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MTH 232
Section 7.1
Decimals and Real Numbers
Objectives
1. Define decimal numbers and represent them
using manipulatives;
2. Write decimals in expanded form (with and
without exponents)
3. Express terminating and repeating decimals
as fractions.
Definition
• A decimal is a base-ten positional numeral,
either positive or negative, in which there are
finitely many digits to a left of a point (called
the decimal point) that represent units (ones),
tens, hundreds, and so on, and a finite or
infinite sequence of digits to the right of the
decimal point that represent tenths,
hundredths, thousandths, and so on.
The Big Idea
• Place values to the left of the decimal point
represent increasingly large powers of 10:
The Big Idea (Continued)
• Place values to the right of the decimal point
represent divisions of 1 into increasingly large
powers of 10:
Another Representation
• Decimals can also be represented, in a
somewhat limited way, by using dollar coins,
dimes, and pennies:
• 10 pennies = 1 dime
• 10 dimes = 1 dollar coin
• Unfortunately, in this overly simplified
representation, nickels and quarters have no
place (value).
Expanded Notation
• When working with whole numbers, we use
place value to expand into increasing detailed
notations:
674  600  70  4
 (6 100)  (7 10)  (4 1)

 
 
 6 10  7 10  4 10
2
1
0

Continued
• The same strategy can be applied to a decimal
number:
38.12  30  8  0.1  0.02
 3 10  8 1  1 0.1  2  0.01
1 
 1 
 3 10  8 1  1    2 

 10   100 
1  
1 

1
0
 3 10  8 10  1 1    2  2 
 10   10 

 

 3 10   8 10   110   2 10 
1
0
1
2
Types of Decimal Numbers
1. Decimal numbers that terminate, or end.
2. Decimal numbers that do not terminate and
have a digit or series of digits that repeat
forever.
3. Decimal numbers that do not terminate but
do not have a digit or series of digits that
repeat forever.
Terminating Decimals
• Terminating decimals can be written as
fractions by adding the fractions associated
with each place value:
0.79  0.7  0.09
 7  0.1  9  0.01
1 
1 

 7   9

 10   100 
7
9
 
10 100
70
9


100 100
79

100
Repeating Decimals
• Repeating decimals can be written as fractions
by algebraic manipulation of the repeating
digit or digits.
• Recall that multiplying by 10 will effectively
move the decimal point in a number one place
to the right:
10  0.5555....  5.5555....
Continued
• Let x = 0.5555…..
• Then 10x = 5.5555….
10 x  x  5.5555.....  0.5555.....
9x  5
9x 5

9 9
5
x
9
Non-terminating, Non-repeating
Decimals
• Decimals that do not terminate but also do
not repeat cannot be written as fractions.
• These decimal numbers are called irrational
numbers.
• The most commonly-referenced irrational
number is pi:
Pi, to 224 Decimal Places