Math 365 Lecture Notes - © J. Whitfield Page 1 of 3

Section 5-1

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 (7/12 = 1/3 + 1/4)

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) as a fraction with the fraction bar is of Hindu origin

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 is any fraction and n  0, then =

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.

Find a value for x so that . x = 60.

2)Simplify fractions by factoring

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

  1. b. c.

= = =

Showing two fractions are equal

1)Write both fractions in simplest form

2)Rewrite fractions with the least common denominator

Since LCM(42,35) = 210, and

3)Rewrite fractions with “any” common denominator

Since 42  35 = 1470 is a common multiple, and .

Properties and Theorems:

Property: Two fractions a/b and c/d are equal iff ad = bc.

Theorem: If a, b, and c are integers and b > 0, then iff ac.

Theorem: If a, b, c, and d are integers and b > 0, d > 0, iff adbc.

Theorem: Let a/b and c/d be any rational numbers with positive denominators where

. Then .

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:

  1. Find two fractions between 5/12 and 3/4.

Since , 6/12 or ½, 7/12, and 8/12 or 2/3 lie between 5/12 and 9/12.

  1. Find two fractions between 2/3 and 3/4.

Since and , 33/48, 34/48, and 35/48 lie between 2/3 and 3/4.

  1. Alternate solution to part b above: As we use bigger common demoninators, we can find more fractions between any two fractions.