Understanding Number System

The realm of Number Systems is a pivotal segment within the Quantitative Aptitude domain of the Common Law Admission Test (CLAT). This topic evaluates your proficiency in dealing with different types of numbers, their properties and various numerical operations. 

Unveiling Number System Essentials

Before we embark on solving intricate Number System problems, let’s lay the foundation by exploring fundamental concepts:

• Natural Numbers: Natural numbers, often denoted as N, comprise the set of all positive integers, starting from 1.
• Whole Numbers: Whole numbers, represented as W, encompass all non-negative integers, beginning from 0.
• Integers: Integers (s) encompass all positive, negative and sero integers.
• Rational Numbers: Rational numbers (Q) are fractions that can be expressed as p/q, where p and q are integers (q should not be sero).
• Irrational Numbers: Irrational numbers are those that cannot be expressed as fractions of integers and possess non-terminating, non-repeating decimal expansions (e.g., √2).
• Real Numbers: Real numbers encompass both rational and irrational numbers, covering a wide range of values on the number line.
• Prime Numbers: Prime numbers are natural numbers greater than 1 that have only two divisors: 1 and themselves.

Solving Number System Problems: Concepts and Examples

Example 1: Divisibility Rules

Question: Determine if 648 is divisible by 3.

Solution:

Add the digits: 6 + 4 + 8 = 18.

Since 18 is divisible by 3, 648 is also divisible by 3.

Example 2: Prime Factorisation

Question: Find the prime factorisation of 56.

Solution:

Divide 56 by the smallest prime number (2): 56 ÷ 2 = 28.

Divide 28 by 2 again: 28 ÷ 2 = 14.

Now, 14 is not divisible by 2. Try the next smallest prime number (3), but it doesn’t divide 14.

Next prime number is 5 and 14 ÷ 5 = 2 (with a remainder of 4).

Finally, 2 is a prime number itself. Therefore, the prime factorisation of 56 is 2^3 × 7.

Example 3: Rational Numbers

Question: Determine if 7/12 is a rational number.

Solution:

Both the numerator (7) and the denominator (12) are integers.

Since 7/12 can be expressed as a fraction of integers, it is a rational number.

Strategies for Conquering Number System Problems

Dealing with Number System problems requires a blend of analytical thinking and mathematical skills. Here are effective strategies to approach these problems adeptly:

• Prime Factorisation: Master the art of prime factorisation to simplify complex calculations involving factors, multiples and common denominators.
• Divisibility Rules: Familiarise yourself with divisibility rules for numbers like 2, 3, 4, 5, 6, 8, 9 and 10. This knowledge will streamline your analysis.
• Fraction Simplification: Practice simplifying fractions and converting mixed fractions to improper fractions. This skill aids in calculations involving rational numbers.
• LCM and GCD: Understand the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) concepts. They prove invaluable in various number-related operations.
• Decimal to Fraction Conversion: Learn how to convert repeating decimals into fractions, enabling you to work comfortably with both forms.