Rational Numbers • Topic 1 of 3

Introduction to Rational Numbers

What are Rational Numbers?

A rational number is any number that can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). The word "rational" comes from the word "ratio".

Examples of rational numbers: \(\frac{1}{2}, \frac{-3}{4}, \frac{5}{1}, 0, -2, \frac{7}{-8}\)

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 Example
Closure PropertyWhen you add/multiply two rational numbers, the result is also a rational number\(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) (rational)\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\) (rational)
Commutative PropertyChanging the order does not change the result\(\frac{1}{4} + \frac{3}{4} = \frac{3}{4} + \frac{1}{4}\)\(\frac{2}{5} \times \frac{1}{3} = \frac{1}{3} \times \frac{2}{5}\)
Associative PropertyChanging the grouping does not change the result\((\frac{1}{2}+\frac{1}{3})+\frac{1}{6} = \frac{1}{2}+(\frac{1}{3}+\frac{1}{6})\)\((\frac{1}{2}\times\frac{1}{3})\times\frac{1}{4} = \frac{1}{2}\times(\frac{1}{3}\times\frac{1}{4})\)
Distributive PropertyMultiplication distributes over addition\(a \times (b + c) = a \times b + a \times c\)-

Real-life Example: When sharing pizza among friends, if you have \(\frac{1}{2}\) pizza and get another \(\frac{1}{3}\) pizza, the total \(\frac{5}{6}\) pizza is also a rational number (closure property).

Important Notes:

  • Subtraction and division are NOT commutative ( \(5 - 3 \neq 3 - 5\) )
  • Subtraction and division are NOT associative ( \((8-4)-2 \neq 8-(4-2)\) )
  • The distributive property connects multiplication and addition beautifully!
The Number System — Where Rational Numbers FitReal Numbers ℝRational Numbers ℚIntegers ℤNon-Integers p/qNatural ℕWhole W-3/407/22-511/3← Examples of Rational Numbers (any p/q where q ≠ 0) →
Solved Examples
1
Worked Example
Example 1: Check if \(\frac{2}{3} + \frac{1}{6}\) is a rational number. Also verify the commutative property for addition.
Solution- Step 1: Add \(\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}\) - Step 2: \(\frac{5}{6}\) is in the form \(\frac{p}{q}\) with \(q \neq 0\), so it is a rational number - Step 3: For commutative property, check \(\frac{1}{6} + \frac{2}{3} = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}\) - Step 4: Both orders give \(\frac{5}{6}\), so commutative property holds
2
Worked Example
Example 2: Verify the associative property of multiplication for the rational numbers \(\frac{1}{2}, \frac{2}{3},\) and \(\frac{3}{4}\).
Solution- Step 1: Left side: \((\frac{1}{2} \times \frac{2}{3}) \times \frac{3}{4}\) - Step 2: First multiply \(\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}\) - Step 3: Then \(\frac{1}{3} \times \frac{3}{4} = \frac{3}{12} = \frac{1}{4}\) - Step 4: Right side: \(\frac{1}{2} \times (\frac{2}{3} \times \frac{3}{4})\) - Step 5: First multiply \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) - Step 6: Then \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\) - Step 7: Both sides equal \(\frac{1}{4}\)
3
Worked Example
Example 3: Using the distributive property, simplify: \(\frac{2}{3} \times (\frac{1}{4} + \frac{1}{2})\)
Solution- Step 1: Apply distributive law: \(a \times (b + c) = a \times b + a \times c\) - Step 2: Here \(a = \frac{2}{3}, b = \frac{1}{4}, c = \frac{1}{2}\) - Step 3: \((\frac{2}{3} \times \frac{1}{4}) + (\frac{2}{3} \times \frac{1}{2})\) - Step 4: First term: \(\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}\) - Step 5: Second term: \(\frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}\) - Step 6: Add: \(\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\) - Step 7: Check by solving inside brackets first: \(\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}\); then \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) ✓

Key Points

  • Closure Property: Sum or product of two rational numbers is always a rational number
  • Commutative Property: \(a + b = b + a\) and \(a \times b = b \times a\) for rational numbers
  • Associative Property: \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\)
  • Distributive Property: \(a \times (b + c) = a \times b + a \times c\)
  • Subtraction and division do NOT follow commutative or associative properties
  • These properties help us simplify complex calculations and solve equations easily
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A rational number is of the form $\tfrac{p}{q}$ where:
Explanation: The denominator cannot be zero.
Q2.The additive identity for rational numbers is:
Explanation: $a+0=a$.
Q3.The reciprocal (multiplicative inverse) of $\tfrac27$ is:
Explanation: $\tfrac27\times\tfrac72=1$.
Q4.Is $0$ a rational number?
Explanation: $0=\tfrac01$.
Practice more MCQs → 📝 Assignment · 35 marks