🔢 Rational Numbers
Closure Property: Sum or product of two rational numbers is always a rational number
▶ Play Door 1Introduction to Rational Numbers
What are Rational Numbers?
A rational number is any number that can be written in the form \(p/q\), where \(p\) and \(q\) are integers and \(q ≠ 0\). The word "rational" comes from the word "ratio".
Examples of rational numbers: \(1/2, -3/4, 5/1, 0, -2, 7/-8\)
What are Properties of Rational Numbers?
Properties are rules that always hold true when we perform operations (like addition, subtraction, multiplication, division) on rational numbers.
The Four Main Properties:
PropertyMeaningAddition ExampleMultiplication ExampleClosure PropertyWhen you add/multiply two rational numbers, the result is also a rational number\(1/2 + 1/3 = 5/6\) (rational)\(2/3 × 4/5 = 8/15\) (rational)Commutative PropertyChanging the order does not change the result\(1/4 + 3/4 = 3/4 + 1/4\)\(2/5 × 1/3 = 1/3 × 2/5\)Associative PropertyChanging the grouping does not change the…
Worked example
Problem: Example 1: Check if \(2/3 + 1/6\) is a rational number. Also verify the commutative property for addition.
- - Step 1: Add \(2/3 + 1/6 = 4/6 + 1/6 = 5/6\)
- - Step 2: \(5/6\) is in the form \(p/q\) with \(q ≠ 0\), so it is a rational number
- - Step 3: For commutative property, check \(1/6 + 2/3 = 1/6 + 4/6 = 5/6\)
- - Step 4: Both orders give \(5/6\), so commutative property holds
Answer: - Step 4: Both orders give \(5/6\), so commutative property holds
Operations on Rational Numbers
What is Additive Inverse?
The additive inverse of a rational number \(a\) is the number that, when added to \(a\), gives zero (the additive identity). For any rational number \(p/q\), its additive inverse is \(-p/q\).
• Formula: \(a + (-a) = 0\)
• Example: Additive inverse of \(3/5\) is \(-3/5\) because \(3/5 + -3/5 = 0\)
What is Multiplicative Inverse (Reciprocal)?
The multiplicative inverse of a non-zero rational number \(a\) is the number that, when multiplied by \(a\), gives 1 (the multiplicative identity). For \(p/q ≠ 0\), its multiplicative inverse is \(q/p\).
• Formula: \(a × 1/a = 1\) (where \(a ≠ 0\))
• Example: Multiplicative inverse of \(2/3\) is \(3/2\) because \(2/3 × 3/2 = 1\)
Real-life Connection: Think of additive inverse like walking forward 5 steps and then backward 5 steps to return to start. Multiplicative inverse is like doubling a recipe…
Worked example
Problem: Example 1: Find the additive inverse and multiplicative inverse of \(-5/7\).
- - Step 1: Additive inverse of \(p/q\) is \(-p/q\)
- - Step 2: Here \(p = -5\), so \(-p = -(-5) = 5\)
- - Step 3: Additive inverse = \(5/7\)
- - Step 4: Check: \(-5/7 + 5/7 = 0\) ✓
- - Step 5: Multiplicative inverse of \(p/q\) is \(q/p\) (where \(p ≠ 0\))
Answer: - Step 5: Multiplicative inverse of \(p/q\) is \(q/p\) (where \(p ≠ 0\))
Comparison, Standard Form & Ordering
What is Standard Form of a Rational Number?
A rational number \(p/q\) is in standard form (or simplest form) when:
• \(q > 0\) (denominator is positive)
• \(p\) and \(q\) have no common factor other than 1 (they are co-prime)
Steps to convert to standard form:
• Make denominator positive (multiply numerator and denominator by -1 if needed)
• Find the HCF (GCD) of numerator and denominator
• Divide both numerator and denominator by the HCF
How to Compare Rational Numbers?
Method 1 (Same denominator): Compare numerators directly
• \(3/7 > 2/7\) because \(3 > 2\)
Method 2 (Different denominators): Use cross-multiplication
• For \(a/b\) and \(c/d\), compare \(a × d\) and \(c × b\)
• If \(a × d > c × b\), then \(a/b > c/d\)
Method 3 (Decimal conversion): Convert to decimals and compare
Ordering of Rational Numbers
Ascending order (smallest to largest): Arrange…
Worked example
Problem: Example 1: Express \(-24/36\) in standard form and compare it with \(-2/3\).
- - Step 1: Denominator is positive (36 > 0) ✓
- - Step 2: Find HCF of 24 and 36
- - Step 3: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- - Step 4: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- - Step 5: HCF = 12
Answer: - Step 5: HCF = 12