Decimal to Binary Conversion

Decimal to binary conversion is done through various methods. One of the methods to convert decimal to binary is by dividing the given decimal number recursively by 2. Then, the remainders are noted down till we get 0 as the final quotient. After this step, these remainders are written in reverse order to get the binary value of the given decimal number. A number system is a mathematical way of representing numbers using a set of digits or symbols.

There are different number systems like the decimal number system, the binary number system, the octal, and the hexadecimal number system. These are identified with the help of the base that they have. Numbers can be easily converted from one base to another using some defined rules.

Decimal

A decimal number is a number expressed in the base 10 numeral system. Decimal numbers' digits have 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each digit of a decimal number counts a power of 10.

Decimal number example:

653₁₀ = 6×10² + 5×10¹ + 3×10⁰

Binary

A binary number is a number expressed in the base 2 numeral system. Binary numbers' digits have 2 symbols: zero (0) and one (1). Each digit of a binary number counts a power of 2.

Binary number example:

1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀

How to Convert Decimal to Binary

Conversion steps:

Divide the number by 2.
Get the integer quotient for the next iteration.
Get the remainder for the binary digit.
Repeat the steps until the quotient is equal to 0.

Example

Understanding Decimal-to-Binary Conversion

Decimal-to-binary conversion involves converting decimal numbers (base 10) into their corresponding binary representations (base 2). Each digit of a decimal number is converted into binary form using division by 2 and noting the remainders.

The general approach to converting decimal numbers to binary includes:

Divide the decimal number by 2 and record the remainder.
Repeat the division until the quotient is 0.
Write the remainders in reverse order to get the binary representation.

Steps for Decimal-to-Binary Conversion

Step 1: Divide the decimal number by 2, noting the quotient and remainder.

Step 2: Repeat the division until the quotient is 0.

Step 3: Write the remainders in reverse order to obtain the binary number.

Example: Converting Decimal to Binary

Convert \( 65 \) to binary:

Step 1: Divide \( 65 \) by \( 2 \): quotient \( 32 \), remainder \( 1 \).
Step 2: Divide \( 32 \) by \( 2 \): quotient \( 16 \), remainder \( 0 \).
Step 3: Divide \( 16 \) by \( 2 \): quotient \( 8 \), remainder \( 0 \).
Step 4: Divide \( 8 \) by \( 2 \): quotient \( 4 \), remainder \( 0 \).
Step 5: Divide \( 4 \) by \( 2 \): quotient \( 2 \), remainder \( 0 \).
Step 6: Divide \( 2 \) by \( 2 \): quotient \( 1 \), remainder \( 0 \).
Step 7: Divide \( 1 \) by \( 2 \): quotient \( 0 \), remainder \( 1 \).
Write the remainders in reverse order: \( 01000001 \).
Final binary result: \( 01000001 \).

Conversion Table for Quick Reference

Here is a quick reference for some decimal numbers and their binary equivalents:

0 → 00000000
1 → 00000001
2 → 00000010
5 → 00000101
10 → 00001010
15 → 00001111
65 → 01000001
255 → 11111111

Applications of Decimal-to-Binary Conversion

Decimal-to-binary conversion is commonly used in:

Encoding numerical data for storage and transmission in computers.
Understanding how numbers are represented in digital systems.
Working with low-level programming and computer architecture.

Practice Problem

Convert \( 103 \) to binary:

Solution:
- Step 1: Divide \( 103 \) by \( 2 \): quotient \( 51 \), remainder \( 1 \).
- Step 2: Divide \( 51 \) by \( 2 \): quotient \( 25 \), remainder \( 1 \).
- Step 3: Divide \( 25 \) by \( 2 \): quotient \( 12 \), remainder \( 1 \).
- Step 4: Divide \( 12 \) by \( 2 \): quotient \( 6 \), remainder \( 0 \).
- Step 5: Divide \( 6 \) by \( 2 \): quotient \( 3 \), remainder \( 0 \).
- Step 6: Divide \( 3 \) by \( 2 \): quotient \( 1 \), remainder \( 1 \).
- Step 7: Divide \( 1 \) by \( 2 \): quotient \( 0 \), remainder \( 1 \).
Write the remainders in reverse order: \( 01100111 \).
Final binary result: \( 01100111 \).

Decimal-to-Binary Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a decimal number to its binary equivalent.	Divide the decimal number by 2. Record the remainder. Repeat the process with the quotient until the quotient is 0. Write the remainders in reverse order to get the binary result.	For \( 72 \): Divide by 2: \( 72 \div 2 = 36 \) remainder \( 0 \). Divide by 2: \( 36 \div 2 = 18 \) remainder \( 0 \). Divide by 2: \( 18 \div 2 = 9 \) remainder \( 0 \). Divide by 2: \( 9 \div 2 = 4 \) remainder \( 1 \). Divide by 2: \( 4 \div 2 = 2 \) remainder \( 0 \). Divide by 2: \( 2 \div 2 = 1 \) remainder \( 0 \). Divide by 2: \( 1 \div 2 = 0 \) remainder \( 1 \). Binary: \( 1001000 \).
Handling Larger Numbers	Converting larger decimal numbers to binary.	Follow the same division-by-2 method, recording each remainder. Write the remainders in reverse order.	For \( 255 \): Divide by 2: \( 255 \div 2 = 127 \) remainder \( 1 \). Divide by 2: \( 127 \div 2 = 63 \) remainder \( 1 \). Divide by 2: \( 63 \div 2 = 31 \) remainder \( 1 \). Divide by 2: \( 31 \div 2 = 15 \) remainder \( 1 \). Divide by 2: \( 15 \div 2 = 7 \) remainder \( 1 \). Divide by 2: \( 7 \div 2 = 3 \) remainder \( 1 \). Divide by 2: \( 3 \div 2 = 1 \) remainder \( 1 \). Divide by 2: \( 1 \div 2 = 0 \) remainder \( 1 \). Binary: \( 11111111 \).
Verifying Conversion	Checking the accuracy of decimal-to-binary conversion.	Convert the binary number back to decimal. Ensure the decimal result matches the original number.	For \( 1001000 \) (binary for \( 72 \)): Binary: \( 1001000 \). Convert back to decimal: \( 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 72 \). Decimal: \( 72 \), matches the original input.
Handling Special Cases	Converting special decimal numbers to binary (e.g., powers of 2).	For powers of 2, the binary representation will have only one 1 followed by zeros.	For \( 64 \): Decimal: \( 64 \). Binary: \( 1000000 \).
Applications	Understanding where decimal-to-binary conversion is used.	Binary encoding in computers for data processing and storage. Converting data for network communication and cryptography.	Example: Converting decimal numbers to binary for use in computer programs or protocols.