MTH 232

Section 7.1Decimals 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:

012 104107106

)14()107()1006(

470600674

Continued

• The same strategy can be applied to a decimal number:

2101

2101

102101108103

10

12

10

11108103

100

12

10

1118103

01.021.0118103

02.01.083012.38

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:

100

79100

9

100

70100

9

10

7

100

19

10

17

01.091.07

09.07.079.0

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:

....5555.5....5555.010

Continued

• Let x = 0.5555…..• Then 10x = 5.5555….

9

59

5

9

9

59

.....5555.0.....5555.510

x

x

x

xx

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