ch27

A rational number is any number that can be expressed in the form \(p/q\), where \(p\) and \(q\) are integers and \(q \neq 0\).

• Integers: \(... -3, -2, -1, 0, 1, 2, 3 ...\)
• Natural Numbers: \(1, 2, 3, ...\)
• Whole Numbers: \(0, 1, 2, 3, ...\)

Key Characteristics

• Form: Always \(p/q\).
• Numerator (p): Can be any integer (positive, negative, or zero).
• Denominator (q): Must be a non-zero integer (positive or negative).
• Standard Form: A rational number is in standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1 (i.e., they are coprime).
• Example: \(2/3\) is in standard form. \(4/6\) is not (common factor 2). \(-2/-3\) is not (denominator negative).

Equivalence of Rational Numbers

• Two rational numbers \(p/q\) and \(r/s\) are equivalent if \(p \times s = q \times r\).
• Multiplying or dividing both the numerator and denominator by the same non-zero integer results in an equivalent rational number.
• Example: \(1/2 = (1 \times 2)/(2 \times 2) = 2/4\). \(2/4 = (2 \div 2)/(4 \div 2) = 1/2\).

Positive and Negative Rational Numbers

• Positive: If both numerator and denominator are positive or both are negative (e.g., \(3/4\), \(-2/-5\)).
• Negative: If one is positive and the other is negative (e.g., \(-3/4\), \(2/-5\)).
• Zero: \(0/1\) or \(0/any\ non-zero\ integer\) is a rational number that is neither positive nor negative.

Decimal Representation

• Rational numbers have either a terminating decimal expansion or a non-terminating repeating (recurring) decimal expansion.
• Terminating: \(1/2 = 0.5\), \(3/4 = 0.75\).
• Non-terminating Repeating: \(1/3 = 0.333...\), \(2/7 = 0.285714285714...\).

Rational numbers exhibit several important properties under different operations. These properties are crucial for simplifying expressions and solving equations.

1. Closure Property

• Definition: A set of numbers is closed under an operation if performing that operation on any two numbers from the set always results in a number that is also in the set.
• For Rational Numbers (Q):
• Addition: Closed. \(a+b\) is always a rational number. (e.g., \(1/2 + 1/3 = 5/6\), which is rational).
• Subtraction: Closed. \(a-b\) is always a rational number. (e.g., \(1/2 - 1/3 = 1/6\), which is rational).
• Multiplication: Closed. \(a \times b\) is always a rational number. (e.g., \(1/2 \times 1/3 = 1/6\), which is rational).
• Division: Not closed. Division by zero is undefined. If we exclude division by zero, then it is closed. (e.g., \(1/2 \div 1/3 = 3/2\), which is rational).

2. Commutativity Property

• Definition: The order of operands does not affect the result.
• For Rational Numbers (Q):
• Addition: Commutative. \(a+b = b+a\). (e.g., \(1/2 + 1/3 = 1/3 + 1/2 = 5/6\)).
• Subtraction: Not commutative. \(a-b \neq b-a\). (e.g., \(1/2 - 1/3 = 1/6\), but \(1/3 - 1/2 = -1/6\)).
• Multiplication: Commutative. \(a \times b = b \times a\). (e.g., \(1/2 \times 1/3 = 1/3 \times 1/2 = 1/6\)).
• Division: Not commutative. \(a \div b \neq b \div a\). (e.g., \(1/2 \div 1/3 = 3/2\), but \(1/3 \div 1/2 = 2/3\)).

3. Associativity Property

• Definition: The grouping of operands does not affect the result for three or more numbers.
• For Rational Numbers (Q):
• Addition: Associative. \((a+b)+c = a+(b+c)\). (e.g., \((1/2+1/3)+1/4 = 5/6+1/4 = 13/12\), and \(1/2+(1/3+1/4) = 1/2+7/12 = 13/12\)).
• Subtraction: Not associative. \((a-b)-c \neq a-(b-c)\). (e.g., \((1/2-1/3)-1/4 = 1/6-1/4 = -1/12\), but \(1/2-(1/3-1/4) = 1/2-1/12 = 5/12\)).
• Multiplication: Associative. \((a \times b) \times c = a \times (b \times c)\). (e.g., \((1/2 \times 1/3) \times 1/4 = 1/6 \times 1/4 = 1/24\), and \(1/2 \times (1/3 \times 1/4) = 1/2 \times 1/12 = 1/24\)).
• Division: Not associative. \((a \div b) \div c \neq a \div (b \div c)\).

4. Distributivity of Multiplication over Addition/Subtraction

• Definition: Multiplication distributes over addition and subtraction.
• For Rational Numbers (Q):
• \(a \times (b+c) = a \times b + a \times c\)
• \(a \times (b-c) = a \times b - a \times c\)
• Example: \(1/2 \times (1/3+1/4) = 1/2 \times 7/12 = 7/24\). Also, \((1/2 \times 1/3) + (1/2 \times 1/4) = 1/6 + 1/8 = 4/24 + 3/24 = 7/24\).

5. Role of Zero (Additive Identity)

• For any rational number \(a\), \(a+0 = 0+a = a\).
• Zero is the additive identity for rational numbers.

6. Role of One (Multiplicative Identity)

• For any rational number \(a\), \(a \times 1 = 1 \times a = a\).
• One is the multiplicative identity for rational numbers.

7. Additive Inverse (Negative of a Number)

• For any rational number \(a/b\), its additive inverse is \(-a/b\) such that \(a/b + (-a/b) = 0\).
• Example: Additive inverse of \(2/3\) is \(-2/3\). Additive inverse of \(-5/7\) is \(5/7\).

8. Multiplicative Inverse (Reciprocal)

• For any non-zero rational number \(a/b\), its multiplicative inverse (reciprocal) is \(b/a\) such that \(a/b \times b/a = 1\).
• Example: Multiplicative inverse of \(2/3\) is \(3/2\). Multiplicative inverse of \(-5/7\) is \(-7/5\).
• Note: The multiplicative inverse of 0 does not exist.

Every rational number can be represented by a unique point on the number line.

Steps for Representation

1. Draw a Number Line: Mark 0 at the center, positive integers to the right, and negative integers to the left.
2. Identify Integers: Locate the two integers between which the rational number lies.

• For \(p/q\) where \(p < q\) and both positive, it lies between 0 and 1.
• For \(p/q\) where \(p > q\) (improper fraction), convert it to a mixed fraction first. E.g., \(7/3 = 2 \frac{1}{3}\), so it lies between 2 and 3.

1. Divide the Segment: Divide the segment between the identified integers into \(q\) (denominator) equal parts.
2. Mark the Point: Count \(p\) (numerator) parts from the left integer to locate the point.

Examples

• Represent \(1/2\):

1. Draw a number line.
2. \(1/2\) is between 0 and 1.
3. Divide the segment between 0 and 1 into 2 equal parts.
4. Mark the first part from 0.

• Represent \(-3/4\):

1. Draw a number line.
2. \(-3/4\) is between -1 and 0.
3. Divide the segment between -1 and 0 into 4 equal parts.
4. Mark the third part from 0 towards -1.

• Represent \(7/3\):

1. Convert to mixed fraction: \(7/3 = 2 \frac{1}{3}\).
2. It lies between 2 and 3.
3. Divide the segment between 2 and 3 into 3 equal parts.
4. Mark the first part from 2.

Unlike integers, there are infinitely many rational numbers between any two given rational numbers. This property is called density.

Method 1: Common Denominator Method

1. Find LCM: Find the Least Common Multiple (LCM) of the denominators of the two given rational numbers.
2. Convert to Equivalent Fractions: Convert both rational numbers to equivalent fractions with the LCM as the common denominator.
3. Check Numerators: If there are enough integers between the new numerators, then the rational numbers with the common denominator and those numerators are the required numbers.
4. Multiply if Needed: If not enough numbers are found, multiply both the numerator and denominator of both equivalent fractions by \(10/10\) (or \(100/100\), etc.) to create more space between them.

Method 2: Mean Method (Average Method)

1. Find the Mean: If \(a\) and \(b\) are two rational numbers, then \((a+b)/2\) is a rational number between them.
2. Repeat: You can repeat this process to find more rational numbers. For example, find the mean of \(a\) and \((a+b)/2\), or \((a+b)/2\) and \(b\).

Example: Find 5 rational numbers between \(1/2\) and \(2/3\)

• Method 1 (Common Denominator):

1. LCM of 2 and 3 is 6.
2. \(1/2 = 3/6\) and \(2/3 = 4/6\).
3. No integers between 3 and 4. Multiply by \(10/10\).
4. \(3/6 = 30/60\) and \(4/6 = 40/60\).
5. Now, we can easily find 5 rational numbers: \(31/60, 32/60, 33/60, 34/60, 35/60\).

• Method 2 (Mean Method):

1. First rational number: \((1/2 + 2/3)/2 = (3/6 + 4/6)/2 = (7/6)/2 = 7/12\).
2. Second rational number (between \(1/2\) and \(7/12\)): \((1/2 + 7/12)/2 = (6/12 + 7/12)/2 = (13/12)/2 = 13/24\).
3. Third rational number (between \(7/12\) and \(2/3\)): \((7/12 + 2/3)/2 = (7/12 + 8/12)/2 = (15/12)/2 = 15/24 = 5/8\).
4. Continue this process to find more.

Performing arithmetic operations on rational numbers follows similar rules to fractions, with careful attention to signs.

1. Addition of Rational Numbers

• Same Denominators: Add the numerators and keep the denominator same.
• \(a/c + b/c = (a+b)/c\)
• Example: \(2/5 + 1/5 = (2+1)/5 = 3/5\)
• Different Denominators: Find the LCM of the denominators, convert to equivalent fractions with the LCM as the new denominator, then add numerators.
• Example: \(1/2 + 1/3\)
• LCM of 2 and 3 is 6.
• \(1/2 = 3/6\), \(1/3 = 2/6\)
• \(3/6 + 2/6 = (3+2)/6 = 5/6\)

2. Subtraction of Rational Numbers

• Same Denominators: Subtract the numerators and keep the denominator same.
• \(a/c - b/c = (a-b)/c\)
• Example: \(4/7 - 1/7 = (4-1)/7 = 3/7\)
• Different Denominators: Find the LCM, convert to equivalent fractions, then subtract numerators.
• Example: \(3/4 - 1/6\)
• LCM of 4 and 6 is 12.
• \(3/4 = 9/12\), \(1/6 = 2/12\)
• \(9/12 - 2/12 = (9-2)/12 = 7/12\)

3. Multiplication of Rational Numbers

• Multiply the numerators together and multiply the denominators together.
• \(a/b \times c/d = (a \times c) / (b \times d)\)
• Example: \(2/3 \times 4/5 = (2 \times 4) / (3 \times 5) = 8/15\)
• Simplification: Always simplify to the lowest terms by cancelling common factors before or after multiplication.
• Example: \(3/4 \times 8/9 = (3 \times 8) / (4 \times 9) = 24/36 = 2/3\). Or, \(3/4 \times 8/9 = (3 \times 2 \times 4) / (4 \times 3 \times 3) = 2/3\).

4. Division of Rational Numbers

• To divide one rational number by another (non-zero) rational number, multiply the first rational number by the reciprocal (multiplicative inverse) of the second rational number.
• \(a/b \div c/d = a/b \times d/c = (a \times d) / (b \times c)\)
• Example: \(2/3 \div 4/5 = 2/3 \times 5/4 = (2 \times 5) / (3 \times 4) = 10/12 = 5/6\)

Word Problems

• Read the problem carefully to identify the operation(s) required.
• Convert mixed numbers to improper fractions before performing operations.
• Pay attention to signs (positive/negative).
• Simplify the final answer to its lowest terms.