Transcript MTH 232
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
3 101 8 10 0 1 10 1 2 10 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