1. Introduction to Rational Numbers
Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. All integers, fractions, and terminating or repeating decimals are rational numbers.
Examples:
- 2/3 is a rational number (p = 2, q = 3)
- -5/7 is a rational number (p = -5, q = 7)
- 4 can be written as 4/1, so it is rational
- 0 can be written as 0/5, so it is rational
2. Properties of Rational Numbers
- Closure: Rational numbers are closed under addition, subtraction, and multiplication.
- Commutativity: Addition and multiplication of rational numbers are commutative.
- Associativity: Addition and multiplication are associative.
- Existence of Identity: 0 is the additive identity, 1 is the multiplicative identity.
- Existence of Inverse: Every rational number except 0 has a multiplicative inverse.
- Distributivity: Multiplication is distributive over addition.
3. Representation on the Number Line
Rational numbers can be represented on the number line. For example, to represent 3/4, divide the segment between 0 and 1 into 4 equal parts and count 3 parts from 0.
4. Standard Form of a Rational Number
A rational number is said to be in standard form if the denominator is positive and the numerator and denominator have no common factors except 1.
- Example: -6/8 can be written as -3/4 in standard form.
5. Comparison of Rational Numbers
To compare rational numbers, convert them to like denominators and then compare the numerators.
- Example: Compare 2/5 and 3/7. Find LCM of 5 and 7 (which is 35), convert both: 2/5 = 14/35, 3/7 = 15/35. So, 3/7 > 2/5.
6. Operations on Rational Numbers
- Addition/Subtraction: Convert to like denominators, then add/subtract numerators.
- Multiplication: Multiply numerators and denominators directly.
- Division: Multiply by the reciprocal of the divisor.
Example:
- 1/2 + 1/3 = (3+2)/6 = 5/6
- 2/5 × 3/7 = 6/35
- 4/9 ÷ 2/3 = 4/9 × 3/2 = 12/18 = 2/3
7. Word Problems
- What is the sum of 3/4 and -2/5?
- Subtract 5/6 from 1/2.
- Multiply -3/7 by 2/9.
- Divide 7/8 by -1/4.
8. Practice Exercises
- Express -12/16 in standard form.
- Compare 5/12 and 7/18.
- Find the multiplicative inverse of -3/8.
- Represent 2/5 on the number line.
9. Summary
- Rational numbers are numbers that can be written as p/q, where q ≠ 0.
- They include integers, fractions, and terminating/repeating decimals.
- They have properties like closure, commutativity, associativity, and distributivity.
- Operations can be performed using rules for addition, subtraction, multiplication, and division.