Exploring Number System

Number System

A Number System is a mathematical method of writing and representing numbers using specific digits or symbols. It defines how a value is represented visually and internally in a system.

In computers, number systems are crucial because electronic circuits are made of transistors that mainly understand only two states:

• ON (Logic 1): Represents high voltage.
• OFF (Logic 0): Represents low voltage.

Therefore, computers primarily use the Binary Number System.

Types of Number Systems

Number System	Base (Radix)	Digits Used
Decimal	10	0 to 9
Binary	2	0, 1
Octal	8	0 to 7
Hexadecimal	16	0 to 9, A to F

1. Decimal Number System

Definition:

The Decimal number system has base 10. It is the positional numeral system we use in our daily lives for calculations.

Place Values:

Powers of 10: ...10³(1000), 10²(100), 10¹(10), 10⁰(1)

Example 1: 34510
= (3 × 10²) + (4 × 10¹) + (5 × 10⁰)
= 300 + 40 + 5 = 345

Example 2: 70910
= (7 × 10²) + (0 × 10¹) + (9 × 10⁰)
= 700 + 0 + 9 = 709

2. Binary Number System

Definition:

Binary number system has base 2. It uses only two digits: 0 and 1. Each digit is called a "Bit" (Binary Digit).

Place Values:

Powers of 2: ...2⁴(16), 2³(8), 2²(4), 2¹(2), 2⁰(1)

Example 1: 10112
= (1×2³) + (0×2²) + (1×2¹) + (1×2⁰)
= 8 + 0 + 2 + 1 = 1110

Example 2: 11012
= (1×2³) + (1×2²) + (0×2¹) + (1×2⁰)
= 8 + 4 + 0 + 1 = 1310

3. Octal Number System

Definition:

Octal number system has base 8. It uses digits from 0 to 7.

Example 1: 178
= (1×8¹) + (7×8⁰)
= 8 + 7 = 1510

Example 2: 258
= (2×8¹) + (5×8⁰)
= 16 + 5 = 2110

4. Hexadecimal Number System

Definition:

Hexadecimal number system has base 16. It uses 16 symbols: 0-9 and A-F.

• A=10, B=11, C=12, D=13, E=14, F=15

Example 1: 2A16
= (2×16¹) + (10×16⁰)
= 32 + 10 = 4210

Example 2: FF16
= (15×16¹) + (15×16⁰)
= 240 + 15 = 25510

Conversion of Number Systems

1. Other Systems → Decimal Number system

A. Binary to Decimal

Rule: Multiply each digit by powers of 2.

Example 1:
1010 = (1×2³)+(0×2²)+(1×2¹)+(0×2⁰)
= 8 + 0 + 2 + 0 = 1010

Example 2:
1101 = 8 + 4 + 0 + 1 = 1310

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B. Octal to Decimal

Example 1:
157 = (1×8²)+(5×8¹)+(7×8⁰)
= 64 + 40 + 7 = 11110

Example 2:
24 = (2×8¹)+(4×8⁰)
= 16 + 4 = 2010

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C. Hexadecimal to Decimal

Example 1:
3F = (3×16¹)+(15×16⁰)
= 48 + 15 = 6310

Example 2:
2A = (2×16¹)+(10×16⁰)
= 32 + 10 = 4210

2. Decimal → Other Number Systems

A. Decimal to Binary

Rule: Divide the number by 2 repeatedly and write remainders from bottom to top.

Example 1:
13 ÷ 2 = 6 rem 1
6 ÷ 2 = 3 rem 0
3 ÷ 2 = 1 rem 1
1 ÷ 2 = 0 rem 1
Answer: 11012

Example 2:
10 ÷ 2 = 5 rem 0
5 ÷ 2 = 2 rem 1
2 ÷ 2 = 1 rem 0
1 ÷ 2 = 0 rem 1
Answer: 10102

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B. Decimal to Octal

Rule: Divide by 8 repeatedly.

Example 1:
65 ÷ 8 = 8 rem 1
8 ÷ 8 = 1 rem 0
1 ÷ 8 = 0 rem 1
Answer: 1018

Example 2:
45 ÷ 8 = 5 rem 5
5 ÷ 8 = 0 rem 5
Answer: 558

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C. Decimal to Hexadecimal

Rule: Divide by 16. Values >9 are written as A–F.

Example 1:
255 ÷ 16 = 15 rem 15 (F)
15 ÷ 16 = 0 rem 15 (F)
Answer: FF16

Example 2:
100 ÷ 16 = 6 rem 4
6 ÷ 16 = 0 rem 6
Answer: 6416

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Questions and Answer

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Multiple Choice Questions

1. In a binary number system, right most digit before the fractional point is called ____________.
✅ (b) LSD
Least Significant Digit (LSD)

2. The total number of digits used in a number sytstem is called its:
✅ (b) Base (Radix)
Explanation: Base tells how many symbols or digits are used in a number system. Example: Decimal has base 10 (0–9), Binary has base 2 (0,1).

3.In hexadecimal number system, the alphabet D represents:
✅ (c) 13
Explanation: Hexadecimal values are: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

4. To convert decimal number into octal number, divide the number by:
✅ (b) 8

Explanation: Octal uses digits 0 to 7, so base = 8.

5. Binary addition, 11 + 10 = ?
✅ (d) 101

Explanation:
11₂ = 3
10₂ = 2
3 + 2 = 5
5 in binary = 101

6. How many digits are used in the hexadecimal number system?
✅ (d) 16

Explanation:
It uses digits 0–9 and letters A–F.
Total symbols = 16

Fill in the Blanks

1. The total number of digits used in a number system is called its base or Radix.
2. The leftmost digit of a number is called MSD (Most Significant Digit).
3. The number system having just 0 and 1 is called Binary.
4. The base of octal system is 8.
5. In binary addition, 1 + 0 = 1.

True / False

1. Binary numbers cannot be converted into hexadecimal number. ✗ False
2. The numbers used in binary number system are 1 and 0. ✓ True
3. The octal number system has the base 16. ✗ False
4. FG92 is a hexadecimal number. ✗ False
5. The processor of computer performs arithmetic operations only on binary numbers. ✓ True

Questions and Answers

1. What do you mean by Number System?
A Number System is a method of writing numbers using digits or symbols according to fixed rules.

2. Briefly Explain Binary Number System.
Binary number system uses only two digits: 0 and 1. Its base is 2. It is used in computers.

3. Define Decimal Number System.
Decimal number system uses digits 0 to 9. Its base is 10.

4. How do you convert Binary Number into Decimal Number?
Multiply each digit by powers of 2 and add the results.
Example: 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11

5. Write the Rules of Subtraction of Binary Numbers.

• 0 − 0 = 0
• 1 − 0 = 1
• 1 − 1 = 0
• 0 − 1 = 1 (Borrow 1)

6. Explain Binary Division with Example.
Binary division works like normal division.
Example: 1100₂ ÷ 10₂ = 110₂