Understanding Rational Numbers

• This chapter introduces rational numbers
• It covers their properties, operations, and real-life applications

The Concept of Rational Numbers

• The chapter aims to enable students to practically understand and express rational numbers as fractions

Rational Numbers on a Number Line

• A fraction represents a portion of a whole
• Numerator (top number)
• Denominator (bottom number)
• Proper fractions have a numerator less than the denominator
• Improper fractions have a numerator greater than or equal to the denominator
• Improper fractions can be expressed as mixed fractions
• Rational number definition: A number in the form a/b, where a and b are integers and b ≠ 0
• The set of rational numbers is denoted by ℚ
• Positive rational numbers are on the right side of zero on a number line
• Negative rational numbers are on the left side of zero on a number line
• Positive proper fractions exist between zero and one on a number line
• Improper fractions are converted into mixed fractions before representation on a number line
• Rational numbers with the same distance from 0 but on opposite sides are opposites:
  • For example 3/2 and -3/2

Relationship among W, ℤ, and ℚ

• A set is a collection of items
• Items in a set are elements, denoted by ∈
• Venn diagrams use intersecting circles to show relationships among sets of numbers
• Natural numbers (ℕ)
• Whole numbers (W)
• Integers (ℤ)

Absolute Value

• The absolute value of a rational number is its distance from zero on a number line, regardless of direction
• The absolute value of a rational number 'xx', denoted by |xx|, is defined as:
  • |xx| = { xx, if xx ≥ 0
  • −xx, if xx < 0
• For equations of the form |xx| = aa:
  • Two solutions exist if aa > 0
  • One solution exists if aa = 0
  • No solution exists if aa < 0

Comparing and Ordering Rational Numbers

• Compare rational numbers by inserting inequalities such as <, =, or > to express the correct relationship
• A rational number a/b can be expressed as a decimal by dividing 'a' by 'b'
• Decimal numbers are compared by comparing the different place values from left to right, starting with the integer part
• When comparing fractions with the same denominator, the fraction with the greater numerator is the greater number
• Equivalent fractions represent the same point on a number line:
  • For any fraction a/b, a/b = (a × m) / (b × m) if m is a rational number not equal to 0
• When comparing fractions with different denominators, use one of two methods:
  • Method 1:
    • Convert the fractions to equivalent fractions with the same denominators
    • Compare the numerators
  • Method 2 (Cross-product method):
    • If a/b and c/d are rational numbers with positive denominators:
    • a/b < c/d if and only if ad < bc
    • a/b > c/d if and only if ad > bc
    • a/b = c/d if and only if ad = bc
• Numbers on the left of the number line are smaller than those on the right
• Every positive rational number is greater than zero
• Every negative rational number is less than zero
• Every positive rational number is greater than every negative rational number
• Among two negative rational numbers, the one with the largest absolute value is smaller than the other
• Ordering rational numbers rearranges numbers in ascending/descending order
• Ordering fractions means rewriting them to have the same denominator or converting to decimals

Operations on Rational Numbers: Addition

• Adding rational numbers with the same denominators: a/b + c/b = (a+c)/b
• To add rational numbers with different denominators:
  • Make all denominators positive.
  • Find the LCM of the denominators.
  • Find equivalent rational numbers with the common denominator.
  • Add the numerators

Rules for Addition of Rational Numbers

• For two negative rational numbers
  • Sign: Negative (-)
  • Take the sum of the absolute values of the addends.
  • Put the sign in front of the sum
• .sign of
• To find the sum of two rational numbers, where the signs of the addends are different:
  • Take the sign of the addend with the greater absolute value.
  • Find the absolute values of both numbers and subtract the addend with smaller absolute value from the addend with greater absolute value.
  • Put the sign in front of the difference.
• Use fraction bars to perform addition of rational numbers graphically

Properties of Addition

• Commutative: a + b = b + a
• Associative: a + (b + c) = (a + b) + c
• Property of 0: a + 0 = a = 0 + a
• Property of opposites: a + (-a) = 0

Subtraction of Rational Numbers

• Follows the same process as that of addition as it is its inverse
• For rational numbers a/b and c/d, subtracting c/d from a/b is equivalent to adding the negative of c/d to a/b: i.e., a/b - c/d = a/b + (-c/d)
• The difference of two rational numbers is always a rational number
• Addition and subtraction are inverse operations of each other

Multiplication of Rational Numbers

• Multiply 2 or more rational number, simply multiply the numerator by numerator and denominator by denominator
• Ensure the answer is in its lowest term

Additional points to note

• If the two factors are both positive, the product is positive
• If the two factors are both negative, the product is positive
• If the factors have the opposite sign, the product is negative
• The product of any factor and zero is zero

Properties of Multiplication

• Commutative: a × b = b × a
• Associative: a × (b × c) = (a × b) × c
• Distributive: a × (b + c) = a × b + a × c
• Property of 0: a × 0 = 0 = 0 × a
• Property of 1: a × 1 = a = 1 × a

Division of Rational Numbers

• When dividing, determine the sign of the quotient
  • If the dividend and the divisor have the same sign i.e. both positive or both negative, then the quotient is positive
  • If the dividend and the divisor have opposite signs i.e. positive and negative then the quotient is negative
• The value of quotient is the absolute values for the dividend over the divisor
• Reciprocal: For any rational number a/b where a ≠ 0, then b/a is called the reciprocal of a/b

Real Life Applications

• Rational numbers can be used to share calculations with friends
• Calculate interest rates on loans and shopping discounts