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9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA
Ordering Rational Numbers
Three equivalent ways (as we did with fractions):
1) Number-Line Approach:
a c ⇣
c a⌘
<
or >
if and only if
b d
d b
a
c
is to the left of on the rational number line.
b
d
2) Common-Positive-Denominator Approach:
a c
<
if and only if a < c and b > 0.
b c
3)Addition approach:
a c
p
< if and only if the is a positive rational number such that
b d
q
a p c
+ = , or equivalently,
b q d
a c
c a
< if and only if
is positive.
b d
d b
5
11
Example. Compare
and
.
6
12
First way:
11 ⇣ 5 ⌘
11 5
11 10
1
=
+ =
+
=
. Then
12
6
12 6
12 12
12
⇣ 5 ⌘ ⇣ 11 ⌘
1
11 ⇣ 5 ⌘
= , so
<
.
6
12
12
12
6