Introduction to Rational Numbers
What are Rational Numbers?
Rational numbers are numbers that can be written in the form
\( \dfrac{p}{q} \), where:
- \( p \) and \( q \) are integers
- \( q \neq 0 \)
Example: \( \dfrac{3}{4}, -\dfrac{7}{2}, 0, \dfrac{-5}{9} \).
Relation with Other Numbers
- All natural numbers, whole numbers, and integers are also rational numbers.
- Example: \( 5 = \dfrac{5}{1} \), \( -2 = \dfrac{-2}{1} \).
Representation on Number Line
Rational numbers can be plotted on a number line just like integers.
For example, \( \dfrac{1}{2} \) lies between 0 and 1.
Standard Form
A rational number is said to be in standard form if:
- The denominator is positive
- Numerator and denominator have no common factor except 1
Example: \( \dfrac{-2}{3} \) is in standard form.
But \( \dfrac{6}{-9} = \dfrac{-2}{3} \) (after simplification).
Operations on Rational Numbers
Addition of Rational Numbers
Example 1: Add \( \dfrac{2}{3} + \dfrac{5}{6} \)
👉 Method 1: Common Denominator
LCM of denominators \(3,6 = 6\)
\( \dfrac{2}{3}\ becomes = \dfrac{4}{6} \)
So, \( \dfrac{4}{6} + \dfrac{5}{6} = \dfrac{4+5}{6} = \dfrac{9}{6} \)
Final Answer after reducing = \( \dfrac{3}{2} \)
👉 Method 2: Cross Multiplication
Formula: \( \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a \times d+b \times c}{b \times d} \)
Here, \( \dfrac{2}{3} + \dfrac{5}{6} = \dfrac{2\times6 + 5\times3}{3\times6} \)
= \( \dfrac{12+15}{18} = \dfrac{27}{18} \)
Final Answer after reducing = \( \dfrac{3}{2} \)
Subtraction of Rational Numbers
Example 2: Subtract \( \dfrac{5}{12} - \dfrac{1}{8} \)
👉 Method 1: Common Denominator
LCM of denominators \(12,8 = 24\)
\( \dfrac{5}{12} = \dfrac{10}{24}, \quad \dfrac{1}{8} = \dfrac{3}{24} \)
So, \( \dfrac{10}{24} - \dfrac{3}{24} = \dfrac{7}{24} \)
Final Answer = \( \dfrac{7}{24} \)
👉 Method 2: Cross Multiplication
Formula: \( \dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad-bc}{bd} \)
Here, \( \dfrac{5}{12} - \dfrac{1}{8} = \dfrac{5\times8 - 1\times12}{12\times8} \)
= \( \dfrac{40-12}{96} = \dfrac{28}{96} \)
Final Answer = \( \dfrac{7}{24} \)
Multiplication of Rational Numbers
Example 3: Multiply \( -\dfrac{2}{3} \times \dfrac{15}{8} \)
Multiply numerators and denominators directly:
\( \dfrac{-2 \times 15}{3 \times 8} = \dfrac{-30}{24} \)
Simplify: \( \dfrac{-30}{24} = \dfrac{-5}{4} \)
Final Answer = \( -\dfrac{5}{4} \)
Division of Rational Numbers
Example 4: Divide \( \dfrac{4}{9} \div \dfrac{2}{3} \)
Rule: Division = Multiply by reciprocal
\( \dfrac{4}{9} \div \dfrac{2}{3} = \dfrac{4}{9} \times \dfrac{3}{2} \)
= \( \dfrac{12}{18} \)
Simplify: \( \dfrac{12}{18} = \dfrac{2}{3} \)
Final Answer = \( \dfrac{2}{3} \)
Properties of Rational Numbers
- Closure: Rational numbers are closed under addition, subtraction, multiplication, and division (except division by 0).
- Commutativity: Addition and multiplication are commutative.
- Associativity: Addition and multiplication are associative.
- Existence of Identity:
- Additive identity: 0 (since \( a + 0 = a \))
- Multiplicative identity: 1 (since \( a \times 1 = a \))
- Existence of Inverse:
- Additive inverse of \( a \) is \( -a \).
- Multiplicative inverse of \( \dfrac{p}{q} \) is \( \dfrac{q}{p} \) (if \( p \neq 0 \)).
- Distributivity: \( a \times (b + c) = ab + ac \).
Decimal Representation
Every rational number can be expressed as a decimal.
- If denominator has factors 2 or 5 only → terminating decimal.
- Otherwise → non-terminating but repeating decimal.
Example:
\( \dfrac{1}{2} = 0.5 \) (terminating)
\( \dfrac{2}{3} = 0.666... \) (repeating)
Summary: Rational numbers form a very important part of the number system.
They include integers, fractions, terminating decimals, and repeating decimals.
Their properties make them the foundation of algebra and higher mathematics.