Decimal Number Conversion

-

 

Decimal Number System Conversion – Complete Guide

Introduction to the Decimal Number System

The decimal number system is the most commonly used number system in our daily life.
It is a base-10 system that uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Each digit’s value depends on its position (place value) and the base (10). For example:
345 = (3 × 10²) + (4 × 10¹) + (5 × 10⁰)

Types of Decimal Number Conversions

We often convert decimal numbers into other number systems in digital electronics, computer science, and mathematics.
The main conversions are:

  1. Decimal to Binary (Base-2)

  2. Decimal to Octal (Base-8)

  3. Decimal to Hexadecimal (Base-16)

  4. Reverse Conversions (Binary/Octal/Hexadecimal to Decimal)

1. Decimal to Binary Conversion

The binary number system uses only two digits: 0 and 1.

Steps for Integer Part:

  • Divide the decimal number by 2.

  • Write down the remainder (0 or 1).

  • Repeat division with the quotient until it becomes 0.

  • Read the remainders from bottom to top.

Steps for Fractional Part:

  • Multiply the fraction by 2.

  • Take the integer part as the binary digit.

  • Repeat multiplication with the fractional part until it becomes 0 (or desired precision).

  • Read digits from top to bottom.

Example: Convert 13.25 to binary

  • Integer: 13 ÷ 2 = 6 R1 → 6 ÷ 2 = 3 R0 → 3 ÷ 2 = 1 R1 → 1 ÷ 2 = 0 R1 → Binary: 1101

  • Fraction: 0.25 × 2 = 0.5 (0), 0.5 × 2 = 1.0 (1) → Binary: 01
    Result: 1101.01

2. Decimal to Octal Conversion

The octal system uses digits 0–7.

Steps for Integer Part:

  • Divide the decimal number by 8.

  • Write down the remainder (0–7).

  • Repeat until quotient is 0.

  • Read from bottom to top.

Steps for Fractional Part:

  • Multiply fraction by 8.

  • Take integer part as octal digit.

  • Repeat with fractional part.

Example: Convert 125 to octal
125 ÷ 8 = 15 R5
15 ÷ 8 = 1 R7
1 ÷ 8 = 0 R1
Result: 175

3. Decimal to Hexadecimal Conversion

Hexadecimal uses 16 symbols: 0–9 and A–F (A=10, B=11, …, F=15).

Steps for Integer Part:

  • Divide decimal number by 16.

  • Write remainder (0–15, convert 10–15 to A–F).

  • Repeat until quotient is 0.

Steps for Fractional Part:

  • Multiply fraction by 16.

  • Take integer part as hexadecimal digit.

  • Repeat with fractional part.

Example: Convert 255 to hexadecimal
255 ÷ 16 = 15 R15 (F)
15 ÷ 16 = 0 R15 (F)
Result: FF

Quick Conversion Table

DecimalBinaryOctalHexadecimal
0000
1111
21022
510155
81000108
10101012A
15111117F
16100002010
3111111371F

Applications of Decimal Number Conversion

  • Digital Electronics – for designing circuits that use binary or hexadecimal representations.

  • Programming – converting between number formats for computations.

  • Networking – IP addressing often requires binary-decimal conversion.

  • Mathematics Education – fundamental concept for number theory and base systems.


Click here to download:



Comments

Post a Comment

Popular posts from this blog

Decimal to BCD Conversion (8421, 5421, 2421 Codes) – Digital Electronics

-
Image
Introduction In Digital Electronics, Binary Coded Decimal (BCD) is a way to represent decimal numbers (0–9) in binary form. Instead of converting the whole decimal number into a binary equivalent, each decimal digit is represented separately by a 4-bit binary code . There are different types of BCD codes depending on the weights assigned to each binary digit: 8421 Code (Natural BCD, Weighted Code) 5421 Code (Weighted Code) 2421 Code (Self-Complementing Weighted Code) Let’s understand each one in detail. 1. 8421 BCD Code (Natural Binary Coded Decimal) This is the most common BCD code . Each digit has positional weights: 8, 4, 2, 1 Decimal numbers 0–9 are represented as 0000–1001       Decimal     8421 BCD 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 Example: Convert Decimal 59 into 8421 BCD 5 → 0101 9 → 1001 So, 59 = 0101 1001 (BCD) 2. 5421 BCD Code In this system, the weights are: 5, 4, 2, 1 Each decima...
Read more

Octal conversion

-
Image
Scroll down to ger pdf What Is the Octal Number System? The Octal (base-8) number system uses digits from 0 to 7 . It's used in computing as a shorthand for binary because 1 octal digit = 3 binary digits (bits) . Example: Octal: 237₈ This is different from Decimal ( 237₁₀ ) or Binary ( 101011₂ ).  1. Octal to Decimal – Explained We convert each octal digit into a power of 8. Decimal (base-10) is what we normally use. So, we want to rewrite the base-8 number into base-10 , by evaluating powers of 8.  Integral Part Let’s take: 237₈ Break it down: 2 × 8² = 2 × 64 = 128 3 × 8¹ = 3 × 8 = 24 7 × 8⁰ = 7 × 1 = 7 Now, add them: 128 + 24 + 7 = 159 So, 237₈ = 159₁₀ Fractional Part Example: 0.52₈ We apply powers of 8 in negative , starting from −1. 5 × 8⁻¹ = 5 × 0.125 = 0.625 2 × 8⁻² = 2 × 0.015625 = 0.03125 Add: 0.625 + 0.03125 = 0.65625 So, 0.52₈ = 0.65625₁₀ 2. Decimal to Octal – Explained We reverse the above process. Integral Part Let’s convert: 1...
Read more