๐ Deep Dive into Rational Numbers
In competitive exams like SSC, IBPS, and Railways, questions on Rational Numbers often appear in the form of decimal conversions, recurring digits, and comparison of fractions. Let’s master the logic behind them!
1. What exactly is a Rational Number?
A number is called Rational if it can be written in the form of p/q, where:
- p and q are integers.
- q ≠ 0 (The denominator can never be zero).
๐ก Pro Tip for Aspirants:
All integers are rational numbers because they can be written as n/1 (e.g., 5 = 5/1). Zero (0) is also a rational number.
2. Decimals: Terminating vs. Recurring
Rational numbers, when expressed in decimal form, follow two patterns:
- Terminating: The division ends after a few steps. (e.g., 1/4 = 0.25)
- Non-Terminating but Recurring: The division never ends, but a digit or a block of digits repeats infinitely. (e.g., 1/3 = 0.333...)
๐ Practice Zone: Rational Numbers
๐ข Beginner Level: The Basics
Q1. Identify which of the following is NOT a rational number: 22/7, √4, 0, or √5?
Solution:
- 22/7 is in p/q form.
- √4 = 2 (which is 2/1).
- 0 can be written as 0/1.
- √5 is a non-terminating, non-recurring decimal. Thus, it is Irrational.
- 22/7 is in p/q form.
- √4 = 2 (which is 2/1).
- 0 can be written as 0/1.
- √5 is a non-terminating, non-recurring decimal. Thus, it is Irrational.
๐ก Intermediate Level: Fractions & Comparison
Q2. Arrange the following rational numbers in ascending order: 2/3, 5/6, and 3/4.
Solution:
Find the LCM of denominators (3, 6, 4) = 12.
- 2/3 = (2×4)/(3×4) = 8/12
- 5/6 = (5×2)/(6×2) = 10/12
- 3/4 = (3×3)/(4×3) = 9/12
Comparing numerators: 8 < 9 < 10.
Ascending Order: 2/3 < 3/4 < 5/6.
Find the LCM of denominators (3, 6, 4) = 12.
- 2/3 = (2×4)/(3×4) = 8/12
- 5/6 = (5×2)/(6×2) = 10/12
- 3/4 = (3×3)/(4×3) = 9/12
Comparing numerators: 8 < 9 < 10.
Ascending Order: 2/3 < 3/4 < 5/6.
๐ด Advanced Level: Recurring Decimals (Bar Questions)
Q3. Convert 0.373737... (or 0.37̅) into a vulgar fraction.
Solution (Shortcut Method):
For purely recurring decimals:
1. Write the repeating number in the numerator: 37.
2. In the denominator, write '9' as many times as there are digits in the repeating block: 99.
Result: 37/99.
For purely recurring decimals:
1. Write the repeating number in the numerator: 37.
2. In the denominator, write '9' as many times as there are digits in the repeating block: 99.
Result: 37/99.
Q4. Simplify: 0.12333... (where only 3 repeats).
Solution (Mixed Recurring Method):
Numerator: (Full number) - (Non-repeating part) = 123 - 12 = 111.
Denominator: A '9' for every repeating digit followed by a '0' for every non-repeating digit after the decimal.
Denominator = 900.
Result: 111/900 (can be simplified to 37/300).
Numerator: (Full number) - (Non-repeating part) = 123 - 12 = 111.
Denominator: A '9' for every repeating digit followed by a '0' for every non-repeating digit after the decimal.
Denominator = 900.
Result: 111/900 (can be simplified to 37/300).
๐ฏ Summary Table
| Property | Rational Numbers |
|---|---|
| Form | p/q (q ≠ 0) |
| Addition/Subtraction | Always Rational |
| Multiplication/Division | Always Rational (if divisor ≠ 0) |
"Mathematics is not about numbers, equations, or algorithms: it is about understanding." — William Paul Thurston
