Rational number: comparative examination with examples

📖 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).

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💡 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:

1. Terminating: The division ends after a few steps. (e.g., 1/4 = 0.25)
2. Non-Terminating but Recurring: The division never ends, but a digit or a block of digits repeats infinitely. (e.g., 1/3 = 0.333...)

📝 Practice Zone: Rational Numbers

🟢 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.

🟡 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.

🔴 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.

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).

🎯 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