Master Binary to Decimal Conversion

In the computer world, all data is processed in binary (base-2). Understanding binary is fundamental to computer science.

What is Binary?

Binary is a number system that uses only two digits: 0 and 1.

Characteristics of Binary

• Base: 2
• Digits used: 0, 1
• Place value: Each position increases by powers of 2 from right to left

Binary to Decimal Conversion Method

Basic Conversion Steps

Calculate the power of 2 for each binary digit and sum the values where 1 appears.

Example 1: Convert 1011

1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰
      = 1×8 + 0×4 + 1×2 + 1×1
      = 8 + 0 + 2 + 1
      = 11₁₀

Example 2: Convert 11111111

11111111₂ = 1×2⁷ + 1×2⁶ + 1×2⁵ + 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰
          = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1
          = 255₁₀

Efficient Calculation Method

For large binary numbers, use this efficient approach:

1. Start from the rightmost bit (least significant bit)
2. Note the power-of-2 values where 1 appears
3. Sum all the values

Powers of 2 Table

Memorize commonly used powers of 2 for efficient conversion:

Real-world Applications

IP Addresses

IPv4 addresses consist of four decimal numbers (0-255) but are internally processed as 32-bit binary numbers.

Example: 192.168.1.1

• 192 = 11000000
• 168 = 10101000
• 1 = 00000001
• 1 = 00000001

File Sizes

Computer file sizes are also calculated on a binary basis:

• 1 KB = 1024 bytes (2¹⁰)
• 1 MB = 1024 KB (2²⁰)
• 1 GB = 1024 MB (2³⁰)

Practice Problems

Try converting these binary numbers to decimal:

1. 101 = ?
2. 1110 = ?
3. 10101010 = ?

Answers

1. 101₂ = 1×4 + 0×2 + 1×1 = 5₁₀
2. 1110₂ = 1×8 + 1×4 + 1×2 + 0×1 = 14₁₀
3. 10101010₂ = 128 + 32 + 8 + 2 = 170₁₀

Summary

Binary to decimal conversion is:

• A fundamental computer science skill
• Important in programming
• Necessary for understanding bit operations

Practice regularly to become proficient at conversion. Use our integer conversion tool to enhance your understanding!