6
Number Notations, Arithmetical Operations, and Calculators
pie, 10
2 means 10 x 10 = 100. The number 2 is the exponent; the number 10 is
the base, which is said to be raised to the second power. Likewise, 2
5 means
2x2x2x2x2 = 32. Here 5 is the exponent, and 2 is the base that is raised
to the fifth power.
It is simple to express any number in exponential form as the product of some
other number and a power of 10. For integral (whole) powers of 10, we have 1 =
1 x 10°, 10 = 1 x 10
1
, 100 = 1 x 10
2 , 1000 = 1 x 10
3
, and so on. If a number is
not an integral power of 10, we can express it as the product of two numbers,
with one of the two being an integral power of 10 that we can write in exponential form. For example, 2000 can be written as 2 x 1000, and then changed to
the exponential form of 2 x 10
3
. The form 2000 is an example of decimal
notation; the equivalent form 2 x 10
3 is an example of scientific notation.
Notice that in the last example we transformed the expression
2000.0 x 10°
into the expression
2.0 x 10
3
Notice that we shifted the decimal point three places to the left, and we also
increased the exponent on the 10 by the same number, three. In changing the
form of a number but not its value, we always follow this basic rule.
1. Decrease the lefthand factor by moving the decimal point to the left the
same number of places as you increase the exponent of 10. An example
is 2000 (i.e., 2000 x 10°) converted to 2 x 10
3
.
2. Increase the lefthand factor by moving the decimal point to the right
the same number of places as you decrease the exponent of 10. An
example is 0.005 (i.e., 0.005 x 10°) converted to 5 x 10~
3
.
PROBLEM:
Write 3,500,000 in scientific notation.
SOLUTION:
The decimal point can be set at any convenient place. Suppose we select the
position shown below by the small x, between the digits 3 and 5. This gives as the
first step
3^00,000
Because we are moving the decimal point 6 places to the left, the exponent on 10°
must be increased by 6, and so the answer is
3.5 x 10
6
Number Notations, Arithmetical Operations, and Calculators
pie, 10
2 means 10 x 10 = 100. The number 2 is the exponent; the number 10 is
the base, which is said to be raised to the second power. Likewise, 2
5 means
2x2x2x2x2 = 32. Here 5 is the exponent, and 2 is the base that is raised
to the fifth power.
It is simple to express any number in exponential form as the product of some
other number and a power of 10. For integral (whole) powers of 10, we have 1 =
1 x 10°, 10 = 1 x 10
1
, 100 = 1 x 10
2 , 1000 = 1 x 10
3
, and so on. If a number is
not an integral power of 10, we can express it as the product of two numbers,
with one of the two being an integral power of 10 that we can write in exponential form. For example, 2000 can be written as 2 x 1000, and then changed to
the exponential form of 2 x 10
3
. The form 2000 is an example of decimal
notation; the equivalent form 2 x 10
3 is an example of scientific notation.
Notice that in the last example we transformed the expression
2000.0 x 10°
into the expression
2.0 x 10
3
Notice that we shifted the decimal point three places to the left, and we also
increased the exponent on the 10 by the same number, three. In changing the
form of a number but not its value, we always follow this basic rule.
1. Decrease the lefthand factor by moving the decimal point to the left the
same number of places as you increase the exponent of 10. An example
is 2000 (i.e., 2000 x 10°) converted to 2 x 10
3
.
2. Increase the lefthand factor by moving the decimal point to the right
the same number of places as you decrease the exponent of 10. An
example is 0.005 (i.e., 0.005 x 10°) converted to 5 x 10~
3
.
PROBLEM:
Write 3,500,000 in scientific notation.
SOLUTION:
The decimal point can be set at any convenient place. Suppose we select the
position shown below by the small x, between the digits 3 and 5. This gives as the
first step
3^00,000
Because we are moving the decimal point 6 places to the left, the exponent on 10°
must be increased by 6, and so the answer is
3.5 x 10
6