6 Chapter Rational Numbers and Proportional Reasoning Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

6-1 The Set of Rational Numbers Students will be able to understand and explain • Different representations for rational numbers. • Equal fractions, equivalent fractions, and the simplest form of fractions. • Ordering of rational numbers. • Denseness property of rational numbers.

Definition Numerator, Denominator numerator denominator

Uses of Rational Numbers

Rational Number Models

Meaning of a Fraction To understand the meaning of any fraction, using the parts-to-whole model, we must consider each of the following: The whole being considered. The number b of equal-size parts into which the whole is divided. The number a of parts of the whole that are selected.

Definition Proper fraction A fraction where Examples are Improper fraction A fraction such that Examples are

Number Line Model What numbers are represented on the number line?

Equivalent or Equal Fractions Equivalent fractions are numbers that represent the same point on a number line.

Fraction Strips

Fundamental Law of Fractions Let be any fraction and n a nonzero whole number, then

Example Find a value for x such that

Simplifying Fractions A rational number is in simplest form if b > 0 and GCD(a,b) = 1; that is, if a and b have no common factor greater than 1.

Example Write each of the following fractions in simplest form if they are not already so: a. b. c. d. cannot be simplified

Example (continued) e. f. g. cannot be simplified

Equality of Fractions Show that Method 1: Simplify both fractions to the same simplest form.

Equality of Fractions Show that Method 2: Rewrite both fractions with the same least common denominator. LCM(42, 35) = 210, so

Equality of Fractions Method 3: Rewrite both fractions with a common denominator. A common multiple of 42 and 35 is 42 · 35 = 1470.

Equality of Fractions Two fractions and , d  0 are equal if and only if ad = bc.

Ordering Rational Numbers If a, b, and c are integers and b > 0, then if and only if a > c. If a, b, c, and d are integers and b > 0, d > 0, then if and only if ad > bc.

Denseness of Rational Numbers Given rational numbers there is another rational number between these two numbers.

Example Find two fractions between Because is between Now find two fractions equal to respectively, but with greater denominators.

Example (continued) are all between so they are between

Example Show that the sequence …, is increasing. Because the nth term of the sequence is the next term is We need to show that for all positive integers n,

Example (continued) The inequality is true, if, and only if, which is true for all n. Thus, the sequence is increasing.

Denseness of Rational Numbers Let be any rational numbers with positive denominators, where Then