jan ‘11
Writing in Scientific Notation
Some sizes of common items.
1. 7, 901 miles
2. 4,800 ft
3. 150 lb
4. .07oz (or 2 grams)
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Convenience of Exponential numbers
A very large numbers consist of extraneous zeros.
These zeros are simply decimal place holder.
These zeros can be express by exponents.
For example in the 1000 can be express in exponential
notation by the following process1000 can be express as 10 x 10 x 10
or 10 multiplied by itself 3 times or 103.
Where 3 is the exponent or the power which indicates how many
times 10 is multiplied by itself.
1000 = 10 x 10 x 10 = 103
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Exponential Notation is based on powers of 10.
The exponential notation system is based on powers of 10s and
the number of times 10 (ten) is multiplied by itself.
Example of scientific notation: For some numbers equal to or
greater than one.
1 = 1 x (10 x 0 ) = 1 • 100
Where 10 is multiplied by itself 0 times.
10 = 1 x 10 = 1 • 101
Where 10 is multiplied by itself one time.
100 = 1 x 10 x 10 = 1 • 102
Where 10 is multiplied by itself twice.
1000 = 1 x 10 x 10 x 10 = 1 • 103
Where 10 is multiplied by itself three times.
100,000 = 1 x 10 x 10 x 10 x 10 x 10 = 1 • 105
Where 5 is the exponent or the power which indicates how many times 10 is
multiplied by itself.
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... powers of 10
For numbers smaller than one i.e., 0.1, 0.001 or 0.00001, are based on
1/10 and the number of times it is multiplied by itself.
Example of scientific notation: For numbers equal to or less than one.
0.1 = 1 x 1/10 = 1 • 1/10
1
= 1 • 10-1
Where 1/10 is multiplied by itself one (1) time.
0.01 = 1 x 1/10 x 1/10 = 1 • 1/102 = 1 • 10-2
Where 1/10 is multiplied by itself two (2) times.
0.0001 = 1 x 1/10 x 1/10 x 1/10 x 1/10 = 1 • 1/104 = 1 • 10-4
Where 1/10 is multiplied by itself 4 time.
0.000001 = 1x 1/10 x 1/10 x 1/10 x 1/10 x 1/10 x 1/10
= 1 • 1/106 = 1 • 10-6
Where 6 is the exponent or the power of the exponent which indicates how
many times 1/10 is multiplied by itself.
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POWER x 10
... powers of 10
http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/
Illustrated is dimension of matter in powers of ten.
Landmass: 1•105 m
Man: 1•100 m
Cells : 1•10-6 m
Subatomic :
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1•10-13
Milky Way: 1•1021 m
m
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Basic Math
Jan ‘11
Exponential Notation
For other numbers expressed in exponential notation. For
other non-unity numbers, instead of initially multiplying by one, the
operation of expressing the number in exponential notation is started
by multiplying by the number and then multiplying by 10.
Exponential notation for some number greater than one.
22400 = 224 x 10 x 10 = 224 x 100 = 224 • 10
2
= 2.24 • 10
4
= 2.2e4
620 000 = 6.2 x 100 000 = 6.2 • 105 = 6.2e5
Exponential notation for some number smaller than one.
0.045 = 4.5 x 1/10 x 1/10 = 4.5 • 1/102 = 4.5 • 10-2 = 4.5e-2
6.33 •10-4 = 6.33 x 1/10 x 1/10 x 1/10 x 1/10 = 6.33 x .0001 = 0.000633
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SideBar: moving decimal places to change exponent
A major source of confusion when learning about exponential notation is how
to change the exponent if the decimal point is moved.
Example1 : Given 760.000 What is the new number if the decimal place is moved
three places to the right? In our example 760 000. •10?
Example2 : Given 760.000 What is the new number if the decimal place is moved
two places to the left? In our example 7.60 000 •10?
Recall the number system:
negative numbers are on the left, positive numbers are to the right.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Working it out
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Basic Math
Jan ‘11
SideBar: moving decimal places to change exponent
A major source of confusion when learning about exponential notation is how
to change the exponent if the decimal point is moved.
Example1 : Given 760.000 What is the new number if the decimal place is moved
three places to the right? In our example 760 000. •10?
Example1 : Given 760.000 What is the new number if the decimal place is moved
two places to the left? In our example 7.60 000 •10?
Recall the number system:
negative numbers are on the left, positive numbers are to the right.
-5
-4
-3
-2
-1
0
+1
+3
+4
+5
Negative portion.
Positive portion.
If the decimal point is moved from the left
(negative) to the right, then the exponent
becomes smaller or negative.
If the decimal point is moved from the right
(positive) to the left, then the exponent
becomes larger.
760.000 = 7.60 000 •102
760.000 = 760 000. •10-3
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Basic Math
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SideBar: converting numbers to exponential notation
Numbers greater than one.
# > 1, positive exponent i.e., 961 000
If a numbers is greater than 1, the number will have a positive exponent when
converted to exponential notation. To converted the number to exponential
notation form, move the decimal from the right to the left, the number of
places the decimal moves is the number used in the exponent.
For example, 961 000 = 961 000 → 9.61000 = 9.61•105
Numbers less than one (non-negative).
# < 1, negative exponent i.e.,0 .000054
If a number is less than one, the number will have a negative exponent
when converted to exponential notation. To convert the number to
exponential form, move the decimal from the left (negative region) to
the right (positive region). The number of places the decimal is moved
is the exponent used.
For example, 0.000054 → 0 0000 54 = 5.4 • 10-5
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Basic Math
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Math operation: Addition-subtraction
Example - keep exponent the same in addition and
subtraction operation
i) 5.6 • 10-2 + 2.3 • 10-3 =
ii)  8.6 • 103 - 2300 =
iii)  7.43 • 105 + 8.44 • 106 =
iv)  8.64 • 10-3 - 7.845 • 10-4 =
Consider the following example: Three individual
who each gave their loose change to the Salvation
Army collection pan. The first had two dollar bills a
quarter and two pennies. The second gave two
quarters and a few pennies. The third donated
three dollar bills and five nickels and one penny.
What is the total amount that was collected?
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Basic Math
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Math operation:
Addition and subtraction.
(I) Addition or subtraction of numbers
expressed in exponential notation:
Keep the exponent consistent in all the numbers
in other words keep exponent the same in
addition and subtraction operation
i) 5.6 • 10-2 + 2.3 • 10-3 = ??
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Basic Math
Working it out
Jan ‘11
...continue:
Addition and subtraction
Example ii
Less precise: Number is
rounded off based on this
value. See Significant
figures rules.
ii) 8.6 • 103 - 2300 = ??
8600
2300
8.6
or
- 2.300 • 103
6.3
6300
= 6.3 ∗103
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€
• 103
Basic Math
• 103
Jan ‘11
...continue:
Addition and subtraction
Example iii
Less precise: Number is rounded
off based on this value.
iii) 7.43 • 105 + 8.44 • 106 = ??
743000.
or
+ 8440000.
9183000.
91.83 ∗105
= 91.8 ∗ 105
= 91.8 •105
18
+
7.43 ∗105
5
84.4 ∗10
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Basic Math
Jan ‘11
...continue:
Addition and subtraction
Less precise: Number is rounded
Example iv
iv) 8.64 • 10-3 - 7.845 • 10-4 = ??
0.00864
off based on this value.
or
-
- 0.0007845
0.0078555
= 7.86 •10-3
8.64 ∗10
−3
0.7845 ∗10
−3
7.855 ∗10 −3
= 7.86 ∗ 10 −3
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Basic Math
Jan ‘11
Math operation:
Multiplication and division
(II) Multiply or division of numbers expressed in exponential notation:
multiply or divide the pre-exponent then add or subtract the exponent of
the number.
i) 5.600 • 10-2 x 2.3 • 10-3 =
ii) 0.6 • 102 x 2.300 • 103 =
iii) 7.431 • 105 ÷ 84.4 • 105 =
iv) 562 000 ÷ 0.00023 =
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Basic Math
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(2) Math operation:
Multiply or division
Example - After multiplying or dividing main number, exponents are
added together (multiplication) or subtracted from each other
(division).
Example i
i) 5.600 • 10-2 x 2.3 • 10-3 = ??
Working it out
Both number has two significant figures.
Answer rounded off based on this value.
See Significant figures rules.
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Basic Math
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...continue:
Multiplication and division
Example ii
ii) 0.6 • 102 x 2.300 • 103 = ??
×
Least number of significant
figures. Answer rounded off
based on this value.
See Significant figures rules.
0.6
• 102
2.300 • 103
138, 000
= 100, 000
= 1 • 105
€
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Basic Math
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...continue:
Multiplication and division
Example iii
iii) 7.431 • 105 ÷ 84.4 • 105 = ??
7.431 • 105
÷
84.4 • 105
8.00•105 has three significant
figures. Answer is rounded off to
0.08804502
= 8.804502 • 10-2
= 8.80 • 10-2
three significant figures.
See Significant figures rules.
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Basic Math
€
Jan ‘11
...continue:
Multiplication and division
Example iv
iv)
562 000 ÷ 0.000230
Both values have three significant
figures. Answer is rounded off to
three significant figures.
562 000
÷
0.000 230
2443478261
= 2.44 • 109
See Significant figures rules.
€
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Basic Math
Jan ‘11
Math operation :
Combinations using addition / subtraction
and multiplication / division
Carry out addition / subtraction operation first before the
multiplication or division operation.
1
€
3
€
4
€
26
2
48.33
(35.2 - 29.0)
= 7.7952 = 7.8
(48.35 - 35.18) ∗ 0.12
(33.792 - 31.426)
(0.0742
=
Working it out
) (
)
× 6.01512 + 0.9258 × 0.190100 =
.446
.01760
(0.742 ∗ 6.01512) + (9.9258 ∗ 0.0090)
4.46321904
0.0893322
4.46
0.089
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Basic Math
= 0.622
=
=
4.5493322
4.55
Jan ‘11
Math operation :
Combinations using addition / subtraction
and multiplication / division
Carry out addition / subtraction operation first before the
multiplication or division operation.
1
2
€
4
€
27
48.33
= 7.7952 = 7.8
(35.2 - 29.0)
(48.35 - 35.18) ∗ 0.12
(33.792 - 31.426)
= 0.668 = 0.67
(0.742 ∗ 6.01512) + (9.9258 ∗ 0.0090)
4.46321904
0.0893322
4.46
0.089
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Basic Math
=
=
4.5493322
4.55
Jan ‘11
Summary
Addition or subtraction operation:
Keep exponent consistent then add or subtract
the pre-exponents. The exponent will remain the
same in the final answer.
Multiplication or division:
Multiply or divide the pre-exponent then:
add the exponents (multiplication)
subtract the exponents (division)
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