Chapter 6: Rational Number Operations and Properties
6.5 Comparing, Ordering, and Connecting Rational Numbers
6.5.1. Comparing rational numbers
6.5.1.1. All of the following help to build notions about comparing rational numbers
6.5.1.1.1. models
6.5.1.1.2. common denominators
6.5.1.1.3. place value
6.5.1.2. Using models to compare rational numbers
6.5.1.2.1. basis for explaining how rational numbers compare
6.5.1.2.2. fraction wall composed of bar fractions – see fig. 6.22, p. 355
6.5.1.2.3. fraction wall can be condensed into a number line
6.5.1.2.4. helps to compare and order rational numbers
6.5.1.3. Using common denominators to compare rational numbers
6.5.1.3.1. Denominators can be used as the basis for comparing fractions
6.5.1.3.2. Can build models to show that if denominator the same, largest numerator is
largest rational number
6.5.1.3.3. Generalization about comparing rational numbers that have like
denominators: For rational numbers ab and cd , where b > 0, ba > cb if and only if
a>c
6.5.1.3.4. To compare 2 rational numbers with different denominators, we use the
fundamental law of fractions
6.5.1.3.5. Generalization about comparing rational numbers that have unlike
denominators: For rational numbers ab and cd , where b > 0 and d > 0, ba > cd ( or
> bc
bd ) if and only if ad > bc
6.5.1.3.6. In other words – look at the results of cross-multiplying –Trichotomy
principle
6.5.1.3.6.1.
ad > bc, then ba > cd
ad
bd
6.5.1.3.6.2.
ad < bc, then
a
b
a
b
<
c
d
c
d
6.5.1.3.6.3.
ad = bc, then =
6.5.2. Ordering Rational Numbers
6.5.2.1. rational numbers are dense
6.5.2.2. Denseness property for rational numbers: for any two rational numbers,
a
c
e
a
e
c
b < d , at least one rational number f exists such that b < f < d
6.5.2.2.1. this is sometimes referred to as the betweenness property of rational
numbers
6.5.3. Additional Notation for Rational Numbers
6.5.3.1. repeating decimals and fractions
6.5.3.2. See p. 360 for examples for converting a repeating decimal to the rational
number form
6.5.3.3. scientific notation: A rational number is expressed in scientific notation when it
is written as a product where one factor is a decimal greater than or equal to 1 and
less than 10 and the other factor is a power of ten
6.5.3.4. See example 6.21 p. 360-61
6.5.4. Connecting rational numbers to whole numbers, integers, and other numbers
6.5.4.1. real numbers = rational numbers + irrational numbers
6.5.4.2. patterned irrational numbers
6.5.5. Problems and Exercises p. 363
6.5.5.1.
Home work: 1-13