Rational Numbers
You have already studied natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, 3, ...), integers (..., -2, -1, 0, 1, 2, ...), and fractions. Rational numbers include all of these and more.
A rational number is any number that can be written in the form p/q, where p and q are integers and q is not equal to zero.
Rational numbers include positive fractions (like 3/4), negative fractions (like -5/7), whole numbers (like 5 = 5/1), integers (like -3 = -3/1), and zero (0 = 0/1). They form a complete number system that is closed under addition, subtraction, multiplication, and division (except division by zero).
This topic is covered in the NCERT Class 7 chapter on Rational Numbers.
What is Rational Numbers?
Definition: A rational number is a number that can be expressed in the form p/q, where:
- p is an integer (called the numerator)
- q is a non-zero integer (called the denominator)
- q ≠ 0
Rational Number = p/q, where p, q are integers and q ≠ 0
Examples of rational numbers:
- 3/4, -5/7, 2/1 (= 2), -8/1 (= -8), 0/1 (= 0)
- 7 is rational because 7 = 7/1
- -3 is rational because -3 = -3/1
- 0.5 is rational because 0.5 = 1/2
Non-examples:
- √2, √3, π are NOT rational numbers (they are irrational).
- Any number that cannot be written as p/q with integer p and non-zero integer q is not rational.
Types of rational numbers:
- Positive rational numbers: p/q where p and q have the same sign. Example: 3/5, -4/-7 = 4/7.
- Negative rational numbers: p/q where p and q have different signs. Example: -3/5, 4/-7 = -4/7.
- Zero: 0 = 0/q for any non-zero q. Zero is neither positive nor negative.
Rational Numbers Formula
Standard Form of a Rational Number:
A rational number p/q is in standard form when:
- The denominator q is positive.
- p and q have no common factor other than 1 (i.e., HCF(|p|, q) = 1).
Steps to convert to standard form:
- If the denominator is negative, multiply both numerator and denominator by -1 to make the denominator positive.
- Divide both numerator and denominator by their HCF.
Equivalent Rational Numbers:
p/q = (p × k) / (q × k) for any non-zero integer k
Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent rational number.
Comparing Rational Numbers:
- Make the denominators equal (find LCM).
- Compare the numerators.
- The rational number with the larger numerator is greater.
Derivation and Proof
The Number System Hierarchy:
Each new number system includes the previous one and adds more numbers:
- Natural Numbers (N): 1, 2, 3, 4, ... (counting numbers)
- Whole Numbers (W): 0, 1, 2, 3, ... (add zero to natural numbers)
- Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ... (add negatives)
- Rational Numbers (Q): All numbers of the form p/q where p, q are integers, q ≠ 0 (add fractions)
Relationship: N ⊂ W ⊂ Z ⊂ Q
Every natural number is a whole number. Every whole number is an integer. Every integer is a rational number.
Why do we need rational numbers?
- Integers cannot represent parts: you cannot express "half" using integers alone.
- Fractions (positive p/q) handle parts but cannot represent negative quantities.
- Rational numbers combine both: they can express any quantity that can be written as a ratio of two integers.
Why q ≠ 0?
- Division by zero is undefined. 5/0 has no meaning.
- If q = 0, the expression p/0 is not a number.
Types and Properties
Rational numbers can be classified as follows:
1. Positive Rational Numbers:
- Both numerator and denominator have the same sign.
- Examples: 3/5, 7/2, -4/-9 = 4/9
- They lie to the right of 0 on the number line.
2. Negative Rational Numbers:
- Numerator and denominator have different signs.
- Examples: -3/5, 4/-7 = -4/7
- They lie to the left of 0 on the number line.
3. Zero:
- 0 = 0/q for any non-zero q.
- Zero is neither positive nor negative.
4. Integers as Rational Numbers:
- Every integer n = n/1.
- Examples: 5 = 5/1, -3 = -3/1, 0 = 0/1
5. Proper Rational Numbers:
- |Numerator| < |Denominator|. They lie between -1 and 1.
- Examples: 2/5, -3/7
6. Improper Rational Numbers:
- |Numerator| ≥ |Denominator|.
- Examples: 7/3, -9/4
Solved Examples
Example 1: Example 1: Identifying rational numbers
Problem: Which of these are rational numbers? (a) 5 (b) -3/7 (c) √5 (d) 0 (e) 2.5
Solution:
- (a) 5 = 5/1. Rational.
- (b) -3/7. p = -3, q = 7, both integers, q ≠ 0. Rational.
- (c) √5. Cannot be written as p/q. Not rational (irrational).
- (d) 0 = 0/1. Rational.
- (e) 2.5 = 5/2. Rational.
Answer: (a), (b), (d), and (e) are rational numbers.
Example 2: Example 2: Converting to standard form
Problem: Write -8/-12 in standard form.
Solution:
Step 1: Make denominator positive
- -8/-12 = 8/12 (both signs cancel)
Step 2: Simplify by dividing by HCF
- HCF of 8 and 12 = 4
- 8/12 = (8 ÷ 4)/(12 ÷ 4) = 2/3
Answer: Standard form = 2/3
Example 3: Example 3: Finding equivalent rational numbers
Problem: Write three equivalent rational numbers for 2/5.
Solution:
Multiply numerator and denominator by the same number:
- 2/5 = (2×2)/(5×2) = 4/10
- 2/5 = (2×3)/(5×3) = 6/15
- 2/5 = (2×4)/(5×4) = 8/20
Answer: 4/10, 6/15, and 8/20 are equivalent to 2/5.
Example 4: Example 4: Comparing rational numbers
Problem: Which is greater: 3/7 or 5/9?
Solution:
Step 1: Find LCM of denominators
- LCM of 7 and 9 = 63
Step 2: Convert to equivalent fractions
- 3/7 = (3×9)/(7×9) = 27/63
- 5/9 = (5×7)/(9×7) = 35/63
Step 3: Compare numerators
- 35 > 27, so 35/63 > 27/63
Answer: 5/9 is greater than 3/7.
Example 5: Example 5: Positive or negative?
Problem: Classify as positive or negative: (a) -5/8 (b) -7/-3 (c) 4/-9 (d) 6/11
Solution:
- (a) -5/8: Different signs → Negative
- (b) -7/-3: Same signs → Positive (= 7/3)
- (c) 4/-9: Different signs → Negative (= -4/9)
- (d) 6/11: Both positive → Positive
Example 6: Example 6: Representing on a number line
Problem: Represent 3/4 on a number line.
Solution:
- 3/4 lies between 0 and 1 (since 0 < 3/4 < 1).
- Divide the segment from 0 to 1 into 4 equal parts.
- Each part = 1/4.
- Count 3 parts from 0: the point is at 3/4.
Answer: 3/4 is located at the third division mark when the segment from 0 to 1 is divided into 4 equal parts.
Example 7: Example 7: Rational number between two numbers
Problem: Find a rational number between 1/3 and 1/2.
Solution:
Method: Average of the two numbers
- Average = (1/3 + 1/2) / 2
- = (2/6 + 3/6) / 2
- = (5/6) / 2
- = 5/12
Verification:
- 1/3 = 4/12, 5/12, 1/2 = 6/12
- 4/12 < 5/12 < 6/12 ✔
Answer: 5/12 is a rational number between 1/3 and 1/2.
Example 8: Example 8: Converting decimal to rational form
Problem: Express 0.75 as a rational number in standard form.
Solution:
- 0.75 = 75/100
- HCF of 75 and 100 = 25
- 75/100 = (75 ÷ 25)/(100 ÷ 25) = 3/4
Answer: 0.75 = 3/4
Example 9: Example 9: Arranging in ascending order
Problem: Arrange in ascending order: -2/3, 1/4, -1/2, 3/8
Solution:
Step 1: Find LCM of denominators 3, 4, 2, 8
- LCM = 24
Step 2: Convert to equivalent fractions
- -2/3 = -16/24
- 1/4 = 6/24
- -1/2 = -12/24
- 3/8 = 9/24
Step 3: Arrange by numerator
- -16/24 < -12/24 < 6/24 < 9/24
Answer: -2/3 < -1/2 < 1/4 < 3/8
Example 10: Example 10: Integer as a rational number
Problem: Express -5 as a rational number with denominator 8.
Solution:
- -5 = -5/1
- Multiply numerator and denominator by 8:
- -5/1 = (-5 × 8)/(1 × 8) = -40/8
Verification: -40/8 = -5 ✔
Answer: -5 = -40/8
Real-World Applications
Temperature: Temperatures can be positive or negative. -3.5°C is a rational number. Temperature differences involve rational number operations.
Money: Financial calculations involve rational numbers. Prices like Rs 24.50 = 49/2, or debts represented as negative amounts.
Measurement: Measurements often result in fractions: 3.75 metres = 15/4 metres. Rational numbers allow precise measurement.
Cooking: Recipes use fractions: "add 2/3 cup of sugar" or "1.5 teaspoons of salt."
Coordinates: Points on a graph can have rational coordinates: (3/2, -5/4). Maps and navigation use rational numbers.
Science: Scientific measurements, concentrations, and ratios are expressed as rational numbers.
Key Points to Remember
- A rational number is of the form p/q where p and q are integers and q ≠ 0.
- Every integer is a rational number (n = n/1).
- Every fraction is a rational number.
- 0 is a rational number (0 = 0/1). It is neither positive nor negative.
- Positive rational numbers: p and q have the same sign.
- Negative rational numbers: p and q have different signs.
- Standard form: denominator is positive, and HCF of |p| and q is 1.
- Equivalent rational numbers: p/q = (pk)/(qk) for any non-zero integer k.
- To compare rational numbers: make denominators equal, then compare numerators.
- There are infinitely many rational numbers between any two rational numbers.
- The number system hierarchy: N ⊂ W ⊂ Z ⊂ Q.
Practice Problems
- Write 5 rational numbers equivalent to -3/5.
- Convert -12/-18 to standard form.
- Arrange in descending order: 2/3, -4/5, 1/6, -1/3.
- Find three rational numbers between 1/4 and 1/3.
- Represent -2/5 on a number line.
- Which is greater: -5/6 or -7/8?
- Express -7 as a rational number with denominator 11.
- Is 0 a positive rational number or a negative rational number?
Frequently Asked Questions
Q1. What is a rational number?
A rational number is a number that can be written as p/q, where p and q are integers and q is not zero. Examples: 3/4, -2/5, 7 (= 7/1), 0 (= 0/1).
Q2. Is every integer a rational number?
Yes. Every integer n can be written as n/1, which is in the form p/q with q = 1 (non-zero). So every integer is a rational number.
Q3. Is every rational number an integer?
No. 3/4 is a rational number but not an integer. Only rational numbers whose denominator divides the numerator evenly are integers (like 6/3 = 2).
Q4. Why can't the denominator be zero?
Division by zero is undefined. p/0 has no meaning in mathematics. So q must not be zero for p/q to be a valid rational number.
Q5. What is the standard form of a rational number?
A rational number p/q is in standard form when the denominator q is positive and p and q have no common factor other than 1. Example: -6/8 in standard form = -3/4.
Q6. How do you compare two rational numbers?
Make the denominators equal by finding the LCM. Then compare the numerators. The number with the larger numerator is greater (when denominators are the same and positive).
Q7. Is 0 a rational number?
Yes. 0 = 0/1 (or 0/any non-zero number). Since 0 and 1 are integers and 1 is not zero, 0 is a rational number. It is neither positive nor negative.
Q8. How many rational numbers are between 0 and 1?
Infinitely many. Between any two rational numbers, there are infinitely many rational numbers. For example, between 0 and 1: 1/2, 1/3, 1/4, 2/5, 3/7, ... the list never ends.
Q9. Is √2 a rational number?
No. √2 cannot be written as p/q for any integers p and q. It is an irrational number. Its decimal expansion (1.41421356...) never terminates and never repeats.
Q10. What is the difference between a fraction and a rational number?
A fraction typically refers to a positive number of the form p/q (with positive p and q). A rational number can be positive, negative, or zero. All fractions are rational numbers, but not all rational numbers are fractions (e.g., -3/5 is rational but not typically called a fraction).
Related Topics
- Equivalent Rational Numbers
- Rational Numbers on Number Line
- Operations on Rational Numbers
- Standard Form of Rational Number
- Comparing Rational Numbers
- Properties of Rational Numbers
- Rational Numbers Between Two Numbers
- Additive Inverse of Rational Numbers
- Multiplicative Inverse of Rational Numbers
- Density Property of Rational Numbers
