Number System

1. What is a Number System?

A Number System is a way to represent numbers using digits or symbols. It is the foundation of arithmetic and is widely used in exams.

Foundation of Arithmetic

Classification of Numbers

Based on Properties

Different Representation Methods

Simple Definition: Number System = Classification of numbers based on their properties and representation.

Examples:

Type of Number	Example	Notes
Natural Numbers	1, 2, 3, …	Counting numbers
Whole Numbers	0, 1, 2, 3, …	Natural numbers + 0
Integers	…, –2, –1, 0, 1, 2, …	Positive & negative numbers
Rational Numbers	1/2, 3/4, 5	Can be expressed as p/q
Irrational Numbers	√2, π, e	Cannot be expressed as fraction
Real Numbers	All rational & irrational	Any point on number line
Complex Numbers	3 + 4i	a + bi form

2. Types of Number Systems

Type	Base	Digits	Notes
Binary	2	0, 1	Used in computers
Octal	8	0–7	Less common
Decimal	10	0–9	Most used
Hexadecimal	16	0–9, A–F	Used in programming

3. Basic Operations

Addition

Combine two numbers → sum

Subtraction

Difference between numbers

Multiplication

Product of numbers

Division

Quotient and remainder

Example:

101₂ + 11₂ = 1000₂ (Binary addition)
Decimal: 5 + 3 = 8

4. Important Concepts

Concept	Formula / Rule	Example
Even/Odd Numbers	Even divisible by 2, Odd remainder 1	4 even, 5 odd
Prime Numbers	Only divisible by 1 and itself	2, 3, 5, 7, 11
Composite Numbers	Not prime	4, 6, 8, 9
Factors & Multiples	Factor divides number, Multiple = product	2×3=6, Factors of 6 =1,2,3,6
HCF	Highest Common Factor	HCF(12, 18) = 6
LCM	Least Common Multiple	LCM(4, 6) = 12
Divisibility Rules	Quick checks	2→last digit even, 3→sum digits divisible by 3

5. Conversion Between Systems

From → To	Method	Example
Binary → Decimal	Multiply each bit by 2^position	101₂ = 1×2² + 0 + 1×2⁰ = 5
Decimal → Binary	Divide by 2, write remainders	13 → 1101₂
Octal → Decimal	Multiply by 8^position	17₈ = 1×8 +7=15
Decimal → Octal	Divide by 8	15 → 17₈
Hex → Decimal	Multiply by 16^position	1A₁₆ = 26
Decimal → Hex	Divide by 16	26 → 1A₁₆

6. Shortcut Tricks

Concept	Shortcut / Trick
Sum of first n natural numbers	n(n+1)/2
Sum of first n odd numbers	n²
Sum of first n even numbers	n(n+1)
Square of number ending with 5	(a*10 + 5)² = a(a+1) × 100 + 25
Divisibility by 3	Sum of digits divisible by 3
Divisibility by 9	Sum of digits divisible by 9

7. Real-Life Analogy

Binary System

Like ON/OFF switches

0=off, 1=on

Prime Numbers

Building blocks of numbers

Fundamental components

HCF & LCM are used in arranging items, clocks, schedules, and many real-world applications involving grouping and timing.

8. Common Delhi Police / SSC Exam Facts

Questions often include HCF, LCM, Divisibility, Prime/Composite numbers

Conversions between binary, decimal, octal, and hexadecimal are important

Quick calculation tricks save time in exams

Number classification questions test fundamental understanding

9. Practice Questions (Exam-Type)

No.	Question	Solution	Answer
1	HCF of 12 & 18	Factors → 1,2,3,6	6
2	LCM of 4 & 6	Multiples → 12	12
3	Convert 1011₂ to decimal	1×2³ +0 +1×2¹ +1 = 11	11
4	Is 29 prime?	Only divisible by 1 & 29	Yes
5	Sum first 10 natural numbers	10×11/2	55

10. Quick Recap

Concept	Summary
Number Types	Natural, Whole, Integers, Rational, Irrational, Real, Complex
Special Numbers	Prime, Composite, Even, Odd
HCF & LCM	Highest factor, Least multiple
Base Systems	Binary, Octal, Decimal, Hexadecimal
Divisibility & Shortcuts	Quick checks for calculations

You've completed Number System Concepts!

Courage Tip: Practice number system problems regularly, focusing on different types like HCF/LCM calculations, base conversions, and number classification. Remember the key formulas and relationships between different number types to solve problems quickly in exams.