 # Thinking Mathematically

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Thinking Mathematically
Number Theory and the Real Number System 5.3 The Rational Numbers

The Rational Numbers The set of rational numbers is the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not equal to 0. The integer a is called the numerator and the integer b is called the denominator. Note that every integer is a rational number. For example 17 can be written 17/1. Where do rationals fit on the number line?

The Fundamental Principle of Rational Numbers
If is a rational number and c is any number other than 0, The rational numbers and are called equivalent fractions

Fraction Basics The fraction denominator (bottom) is the size of the piece. The fraction numerator (top) is the number of those pieces. A mixed number combines a whole number with a fraction. In a proper fraction, the numerator is smaller than the denominator. Not so in an improper fraction.

Lowest Terms Lowest Terms – GCD of numerator and denominator is 1 A rational number can be reduced to its lowest terms by dividing the numerator and denominator by the greatest common divisor. Exercises 1-12 Exercise Set 5.3 #7 Reduce to lowest terms

Converting a Mixed Number to an Improper Fraction
Multiply the denominator of the rational number by the integer and add the numerator to this product. Place the sum in step 1 over the denominator in the mixed number. Exercises 13-18 Exercise Set 5.3 #15 Convert to an improper fraction

Converting a Improper Fraction to a Mixed Number
Divide the denominator into the numerator. Record the quotient and the remainder. Write the mixed number using the following form: Exercises 19-24 Exercise Set 5.3 #21 Convert to a mixed number

Rational Numbers and Decimals
Any rational number can be expressed as a decimal. Meaning of decimals? The resulting decimal will either terminate, or it will have a digit that repeats or a block of digits that repeat. Exercises 25-36 Exercises 37-48 Exercise Set 5.3 #27, #31 Express as a decimal

Expressing a Repeating Decimal as a Quotient of Integers
Step 1 Let n equal the repeating decimal. Step 2 Multiply both sides of the equation in step 1 by 10 if one digit repeats, by 100 if two digits repeat, by 1000 if three digits repeat, and so on. Step 3 Subtract the equation in step 1 from the equation in step 2. Step 4 Divide both sides of the equation in step 3 by an appropriate number and solve for n. Exercises 49-56

Example: Decimal to Fraction
Exercise Set 5.3 #39, #53 Express as a quotient of integers

Multiplying Rational Numbers
The product of two rational numbers is the product of their numerators divided by the product of their denominators. Cancel/reduce to achieve a result in lowest terms Exercises 57-64 Exercise Set 5.3 #57, #61

Dividing Rational Numbers
The quotient of two rational numbers is the product of the first number and the reciprocal of the second number. If a/b and c/d are rational numbers, and c/d is not 0, then Exercises 65-70 How else can a fraction divided by a fraction be written?

Example: Dividing Rational Numbers
Exercise Set 5.3 #67

Adding and Subtracting Rational Numbers with Identical Denominators
The sum (difference) of two rational numbers with identical denominators is the sum (difference) of their numerators over the common denominator. If a, b, and c are integers (b not 0) numbers, then Exercises 71-76

Example: Subtracting Rational Numbers (same denominator)
Exercise Set 5.3 #73

Adding and Subtracting Rational Numbers – Differing Denominators
The least common multiple of the two denominators is called the least common denominator (LCD) Replace each fraction with an eqivalent fraction in which the denominator is the LCD Exercises 77-92

Example: Adding and Subtracting Rational Numbers – Differing Denominators
Exercise Set 5.3 #79, #83

Density of Rational Numbers
If r and t represent rational numbers, with r<t, then there is a rational number s such that s is between r and t. r < s < t. Between any two rational numbers, there is another rational number. Between any two rational numbers, there is an infinite number of rational numbers. Exercises 93-98

Thinking Mathematically
Number Theory and the Real Number System 5.3 The Rational Numbers