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Appendix B
Exponential Notation
1. Writing Large and Small Numbers in Exponential Notation
Many of the numbers used in science are very large or very small, and are not conveniently
written in the sort of notation used for normal-sized numbers. For example, the number of
atoms of carbon present in 12.0 g of carbon is
602,000,000,000,000,000,000,000
Clearly, in a number of this magnitude, most of the zeros indicated are merely place-holders, used to locate the decimal point in the correct place.
With very large and very small numbers, it is usually most convenient to express the
number in scientific or exponential notation. For example, the number of atoms of carbon
could be written in exponential notation as
6.02 x 1023
This expression indicates that 6.02 is to be multiplied by 10 twenty-three times. To see
how to convert a number to exponential notation, consider the following example.
Example:
Write 125000 in exponential notatioa
The standard format for exponential notation uses a factor between 1 and 10 that gives
all the appropriate significant digits of the number, multiplied by a power of ten that
locates the decimal point and indicates the order of magnitude of the number.
The number 125000 can be thought of as 1.25 x 100000.
100000 is equivalent to 10 x 10 x 10 x 10 x 10 = 105.
125000 in exponential notation is then 1.25 x 105.
A simple method for converting a large number to exponential notation is as follows. Move
the decimal point of the given number to the position after the first significant digit of the
number. This gives the multiplying factor for the number when it has been written in exponential notation. Count the number of places the decimal point has been moved from its
original position (at the end of the number) to its new position (after the first significant
576
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Appendix B
Exponential Notation
577
digit). The number of places the decimal point has been moved represents the exponent of
the power of ten of the number when expressed in scientific notation.
Example:
Write the following in exponential notation:
1723
Move decimal point to position after the first significant digit:
1.723
Decimal point has been moved three spaces to the left from the original
1723 to givel.723. Exponent will be 3:
1723 = 1.723 x 103
Example:
Write the following in exponential notation:
7,230,000
Move decimal point to position after the first significant digit:
7.230000
Decimal point has been moved six spaces to the left from its
original position. Exponent will be 6:
7,230,000 = 7.23 x 106
Basically the same methods are used when attempting to express very small numbers in
exponential notation For example, consider this number:
0.0000359
This number is equivalent to 3.59 x 0.00001, and since 0.00001 is equivalent to 10~5:
0.0000359 = 3.59 x 10~5
A simple method for converting a small number to exponential notation is as follows. Move
the decimal point of the given number to the position after the first significant digit of the
number. This gives the multiplying factor for the number when it has been written in exponential notation Count the number of places the decimal point has been moved from its
original position (at the beginning of the number) to its new position (after the first significant digit). The number of places the decimal point has been moved represents the exponent
of the power of ten of the number when expressed in scientific notatioa Because the number
is small, the exponent is negative.
Example:
Write the following in exponential notation:
0.0000000072
Move decimal point to position after the first significant digit: 7.2
Decimal point has been moved nine spaces from its original position at
the beginning of the number to after the first significant digit. Exponent
will thus be-9:
0.0000000072 = 7.2 x 10"9
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578
Appendix B
Exponential Notation
Example:
Write the following in exponential notation. 0.00498
Move decimal point to position after first significant digit:
4.98
Decimal point has been moved three spaces from its original location.
Exponent will be -3:
0.00498 = 4.98 x 10"3
2. Arithmetic with Exponential Numbers
Oftentimes in chemistry problems, it becomes necessary to perform arithmetical operations
with numbers written in exponential notation. Such arithmetic is basically no different from
arithmetic with normal numbers, but certain methods for handling the exponential portion
of the numbers are necessary.
A. Addition and Subtraction of Exponential Numbers
Numbers written in exponential notation can be added together or subtracted only if they
have the same power of ten. Consider this problem:
1.233 x 103 + 2.67 x 102 + 4.8 x 101
It is not possible to just add up the coefficients of these numbers. This problem, if written
in normal (nonexponential) notation would be
1233
267
+ 48
1548
Before numbers written in exponential notation can be added, they must all be converted to
the same po>
power of tea For example, the numbers could all be written so as to have 10 as
their power.
,
1233 = 1.233 x 103
2.67 x 102 = 0.267 x 103
4.8 x 101 = 0.048 x 103
and the sum as (1.233 + 0.267 + 0.048) x 103 = 1.548 x 103 = 1548.
B. Multiplication and Division of Exponential Numbers
According to the commutative law of mathematics, the expression (A x B)(C x D) could
be written validly as (A x C)(B x D). When this is applied to numbers written in exponential notation, it becomes very straightforward to find the product of two such numbers.
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Appendix B
Exponential Notation
579
Example:
Evaluate:
(4.2 x 106X1.5 x 104)
(4.2 x 106)(1.5 x 104) = (4.2 x 1.5X106 x 104)
(4.2 x 1.5) = 6.3
(10 6 xloVlO (6+4) =10 10
(4.2 x 106)(1.5 x 104) = 6.3 x 1010
When dividing numbers written in exponential notation, a similar application of the commutative law of mathematics is made: (E x F)/(G x H) = (E/G) x (F/H)
Example:
Evaluate:
(4.8 x 109)/(1.5 x 107)
(4.8 x 109)/(1.5 x 107) = (4.8/1.5) x (109/107)
(4.8/1.5) = 3.2
(109/107) = lO^-7* = 102
(4.8 x 109)/(1.5 x 107) = 3.2 x 102
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