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Math 365 Lecture Notes - © J. Whitfield
Section 5-1
Page 1 of 3
Math 365 Lecture Notes
Section 5.1 – The Set of Rational Numbers
Definitions
1) Rational Number – Q = {a/b | a and b are integers and b  0}
2) Numerator – In the rational number a/b, a is the numerator.
3) Denominator – In the rational number a/b, b is the denominator
4) Fraction – derived from the Latin word fractus meaning “to break.”
5) Proper Fraction – A fraction a/b, where 0  |a| < |b|.
6) Improper Fraction – A fraction a/b, where |a|  |b| > 0.
Early Fractions
1) Egyptians used fractions in the form
1
(7/12 = 1/3 + 1/4)
a
2) Babylonian notation for fractions: 12,35 (meant 12 + 35/60)
3) Note the similarity to 23 42 16 13 + 19/60 + 47/602
4)
a
as a fraction with the fraction bar is of Hindu origin
b
Ways to Use Fractions
1) Division problem: The solution to 2x = 3 is 3/2.
2) Part of a whole: Joe received ½ of Mary’s salary each month for alimony
3) Ratio: The ratio of Republicans to Democrats in the Senate is five to four.
4) Probability: When you toss a fair coin, the probability of getting heads is ½.
Fundamental Law of Fractions:
If
a
a
an
is any fraction and n  0, then =
b
b
bn
Math 365 Lecture Notes - © J. Whitfield
Section 5-1
Page 2 of 3
Activity 1
Activity 2
By the end of this section you should be able to:
1. Explain the meaning of a/b.
2. Solve equations with fractions
3. Simplify Fractions
Solve equations with fractions
1) Use the Fundamental Law of Fractions to rewrite each fraction with the same
denominator.
12
x
12  5
x
60
x



Find a value for x so that
. 

 x = 60.
42 210
42  5 210 210 210
2) Simplify fractions by factoring
60 2  2  3  5 2


210 2  3  5  7 7
3) Simplest Form: A rational number a/b is in simplest form if a and b have no common
factor greater than 1, that is, if a and b are relatively prime.
4) Practice Problems
a.
=
x2  x
x  x3
x( x  1)
x 1
 2
2
x(1  x ) x  1
b.
=
5  5x
5x 2
5(1  x) x  1
 2
5x 2
x
c.
=
a 2  b2
ab
(a  b)( a  b) a  b

 ab
( a  b)
1
Math 365 Lecture Notes - © J. Whitfield
Section 5-1
Page 3 of 3
Showing two fractions are equal
1) Write both fractions in simplest form
12 2  2  3 2
10 2  5 2




42 2  3  7 7
35 5  7 7
2) Rewrite fractions with the least common denominator
12  5 60
10  6 60


Since LCM(42,35) = 210,
and
42  5 120
35  6 120
3) Rewrite fractions with “any” common denominator
12  35 420
10  42 420


Since 42  35 = 1470 is a common multiple,
and
.
42  35 1470
35  42 1470
Properties and Theorems:
Property: Two fractions a/b and c/d are equal iff ad = bc.
a c
Theorem: If a, b, and c are integers and b > 0, then  iff a > c.
b b
a c
 iff ad > bc.
Theorem: If a, b, c, and d are integers and b > 0, d > 0,
b d
Theorem: Let a/b and c/d be any rational numbers with positive denominators where
a c
a ac c
 . Then 
 .
b d
b bd d
Ordering fractions
Order the fractions 3/4, 9/16, 5/8, and 2/3 from least to greatest.
Equivalent fractions with LCD=48 are: 36/48, 27/48, 30/48, and 32/48.
Ordering these we get, 27/48 < 30/48 < 32/48 < 36/48. So 9/16 < 5/8 < 2/3 < ¾.
Denseness Property
1) Definition
Denseness Property for Rational Numbers: Given rational numbers a/b and c/d,
there is another rational number between these two numbers.
2) example:
a. Find two fractions between 5/12 and 3/4.
3 9
Since  , 6/12 or ½, 7/12, and 8/12 or 2/3 lie between 5/12 and 9/12.
4 12
b. Find two fractions between 2/3 and 3/4.
2 32
3 36
Since 
and 
, 33/48, 34/48, and 35/48 lie between 2/3 and 3/4.
3 48
4 48
c. Alternate solution to part b above: As we use bigger common demoninators,
we can find more fractions between any two fractions.